Free curves on varieties
Frank Gounelas
We study various generalisations of rationally connected varieties, allowing the connecting curves to be of higher genus. The main focus will be on free curves $f:C\to X$ with large unobstructed deformation space as originally defined by Koll\'ar, but we also give definitions and basic properties of varieties $X$ covered by a family of curves of a fixed genus $g$ so that through any two general points of $X$ there passes the image of a curve in the family. Giving examples along the way we prove that the existence of a free curve of genus $g\geq1$ implies the variety is rationally connected in characteristic zero.
1208.4055.tex
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\address{Frank Gounelas, Mathematical Institute, University of Oxford, 24-29 St Giles'
Oxford, OX1 3LB, United Kingdom}
\author{Frank Gounelas}
\title{Free curves on varieties}
\date{\today}

\begin{document}
\maketitle
\begin{abstract}
    We study various generalisations of rationally connected varieties, allowing the
    connecting curves to be of higher genus. The main focus will be on free curves $f:C\to
    X$ with large unobstructed deformation space as originally defined by Koll\'ar, but we
    also give definitions and basic properties of varieties $X$ covered by a family of
    curves of a fixed genus $g$ so that through any two general points of $X$ there passes
    the image of a curve in the family. Giving examples along the way we prove that the
    existence of a free curve of genus $g\geq1$ implies the variety is rationally
    connected in characteristic zero.
\end{abstract}

\section{Introduction}

    Let $k$ be an algebraically closed field. A smooth projective rationally connected
    variety, originally defined in \cite{kmm92c} and \cite{campana92}, is a variety such
    that through every two general points there passes the image of a rational curve. In
    characteristic zero this is equivalent to the notion of a separably rationally
    connected variety, given by the existence of a rational curve $f:\bP^1\to X$ such that
    $f^*T_X$ is ample. In characteristic $p$ however one has to distinguish between these
    two notions. Deformations of a morphism $f:\bP^1\to X$ are controlled by the sheaf
    $f^*\sT_X$, hence studying positivity conditions of this bundle is intimately tied to
    deformation theory and the existence of many rational curves on $X$. A classical
    theorem of Grothendieck \cite{grothendieck} gives us easy numerical criteria for
    testing positivity statements for $f^*\sT_X$ on $\bP^1$.

    \lem{(\cite{kollar} II$.3.8$)
        A locally free sheaf $\cE$ of rank $n$ on $\bP^1$ is called semi-positive (respectively
        ample) if any of the following equivalent conditions are satisfied
        \begin{enumerate}
            \item $H^1(\bP^1, \cE(-1))=0$ (respectively $H^1(\bP^1, \cE(-2))$),
            \item $\cE$ (respectively $\cE(-1)$) is generated by global sections,
            \item $\cE \cong \oplus_{i=1}^n \cO_{\bP^1}(a_i)$ with $a_i\geq0$ (respectively
                $\geq1$) for all $i$,
            \item $H^0(\bP^1, \cE)\to \cE\otimes k(p)$ (respectively $H^0(\bP^1, \cE(-1))\to
                \cE(-1)\otimes k(p))$ is surjective for some $p\in\bP^1$,
            \item $\cO_{\bP(\cE)}(1)$ is nef (respectively ample) as a line bundle on
                $\bP(\cE)$.
        \end{enumerate}
        \label{vbp1}
    }

    Rationally connected varieties have especially nice properties. Note in particular the
    following important theorem which we will make repeated use of throughout this paper.

    \theo{(Graber-Harris-Starr \cite{ghs}, de Jong-Starr \cite{djs})
        Over $k$ an arbitrary algebraically closed field, let $C$ be a smooth irreducible
        projective curve and $f:X\to C$ a proper flat morphism whose geometric generic
        fibre is normal and separably rationally connected. Then $f$ has a section.
        \label{ghs}
    }

    Starting with an arbitrary smooth variety one can form a fibration (the MRC fibration)
    over a smooth base such that the general fibre is rationally chain connected, or in
    other words that through every two points there is a chain of rational curves (i.e. a
    connected genus $0$ curve). Smoothing of combs techniques (see \cite{kollar} II$.7$)
    make the notions of rationally chain connected and rationally connected varieties
    equivalent in characteristic zero so in this case the MRC fibration has rationally
    connected fibres. In positive characteristic it is not known whether these two notions
    differ however. A consequence of the above theorem in characteristic zero is that if
    $X\to Y$ is a morphism of smooth projective varieties such that the general fibre and
    $Y$ are rationally connected, then so is $X$. Another consequence is the following
    in characteristic zero which we will make repeated use of, noting that it is
    equivalent to the Graber-Harris-Starr Theorem.

    \cor{(\cite{kollar} Proposition IV$.5.6.3$)
        Let $X$ be a variety over an algebraically closed field $k$ of characteristic zero
        and let $X\dashrightarrow R(X)$ be the MRC fibration. Then $R(X)$ is not uniruled.
        \label{ghscor}
    }

    One of the big open conjectures in the area gives a numerical characterisation of when
    a variety can be rationally connected.

    \conj{(Mumford)
        \label{mumfordconj}
        A smooth projective variety $X$ over an algebraically closed field of
        characteristic zero is uniruled if and only if
        \eqn{
            H^0(X, \omega_X^{\otimes m})=0 \text{ for all }m>0
        }
        whereas it is rationally connected if and only if
        \eqn{
            H^0(X, (\Omega_X^1)^{\otimes m}) = 0 \text{ for all }m>0.
        }
    }

    Note that the reverse directions of both conjectures hold (see \cite{debarre}
    Corollary $4.12$), whereas both are true in dimensions two and three.

    \n
    In this paper we study the various ways in which a variety can be connected by higher
    genus curves. After an introductory section with preliminary results on vector bundles
    on curves and Frobenius, we consider first varieties which admit a morphism from a
    family of curves of fixed arithmetic genus $g$ whose product with itself dominates the
    product of the variety with itself and call these varieties ``genus $g$ connected'',
    generalising the notion of there being a rational curve through two general points.
    We also consider $C$-connected varieties, where there exists a family $C\times U \to
    X$ of a single smooth genus $g$ curve $C$ such that $C\times C\times U\to X\times X$
    is dominant. Mori's important bend and break result allows us to produce rational
    curves going through a fixed point given a higher genus curve which has large enough
    deformation space.

    \theo{(Bend and break, \cite{debarre} $3.1$)
        Let $X$ be a projective variety over an algebraically closed field $k$, $f:C\to X$
        a morphism from a smooth projective curve and $c\in C$. If $\dim_{[f]} \Hom(C, X;
        f|_{\{c\}})\geq 1$, then there exists a rational curve on $X$ through $f(c)$.
        \label{bendandbreak}
    }

    In Proposition \ref{cconisuniruled} we show that over any characteristic, if for any
    two general points of a smooth projective variety $X$ with $\dim X\geq2$ there passes
    the image of a morphism from a fixed curve $C$ of genus $g$, then $X$ is uniruled or
    in other words there is a rational curve through a general point. In chapter $2.2$ we
    prove basic properties of such varieties and give examples but one quickly has to
    restrict to a less general setting.
    
    \n
    A stronger condition than the aforementioned is a generalisation of the existence of a
    morphism from a curve which deforms a lot, as discussed for separably rationally
    connected varieties above. Namely, we introduce the existence of a free curve $f:C\to
    X$ first studied by Koll\'ar \cite{kollar} where now $C$ can be of any genus $g$. We
    call a morphism free if $f^*\sT_X$ is globally generated as a vector bundle on $C$ and
    also $H^1(C, f^*\sT_X)=0$. In the case of genus $g=0$ one must distinguish between
    free and very free curves, geometrically meaning that $f:\bP^1\to X$ deforms so that
    its image covers all points in $X$ or whether it can do so after fixing a point $x\in
    f(\bP^1)$ respectively. In other words a variety with a free rational curve is
    uniruled whereas a variety with a very free curve is rationally connected. If $g\geq1$
    however, after defining an $r$-free curve to be one which deforms keeping any $r$
    points fixed, we show that the notions of the existence of a free ($0$-free) and very
    free ($1$-free) curve coincide and in fact are equivalent with the existence of a
    curve $f:C\to X$ such that $f^*\sT_X$ is ample.

    \theon{(Theorem \ref{rfreeiffample})
        Let $X$ be a smooth projective variety and $C$ a smooth projective curve of genus
        $g\geq1$ over an algebraically closed field $k$. Then for any $r\geq0$, there
        exists an $f:C\to X$ which is $r$-free if and only if there exists a morphism
        $f':C\to X$ such that $f'^*\sT_X$ is ample.
    }

    Work of Bogomolov-McQuillan (see Theorem \ref{bm}) and more recently a proof of the
    same result by Kebekus-Sol\'a Conde-Toma (see \cite{bm}, \cite{ksct}) on foliations
    which restrict to an ample bundle on a smooth curve sitting inside a complex variety
    $X$ show that the leaves of such a foliation are not only algebraic but in fact have
    rationally connected closures. We reprove this result in the case of the foliation
    $\sF=\sT_X$ thus assuming the existence of a free curve $f:C\to X$, complementing the
    currently known connections between existence of curves with large deformation spaces
    and rationally connected varieties. Our proof is in the same vein as that in
    \cite{ksct} but emphasises the use of free curves.
    
    \theon{(Corollary \ref{mrctowerispointcor})
        \label{maintheointro}
        Let $X$ be a smooth projective variety over an algebraically closed field of
        characteristic zero and let $f:C\to X$ be a smooth projective curve of genus
        $g\geq1$ such that $f^*\sT_X$ is globally generated and $H^1(C, f^*\sT_X)=0$. Then
        $X$ is rationally connected.
    }
    We give examples along the way and prove various useful properties of $C$-connected
    varieties and varieties admitting a free morphism from a curve of any genus $g$.
    Notably, in characteristic zero, this last class of varieties is the same as that of
    rationally connected varieties. One would hope that the existence of a higher genus
    free curve could on occasion be easier to prove, thus meaning our variety is
    rationally connected, however the author has not been able to find such a non-trivial
    example.

    \n
    In the third section, we study the particular case of elliptically connected varieties
    (i.e.\ genus $1$ connected varieties) where, even allowing a covering family of genus
    $1$ curves to vary in moduli, one can in fact prove the following theorem.

    \theon{(Theorem \ref{elconclassification})
        Let $X$ be a smooth projective variety over a field of characteristic zero. Then
        the following two statements are equivalent
        \begin{enumerate}
            \item There exists $\cC\to U$ a flat projective family of irreducible genus
                $1$ curves with a map $\cC\to X$ such that $\cC\times_U\cC\to X\times X$ is
                dominant.
            \item $X$ is either rationally connected or a rationally connected fibration
                over a curve of genus $1$.
        \end{enumerate}
    }

    In positive characteristic, at this point we have not been able to prove that the
    existence of a higher genus free curve implies the existence of a very free rational
    curve (which would mean that $X$ is separably rationally connected). We work however
    in this direction, establishing this result in dimensions two (with a short discussion
    about dimension three) and furthermore by studying algebraic implications of the
    existence of a free higher genus curve, such as the vanishing of pluricanonical forms
    and triviality of the Albanese variety. In the final section we give an example of a
    threefold in characteristic $p$ whose MRC quotient is rationally connected and which
    has infinite fundamental group.


    \acknowledgements{
        The contents of this paper are from the author's thesis under the supervision of
        Victor Flynn, which the author would like to thank for his continuous
        encouragement and help over the years. I am also hugely indebted to Damiano Testa
        for the countless hours he patiently spent answering questions and educating me
        and to Johan de Jong not only for the hospitality at Columbia University during my
        final year but also for helping significantly improve the contents of this paper
        and sharing his ideas. I would also like to thank Jason Starr for helpful
        discussions and suggestions and to Yongqi Liang for comments and pointing out some
        errors in an early draft. This research was completed under the support of EPSRC
        grant number EP/F$060661/1$. 
    }

\section{Ample vector bundles and Frobenius}

    We begin with some results concerning positivity of vector bundles on curves. Unless
    otherwise mentioned, assume that a variety $X$ is over an algebraically closed field
    and if projective, $\sO_X(1)$ is a fixed ample vector bundle.

    \defn{
        A locally free sheaf $\sE$ on a scheme $X$ is called ample if $\sO_{\bP(\sE)}(1)$
        has this property.
    }

    Ampleness was originally discussed in \cite{hartcurveample}, \cite{hartample},
    \cite{hartamplesubvar} where the definition was given as follows: $\sE$ is ample if
    for every coherent sheaf $\sF$ on $X$, there exists a positive integer $n_0$ such that
    for all $n>n_0$, the $\sO_X$-module $\sF\otimes S^n(\sE)$ is generated by global
    sections. This definition is equivalent with the one given above.

    \lem{
        Let $C$ be a smooth projective curve of genus $g\geq2$ over an algebraically
        closed field of characteristic zero and $\sE$ a locally free sheaf on $C$ such that
        $H^1(C, \sE)=0$. It follows that $\sE$ is ample.
        \label{fujitalemma}
    }
    \proof{
        From \cite{hartcurveample} Theorem $2.4$, it suffices to show that every
        non-trivial quotient locally free sheaf of $\sE$ has positive degree. Let $\sE\to
        \sE'\to0$ be a quotient. From the long exact sequence in cohomology we see that
        $H^1(C,\sE')$ is also $0$. From the Riemann-Roch formula $\chi(\sE') = \deg \sE'
        + \rk \sE'\chi(\sO_X)$ from which we obtain $h^0(C, \sE') = \deg \sE' + (\rk
        \sE')(1-g)$ and since $g\geq2$ we deduce that $\deg \sE' >0$.
    }

    More generally, over any characteristic if we further assume that our locally free
    sheaf is globally generated then the same result holds so long as the genus is at
    least $1$.

    \prop{
        Let $C$ be a smooth projective curve of genus $g\geq1$ over an algebraically closed
        field $k$ and $\sE$ a globally generated locally free sheaf on $C$ such that $H^1(C,
        \sE)=0$. Then $\sE$ is ample.
        \label{ggandh1impliesample}
    }
    \proof{
        Since $\sE$ is globally generated, there exists a positive integer $n$ such that
        $\sO_C^{\oplus n} \to\sE\to 0$ is exact. This gives (see \cite{hartshorne} ex.
        II$.3.12$) a closed immersion of the respective projective bundles $\bP(\sE)
        \hookrightarrow \bP^{n-1}_C$. By projecting onto the first factor we have the
        following diagram
        \eqn{
            \xymatrix{
                \bP(\sE) \ar@{^{(}->}^-i[r]\ar[dr]^-{\pi} & \bP^{n-1}\times
                C\ar[d]^{\pr_2}\ar[r]^-{\pr_1} & \bP^{n-1} \\ & C &
            }
        }
        and from \cite{hartshorne} II$.5.12$ we have $\pr_1^*\sO_{\bP^{n-1}}(1) =
        \sO_{\bP^{n-1}_C}(1)$. Also, since $i$ is a closed immersion it follows that
        $i^*\sO_{\bP^{n-1}_C}(1) = \sO_{\bP^{n-1}_C}(1)|_{\bP(\sE)} = \sO_{\bP(\sE)}(1)$
        which concludes that $i^*\pr_1^*\sO_{\bP^{n-1}}(1) = \sO_{\bP(\sE)}(1)$. To show
        that $\sE$ is an ample locally free sheaf on $C$ it is enough to show that this
        invertible sheaf is ample. Since we know that $\sO_{\bP^{n-1}} (1)$ is ample
        though, it is sufficient to show that $i\circ\pr_1$ is a finite morphism. Since it
        is projective, we need only show that it is quasi-finite. Hence assuming that the
        fibre of $i\circ\pr_1$ over a general point $p\in\bP^{n-1}$ is not finite, it must
        be the whole of $C$. We now embed this fibre $j:C\to\bP(\sE)$ as a section to
        $\pi$ and pull back the surjection $\pi^*\sE \to \sO_{\bP(\sE)}(1)$ via $j$,
        obtaining $j^*\sO_{\bP(\sE)}(1)$ as a quotient of $j^*\pi^*\sE = \sE$ (see
        \cite{hartshorne} II.$7.12$). However $\pr_1\circ i\circ j:C\to \bP^{n-1}$ is a
        constant map so $j^*\sO_{\bP(\sE)}(1) = \sO_C$. Taking cohomology of the
        corresponding short exact sequence given by this quotient, we obtain a
        contradiction since $H^1(C,\sE) = 0$ whereas $H^1(C,\sO_C)$ is not trivial for
        $g\geq1$.
    }

    In Proposition \ref{ampleisvgincharp} below we will prove that given an ample bundle
    on a curve in positive characteristic, then after pulling back by Frobenius, we can
    make this bundle be globally generated and have vanishing first cohomology.

    \lem{
        Let $C$ be a smooth projective curve over an algebraically closed field $k$,
        $d\geq0$ an integer and $\sE$ a locally free sheaf on $C$. If $H^1(C, \sE(-D))=0$
        for all effective divisors $D$ of fixed degree $d$ then for $d'\leq d$ it follows
        that $H^1(C, \sE(-D'))=0$ and $\sE(-D')$ is globally generated for all
        effective divisors $D'$ of degree $d'$.
        \label{appendixvfimpliesf}
    }
    \proof{
        The first result follows from the short exact sequence
        \eqn{
            0\to \sE(-D'-R) \to \sE(-D') \to \sE(-D')|_R \to 0
        }
        where $R$ is an effective divisor of degree $d-d'$. For the second, let $p\in C$.
        From the first part we have $H^1(C,\sE(-D'-p))=0$ since $D'+p$ is an effective
        divisor of degree $d'+1\leq d$ so the following sequence is exact
        \eqn{
            0\to H^0(C,\sE(-D'-p)) \to H^0(C,\sE(-D')) \to \sE(-D')\otimes k(p) \to 0.
        }
        Hence $\sE(-D')$ is globally generated at $p$ and the result follows.
    }

    \lem{
        Let $C$ be a smooth projective curve and $\phi:\sE\to\sF$ a morphism between two
        locally free sheaves. Assume that the image of $\phi$ has full rank (i.e.\ that
        the image of $\phi$ has rank $\rk\sF$), that $\sE$ is globally generated and
        $H^1(C,\sE)=0$. Then $\sF$ is also globally generated and $H^1(C,\sF)=0$.
        \label{fullrkggh1}
    }
    \proof{
        Let $\sF'$ be the locally free sheaf associated to the image of $\phi$. For a
        torsion sheaf $\sQ$ and a locally free sheaf $\sK$ we have the following short exact
        sequences of sheaves
        \eqn{
            &0\to\sF'\to\sF\to \sQ\to 0,\\
            &0\to \sK\to \sE\to\sF'\to0.
        }
        As $\sF'$ is a quotient of $\sE$ we have that it is globally generated and
        $H^1(C,\sF')=0$. From the cohomology of the first exact sequence above we obtain
        that $H^1(C,\sF)=0$ too. Now for $p\in C$, from the short exact sequence
        \eqn{
            0\to\sF'(-p)\to\sF'\to\sF'\otimes k(p)\to0
        }
        we obtain that $H^1(C, \sF'(-p))=0$ for any $p\in C$. Twisting the first short
        exact sequence above by $\sO_C(-p)$ we similarly obtain that $H^1(C,\sF(-p))=0$
        and from $0\to\sF(-p)\to\sF\to\sF\otimes k(p)\to0$ we see that $\sF$ is globally
        generated at $p$.
    }

    Let $k$ be an algebraically closed field of characteristic $p>0$ and $F_X:X\to X$ be
    the absolute Frobenius morphism of a $k$-scheme $X$ which leaves the topological space
    $X$ unchanged but raises functions to the $p$-th power. More generally, consider an
    $S$-scheme $X$ where $S$ is a scheme over $k$. Pulling back the map $X\to S$ via $F_S$
    we obtain
    \eqn{
        \xymatrix{
            X\ar@{-->}[dr]^{F_{X/S}}\ar@/_1pc/[ddr]\ar@/^1pc/[drr]^{F_X} & & \\
            & X^{(1)} \ar[r]^\alpha\ar[d] & X\ar[d] \\
            & S\ar[r]^{F_S} & S
        }
    }
    where $F_{X/S}:X\to X^{(1)}$ is the relative Frobenius morphism which we get from the
    universality of the pull-back diagram. In the case where $S=\Spec k$ and $X$ is a
    variety the relative Frobenius morphism $F_{X/k}$ raises variables of the equation of
    $X$ to the $p$-th power, $\alpha$ is called the $k$-linear Frobenius morphism and
    raises the coefficients to the $p$-th power whereas $F_X$ raises everything to the
    $p$-th power. Note that if $C$ is a curve then $C^{(1)}$ has the same genus as $C$ and
    $\a$ is a finite morphism of degree $p$ so the pull back of a bundle under $\a$ has
    degree multiplied by $p$, a fact which has many applications in the deformation theory
    of curves and Mori theory.

    \n
    A partial converse to Proposition \ref{ggandh1impliesample} in characteristic $p$ is
    given by the following, using $\bQ$-twisted vector bundles as in \cite{lazarsfeld2}
    II$.6.4$.

    \prop{(\cite{ksct} Proposition $9$) 
        Let $C$ be a smooth projective curve of genus $g$ over a field of characteristic
        $p>0$, $\sE$ a locally free sheaf of rank $r$ and $\delta\in\bQ_+$. Assume
        $\sE\gen{-\delta}$ is ample and $b_p(\delta) := p \delta-2g+1\geq0$. Then for
        $F:C^{(1)}\to C$ the $k$-linear Frobenius morphism and every subscheme $B\subset
        C^{(1)}$ of length smaller than or equal to $b_p(\delta)$ we have
        \eqn{
            H^1(C^{(1)}, F^*\sE\otimes \sI_B) = 0
        }
        and $F^*\sE\otimes \sI_B$ is globally generated.
    }

    We prove the following different version of this result.

    \prop{
        Let $C$ be a smooth projective curve of genus $g$ over an algebraically closed
        field $k$ of characteristic $p$ and let $\sE$ be an ample locally free sheaf on
        $C$. Let $B\subset C$ be a closed subscheme of length $b$ and ideal sheaf $\sI_B$.
        Then there exists a positive integer $n$ such that $H^1(C^{(n)}, F_n^*\sE\otimes
        \sI_B)=0$ and $F_n^*\sE\otimes \sI_B$ is globally generated on $C^{(n)}$ where
        $F_n:C^{(n)}\to C$ the $n$-fold composition of the $k$-linear Frobenius morphism.
        \label{ampleisvgincharp}
    }
    \proof{
        We proceed by induction. First, assume we can write $\sE$ as an extension
        \eqn{
            0\to \sO_C(p)\to \sE\to \sQ\to 0
        }
        where $p\in C$. If $\sQ$ is not torsion free, consider the saturation of $\sO_C(p)$
        in $\sE$ instead and take $\sQ$ as that quotient. Since $\sE$ is ample, so is its
        quotient $\sQ$ and indeed so is $\sO_C(p)$ as a degree $1$ invertible sheaf on a curve.
        Note also that the rank of $\sQ$ is one less than that of $\sE$ and that if we can
        prove the result for $\sQ$ then we will have it for $\sE$ too by considering
        cohomology of the appropriate exact sequences. We thus reduce to the case of
        $\sE=\sL$ an invertible sheaf of positive degree (since it is ample). We now know
        that an invertible sheaf $\sL$ pulls back under the $n$-fold composition of the
        linear Frobenius morphism to an invertible sheaf $F_n^*\sL$ of degree $p^n\deg
        \sL$. To show that $H^1(C^{(n)}, F_n^*\sL\otimes \sI_B)=0$, it is equivalent by
        Serre duality to show that $\Hom_{C^{(n)}}(F_n^*\sL, \sO_{C^{(n)}}(B)\otimes
        \omega_{C^{(n)}})=0$. Since the invertible sheaf $\sO_{C^{(n)}}(B)\otimes
        \omega_{C^{(n)}}$ has degree $b+2g-2$ and by picking $n$ large enough, we can
        ensure $p^n\deg \sL > b+2g-2$ from which we obtain $H^1(C^{(n)}, F_n^*\sL\otimes
        \sI_B)=0$ and hence $H^1(C^{(n)}, F_n^*\sE\otimes \sI_B)=0$ for a locally free
        sheaf of any rank.

        \n
        To show that $F_n^*E\otimes \sI_B$ is globally generated, pick a point $q\in C$.
        Then $\sI_B\otimes \sI_q$ has length $b+1$ and from the discussion above $H^1(C^{(n)},
        F_n^*E\otimes \sI_B\otimes \sI_q)$ vanishes when $p^n\deg L > b+1+2g-2$ so we can just
        pick $n$ large enough to fit this condition. Now, by taking the long exact
        sequence in cohomology of 
        \eqn{
            0\to F_n^*E\otimes \sI_B\otimes \sI_q \to F_n^*E\otimes \sI_B \to (F_n^*E\otimes
            \sI_B)\otimes k(q)\to 0
        }
        we conclude that $F_n^*E\otimes \sI_B$ is globally generated.

        \n
        That $\sE$ can not be written as an extension of $\sO_C(p)$ and a quotient locally
        free sheaf $\sQ$ is equivalent to $H^0(C, \sE(-p))=0$. However there exists a
        positive integer $m$ for which $H^0(C^{(m)},
        (F_m^*\sE)\otimes\sO_{C^{(m)}}(-p))\neq0$ and we proceed as before with this
        sheaf.
    }

\section{Definition of curve connectedness: Part I - Covering families}

    We now define various ways in which a variety can be covered by curves, generalising
    the classical notions of ruled, uniruled and rationally connected varieties. 

    \n
    A preliminary remark should be made about the Rigidity Lemma of Mumford from
    \cite{mumford} (or \cite{git} Proposition $6.1$, \cite{kollarmori} Lemma $1.6$). The
    underlying principle is that if one of the fibres of a proper flat family is
    contracted under a morphism to another variety, then every fibre is contracted. We
    will often subsume this result whenever we have a family of varieties mapping to
    another variety, thus often without explicitly stating that one of the fibres does not
    get contracted so that the family is not trivial under the morphism. 

    \defn{
        Let $X$ be a variety of dimension $n$ over a field $k$. We say that $X$ is ruled
        (resp. uniruled, separably uniruled) if there is a scheme $Y$ over $k$ of
        dimension $n-1$ and a map $\bP^1_Y \dashrightarrow X$ which is birational (resp.
        dominant, dominant and smooth at the generic point).
    }

    \defn{(\cite{kollar} IV$.3.2$)
        Let $X$ be a variety over a field $k$. We say that $X$ is rationally chain
        connected if there is a family of proper and connected algebraic curves $g:U\to Y$
        over a variety $Y$ whose geometric fibres have only rational components, and a map
        $u:U\to X$ such that 
        \eqn{
            u^{(2)}: U\times_Y U\to X\times_k X
        }
        is dominant. We say that $X$ is rationally connected if the geometric fibres of
        $g$ are furthermore assumed to be irreducible rational curves. If we assume that
        $u^{(2)}$ is also smooth at the generic point then $X$ is called separably
        rationally (chain) connected.
    }

    We can extend the definition of a uniruled variety to allow a covering family of
    morphisms from an arbitrary curve of higher genus as follows.

    \defn{
        Let $X$ be a variety over a field $k$ and $C$ a smooth projective curve over $k$.
        We say that $X$ is $C$-ruled if there exists a variety $Y$ over $k$ and a dominant
        morphism $u:C\times Y\to X$.
        \label{cruled}
    }
    
    We similarly extend the definition of rationally connected varieties to include curves
    of higher genus as follows.

    \defn{
        We say that a variety $X$ over a field $k$ is connected by genus $g\geq0$ curves
        (resp. chain connected by genus $g$ curves) if there exists a proper flat morphism
        $\cC\arrow Y$, for a variety $Y$, whose geometric fibres are irreducible genus $g$
        curves (resp. connected genus $g$ curves) such that there is a morphism $u: \cC
        \rightarrow X$ making the induced morphism $u^{(2)}: \cC\times_Y \cC \rightarrow X
        \times_k X$ dominant.
        \label{defngcon}
    }

    \defn{
        Keeping notation as in Definitions \ref{cruled} and \ref{defngcon} we say that a
        variety $X$ is separably (chain) connected by genus $g$ curves if the morphism
        $\cC\times_Y\cC\arrow X\times_k X$ is also smooth at the generic point. One
        likewise defines a separably $C$-ruled variety.
    }

    A genus $0$ connected variety is rationally connected. A variety which is connected by
    genus $1$ curves will be called (with a slight abuse of notation) elliptically
    connected and similarly we have rationally and elliptically chain connected varieties.
    Even though we will often assume that a genus $g$ connected variety is of dimension at
    least $2$, note that we allow $Y$ to be a point in the above definition. Consider for
    example that $\bP^1$ is an elliptically connected variety.

    \n
    Note that the notions of separably and non-separably genus $g$ connected varieties
    coincide when $k$ is of characteristic zero due to generic smoothness.
    
    \rem{
        In the case of rationally connected varieties over an algebraically closed field,
        it is enough to consider trivial families $\bP^1\times Y\arrow X$ in our
        definition, as rational curves are all isomorphic to $\bP^1$ over an algebraically
        closed field. Note that a non-constant family $u: E\times Y\to X$ where $E$
        is an elliptic curve will have the property that the image $u(E_y)$ for $y\in
        Y$ is isogenous to $E$.
        \label{elisogrem}
    }

    The relevant moduli spaces which we will be considering are the following. First (see
    \cite{kollar} II$.1.5$ and \cite{mori}), the space $\Hom_S(\cC, X, g)$ parametrising
    $S$-morphisms from an $\cC\to S$ a flat projective curve over an irreducible scheme
    $S$ to $X\to S$ a smooth quasi-projective scheme with fixed points $g:B\to X$ an
    $S$-morphism from some $B\subset\cC$ closed subscheme which is flat and finite over
    $S$. We now restrict the above to the case where $S$ is the spectrum of an
    algebraically closed field $k$ and fix some notation of the following evaluation
    morphisms to be used in later sections
    \eqn{
        F:C\times\Hom(C,X,g)&\arrow& X \\
        F^{(2)}: C\times C\times\Hom(C,X,g) &\arrow& X\times X \\
        \phi(p,f): H^0(C, f^*\sT_X\otimes \sI_B) &\arrow& f^*\sT_X\otimes k(p) \\
        \phi^{(2)}(p,q,f): H^0(C, f^*\sT_X\otimes \sI_B) &\arrow& f^*\sT_X\otimes k(p)\oplus
            f^*\sT_X\otimes k(q).
    }
    For a morphism $f:X\arrow Y$ and $x\in X$ denote the map on tangent spaces by $df(x):
    \sT_X\otimes k(x)\arrow \sT_Y\otimes k(f(x))$. Secondly we consider the relative
    moduli space of genus $g$ degree $d$ stable $Q$-marked curves with base point $t:P\to
    X$, denoted by $\bar{\cM}_{g,Q}(X/S, d, t)$ as in \cite{ak}.

    \lem{
        Let $X$ be a genus $g$ (chain) connected smooth projective variety over an
        algebraically closed field $k$. Then if $g'\geq 2g-1$, $X$ is also genus $g'$
        (chain) connected.
        \label{genusgconnisgconforbiggerg}
    }
    \proof{
        Let $\cC/Y\to X$ be the family making $X$ a genus $g$ (chain) connected variety
        with notation as in Definition \ref{defngcon}. From Theorem $50$ of \cite{ak} we have
        a projective algebraic space $Y'=\bar{\cM}_{g',Y}(\cC/Y,d)$ of finite type over
        $Y$ parametrizing stable families of degree $d$ curves of genus $g'$ over $\cC\to
        Y$. The condition $g'\geq2g-1$ coming from the Riemann-Hurwitz formula ensures
        that this moduli space is non-empty. From \cite{acgh} $12.9.2$ there exists a
        normal scheme $Z$ finite and surjective over $Y'$ and a flat and proper family
        $\cX\to Z$ of stable genus $g$ curves of degree $d$. Restricting to a suitable
        open subset $W\subset Z$ parametrizing irreducible curves we compose the family
        $\cX|_W\to W$ with the evaluation morphism to $X$ and the result follows.
    }

    An example of an elliptically connected variety over a non-algebraically closed field
    is given after the proof of Theorem \ref{elconclassification}.
    
    \n
    Instead of assuming we have a possibly non-isotrivial family of curves dominating a
    variety $X$ we may want to restrict to a single curve.

    \defn{
        We say that a variety $X$ over a field $k$ is $C$-connected for a curve $C$ if
        there exists a variety $Y$ and a map $u:C\times Y\to X$ such that the induced map
        $u^{(2)}: C\times C\times Y\to X\times X$ is dominant. If $u^{(2)}$ is also
        smooth at the generic point, then we say that $X$ is separably $C$-connected.
        \label{defncconn}
    }

    Obviously a $C$-connected variety is genus $g$ connected where $g$ is the genus of the
    curve $C$. Projective space is $C$-connected for every smooth projective curve $C$. An
    example of a $C$-connected variety which is not rationally connected is $C \times
    \bP^n$. To see this let $(c_1,x_1),(c_2,x_2)$ be any two points in $C \times \bP^n$
    and let $f:C\to \bP^n$ a morphism which sends $c_i\mapsto x_i$. Considering the graph
    of $f$ in $C\times\bP^n$ we have found a curve isomorphic to $C$ which goes through
    our two points. Using part $(4)$ and $(5)$ from Lemma \ref{gconproperties} below, the
    result follows. More generally, examples can also be constructed from Proposition
    \ref{rcfibrationovercurveiscconnected} below.

    \lem{
        Let $X$ be a smooth projective variety over an algebraically closed field $k$.
        There exists an integer $g\geq0$ such that $X$ is connected by genus $g$ curves.
        \label{allvarsaregconnforsomeg}
    }
    \proof{
        Let $d=\dim X$ and consider $\bP^N$ as the space parametrizing hypersurfaces of
        high degree for some $N\gg d$. Let $T=\bP^N\times\ldots\times\bP^N$ be the product
        of $d-1$ copies of $\bP^N$. Consider the family $\cC\to T$ such that 
        \eqn{
            \cC = \{(x, H_1, \ldots, H_{d-1}) : x\in H_i\text{ for all
            }i=1,\ldots,d-1\}\subset X\times T. 
        }
        The fibre $\cC_t$ above a point $t\in T$ is given by the curve $H_1\cap\ldots\cap
        H_{d-1}$ and by Bertini's Theorem \cite{hartshorne} II$.8.18$ and generic flatness
        we know that there is a non-empty locus $Y\subset T$ over which the fibres are
        smooth and the map to $Y$ is flat. Finally, we replace $\cC\to T$ with the
        irreducible component of $\cC$ over $Y$ which dominates $X$ under the projection
        map onto the first factor.
    }

    The following are mostly straight-forward generalisations of various results in
    \cite{kollar} IV$.3$.
    \lem{
        The following statements hold for a variety $X$ over a field $k$ and $C$ a smooth
        projective curve.
        \begin{enumerate}
            \item If $X$ is genus $g$ connected and $X\dashrightarrow Y$ a dominant
                rational map to a proper variety $Y$, then $Y$ is also genus $g$
                connected. The same holds if $X$ is $C$-connected.
            \item If $X$ is proper then it is $C$-connected if and only if there is a
                proper variety $W$, closed in $\Hom(C,X)$ such that $u^{(2)}: C\times
                C\times W\to X\times X$ is dominant.
            \item If $X$ is proper and $C$-connected over an algebraically closed field,
                then for arbitrary points $x_1,x_2\in X$ there exists a morphism $C\to X$
                whose image contains $x_1,x_2$.
            \item If $X$ is defined over a field $k$ and $K/k$ an extension of fields,
                then $X_K := X\times_k K$ is $C$-connected if and only if $X_k$ is.
            \item A variety $X$ over an uncountable algebraically closed field is
                $C$-connected if and only if for all very general $x_1,x_2\in X$ there
                exists a morphism $C\to X$ which passes through $x_1,x_2$.
            \item A variety $X$ over an uncountable algebraically closed field is genus
                $g$ connected if and only if for all very general $x_1,x_2 \in X$ there
                exists a smooth irreducible genus $g$ curve containing them. 
            \item Being rationally connected is closed under connected finite \'etale
                covers for proper varieties.
            \item Being elliptically connected is closed under connected finite \'etale
                covers for proper varieties.
        \end{enumerate}
        \label{gconproperties}
    }
    \proof{
        To prove $(1)$, let $u:\cC/M\to X$ be the family making $X$ genus $g$ connected
        and denote by $u':\cC/M\dashrightarrow Y$ the composition. Restricting $u'$ to the
        generic fibre $\cC_{k(M)}$ we have a rational map $\phi: \cC_{k(M)}
        \dashrightarrow Y$. Since $Y$ is proper, by a corollary to the valuative criterion
        of properness (see \cite{ega} II Proposition $7.4.9$) we can extend $\phi$ to a
        morphism $\phi: \cC_{k(M)} \to Y$. By spreading out to an open subset $M'\subseteq
        M$ (see \cite{ega} IV$_3$ $8.10.5$ for properness and $11.2.6$ for flatness of the
        family) we obtain a family $\cC|_{M'}\to M'$ which makes $Y$ also genus $g$
        connected.
        
        \n
        For $(2)$, consider $\Hom(C,X) = \cup R_i$ the decomposition into irreducible
        components. One direction of the statement is obvious, whereas for the other let
        $C\times W\to W$ be a family which makes $X$ a $C$-connected variety. If
        $u_i:C\times R_i\to X$ the evaluation morphism, then for some $i$ there is a
        morphism $h:W \to R_i$ such that $h(w) = [C_w\to X]$ for general $w\in W$. This
        implies that $u_i^{(2)}: C\times C\times R_i \to X\times X$ is also dominant.
        Statement $(3)$ follows from $(2)$ by noting that since the composition of a
        proper morphisms is proper, we have that $W$ is proper over $k$, and so $C\times
        C\times W \to X\times X$ is also proper. Now the result follows from the fact that
        a proper dominant morphism is surjective.

        \n
        For the next statement $(4)$, we may compactify $X$ and also assume that it is
        projective. One direction is simply a pullback under the morphism $\Spec K\to
        \Spec k$. For the other, if $X_K$ is $C_K$-connected then from $(2)$ there is a
        positive integer $d$ such that the evaluation morphism
        \eqn{
            \ev_K^d: C_K\times C_K\times\Hom_d(C_K, X_K) \to X_K \times X_K
        }
        is dominant. Because of the universal property of the $\Hom$ scheme, we have that
        $\Hom(C,X) \times_k K = \Hom(C_K, X_K)$ and $(\ev^d)_K = \ev_K^d$ so $ev^d:
        C\times C\times\Hom(C,X)\to X\times X$ is also dominant.
        
        \n
        For $(5)$ and $(6)$, again we may assume $X$ is projective. If through every two
        very general points there passes the image of $C$ under some morphism, then the
        map $u^{(2)}: C\times C\times\Hom(C,X)\to X\times X$ is dominant. Since
        $\Hom(C,X)$ has at most countably many irreducible components and $X$ is the union
        of uncountably many proper subvarieties, the restriction of $u^{(2)}$ to at least
        one of the components $R_i$ must be dominant, which proves $(5)$. Similarly for
        $(6)$, after fixing an ample line bundle, let 
        \eqn{
            \cM_{g,0}(X) = \bigsqcup_{n\geq 0} \cM_{g,0}(X, d) \subset \bigsqcup_{n\geq 0}
            \bar{\cM}_{g,0}(X, d) 
        }
        be the subspace of smooth curves of the moduli space from Theorem $50$ of
        \cite{ak}, where we have taken $P,Q$ to be trivial. By taking a one point marking
        for $Q$ we have a family $\cM_{g,1}(X)\to \cM_{g,0}(X)$ with an evaluation map to
        $X$. Since for every two very general points in $X$ we can find a smooth
        irreducible curve of genus $g$ passing through them, the map
        \eqn{
            \xymatrix{
                \cM_{g,1}(X) \times_{\cM_{g,0}(X)} \cM_{g,1}(X)\ar[d]\ar[r] & X\times X \\
                \cM_{g,0}(X) &
            }
        }
        must be dominant. Arguing now with this family as we did for $(5)$, the result
        follows.

        \n
        For $(7)$ and $(8)$ we may assume that $X$ is projective from Chow's Lemma. For
        $(7)$ the proof is contained for example in \cite{debarre} $4.4.(5)$. For $(8)$,
        let $\cC\to U$ be a family which makes $X$ elliptically connected and let $X'\to
        X$ be a connected finite \'etale cover.  Consider the pullback diagram and
        $\cC'\to U'\to U$ the Stein factorisation
        \eqn{
            \xymatrix{
                & \cC'=\cC\times_X X'\ar[dl]\ar[r]\ar[d] & X'\ar[d] \\
                U'\ar[dr] & \cC\ar[d]\ar[r] & X \\
                & U. &
            }
        }
        After possibly restricting $U'$ to the open subset of curves in $\cC'$ which are
        irreducible, the family $\cC'\to U'$ makes $X'$ elliptically connected.
    }

    \rem{
        Note from part $(1)$ above that being genus $g$ connected or $C$-connected is a
        birational invariant.
    }

    \prop{
        Let $X$ be a smooth projective variety over an algebraically closed field $k$ and
        $f:X\to C$ a flat morphism to a smooth projective curve whose geometric generic
        fibre is separably rationally connected. Then $X$ is $C$-connected.
        \label{rcfibrationovercurveiscconnected}
    }
    \proof{
        From Theorem \ref{ghs}, there is a section $\s:C\to X$ to $f$. Now from Theorem
        $2.13$ of \cite{kmm92c} we can find a section to $f$ passing through any two
        points in different fibres over $C$, hence we can find a copy of $C$ passing
        through two very general points. The result now follows from Lemma
        \ref{gconproperties} parts $(5)$ and $(6)$ above after possibly passing to an
        uncountable extension $K/k$.
    }

    \prop{
        Let $X$ be a $C$-connected variety of dimension at least $2$ over an algebraically
        closed field $k$. Then $X$ is uniruled.
        \label{cconisuniruled}
    }
    \proof{
        We can compactify $X$ and also assume it is projective.  Let $u:C\times Y\to X$ be
        a family such that $u^{(2)}: C\times C\times Y\to X\times X$ is dominant. By
        dimension considerations, we have
        \eqn{
            \dim Y + 2 \geq 2\dim X
        }
        from which $\dim Y\geq 2\dim X-2$ and so if $\dim X\geq 2$ we obtain $\dim Y \geq
        2$. Now, pick a general point $x\in X$ and denote by $Z\subset Y$ the locus of
        curves $u_z: C_z\to X$ such that $x\in u_z(C_z)$ for all $z\in Z$. We have that
        $\dim Z \geq \dim Y-(\dim X-1)$ and so for $\dim X\geq2$, $\dim Z\geq 1$. From the
        Bend and Break Lemma \ref{bendandbreak} we obtain a rational curve through every
        general point. After possibly an extension to an uncountable algebraically closed
        field this implies that $X$ is uniruled (see \cite{debarre} Remark $4.2(5)$).
    }

    Denoting by $X\dashrightarrow R(X)$ the MRC fibration, we let $R^0(X)=X$, $R^1(X) =
    R(X)$, $R^2(X) = R(R(X))$ and so on the successive MRC quotients and obtain a tower of
    MRC fibrations 
    \eqn{
        X\dashrightarrow R^1(X) \dashrightarrow \cdots \dashrightarrow R^n(X).
    }
    Since $X$ is finite dimensional this tower must eventually stabilise and if $R^i(X)$
    is uniruled then $\dim R^{i+1}(X) < \dim R^i(X)$. In characteristic zero, we in fact
    have $R(X) = \ldots = R^n(X)$ (see discussion bellow). In positive characteristic it
    can be that the tower has length greater than one - see the example given in the last
    section of this paper.

    \prop{
        Let $X$ be a normal and proper $C$-connected variety over an algebraically closed
        field where $C$ is a smooth projective curve. Then the tower $X \dashrightarrow
        R^1(X)\dashrightarrow \cdots \dashrightarrow R^n(X)$ of MRC quotients terminates
        in either a point or a curve.
        \label{cconmrctopointiorcurve}
    }
    \proof{
        Let $C\times Y\to X$ be the family which makes $X$ a $C$-connected variety. After
        replacing $R(X)$ by the closure of the image of $\pi^0:X\dashrightarrow R(X)$ we
        can assume that $\pi^0$ is dominant. From Lemma \ref{gconproperties} part $(1)$ it
        follows that $R(X)$ is also $C$-connected. Using the same method we obtain that
        $R^i(X)$ are all $C$-connected. Hence from Proposition \ref{cconisuniruled} if
        $\dim R^i(X) \geq2$, it is uniruled. This implies that $R^{i+1}(X)$ must have
        dimension strictly less than $R^i(X)$ and so the result follows.
    }

    Note that if $k$ is algebraically closed of characteristic zero then we know from the
    Graber-Harris-Starr Theorem \ref{ghscor} that the MRC quotient $R^1(X)$ is not
    uniruled, so if $X$ is $C$-connected as in Proposition \ref{cconmrctopointiorcurve},
    $R^1(X)$ must be either a curve or a point and one does not need to resort to taking
    successive quotients. Also, again in characteristic zero, from Theorem \ref{ghs} it
    follows that the composition of two rationally connected fibrations is a rationally
    connected fibration, so in combination with the fact that rationally chain connected
    are rationally connected in characteristic zero, we obtain that if the tower of MRC
    fibrations ends in a point then $X$ is rationally connected.

    \n
    Hence if $X$ is $C$-connected over an algebraically closed field of characteristic
    zero, we either have that $X$ is rationally connected or that it is a rationally
    connected fibration over a curve. From Proposition \ref{rcfibrationovercurveiscconnected} 
    the converse holds too. 

    \rem{
        As observed in \cite{occhetta} Remark $4$, over an algebraically closed field of
        characteristic zero, if the MRC quotient of a smooth projective variety $X$ is a
        curve $C$, then the MRC fibration extends to the whole variety and coincides with
        the Albanese map. To see this, let $\pi:X\dashrightarrow C$ be the MRC fibration.
        We know that the Albanese map contracts all rational curves so we have a
        commutative diagram
        \eqn{
            \xymatrix{
                X\ar[dr]\ar@{-->}[d] & \\
                C\ar@{-->}[r] & B
            }
        }
        where $B\subset \Alb X$ is the image of the Albanese morphism. Hence $B$ must be a
        curve, and so the fibres of the Albanese morphism $X\to\Alb X$ are connected (see
        \cite{ueno75} $9.19$). We conclude that $B$ is isomorphic to $C$.
    }

\section{Definition of curve connectedness: Part II - Free morphisms}


    We work with varieties over an algebraically closed field $k$ of arbitrary
    characteristic. In this section we define ways in which a morphism from a curve $C$ to
    a variety $X$ can deform enough to give a large family of morphisms from $C$ so as to
    cover $X$.

    \n
    A notion studied extensively by Hartshorne \cite{hartamplesubvar} is that of a (local
    complete intersection) subvariety $Y$ in a smooth projective variety $X$ such that the
    normal bundle $\sN_{Y/X}$ is ample. In the case of curves we have the following lemma.

    \lem{
        Let $X$ be a smooth projective variety over an algebraically closed field $k$.
        Then for some $g\geq0$ there exists a curve $C\subset X$ of genus $g$ such that
        $\sN_{C/X}$ is ample.
        \label{defnamplecurve}
    }
    \proof{
        See \cite{hartamplesubvar} III$.4$. By taking successive ample hyperplane sections
        $H_1\cap\ldots \cap H_r$ given by Bertini's Theorem we obtain a curve $C$ whose normal
        bundle decomposes as
        \eqn{
            \sN_{C/X} = \oplus_i \sN_{H_i/X}|_C.
        }
        Since each of the summands is ample, so is $\sN_{C/X}$.
    }

    \rem{
        In \cite{ottem}, Ottem defines an ample closed subscheme $Y\subset X$ of
        codimension $r$ to be one where the exceptional divisor $\sO(E)$ of the blowup
        $\Bl_Y X$ of $X$ along $Y$ is an $(r-1)$-ample line bundle in the sense that for
        every coherent sheaf $\sF$ there is an integer $m_0>0$ such that $H^i(X,
        \sF\otimes \sO(E)^m)=0$ for all $m>m_0$ and $i>r-1$. In the case where $Y$ is a
        Cartier divisor, this notion coincides with the usual notion of $Y$ being an ample
        divisor. One can then prove that if $Y$ is a local complete intersection subscheme
        of $X$ which is ample, then the normal bundle $\sN_{Y/X}$ is an ample bundle.
    }

    We impose the following stronger positivity condition than that of Hartshorne above.

    \defn{(\cite{kollar} II$.3.1$)
        Let $C$ be a smooth proper curve and $X$ a smooth variety over a field $k$. Let
        $f:C\arrow X$ a morphism and $B\subset C$ a closed subscheme with ideal sheaf
        $\sI_B$ and $g=f|_B$. The morphism $f$ is called free over $g$ if it is non-constant
        and one of the following two equivalent conditions is satisfied:
        \begin{enumerate}
            \item for every $p\in C$ we have $H^1(C, f^*\sT_X\otimes \sI_B(-p))=0$ or,
            \item $H^1(C, f^*\sT_X\otimes \sI_B)=0$ and $f^*\sT_X\otimes \sI_B$ is generated by
                global sections.
        \end{enumerate}
        \label{freeoverBdefn}
    }

    That the two conditions are equivalent follows by taking cohomology of the short exact
    sequence
    \eqn{
        0 \to f^*\sT_X\otimes \sI_B(-p) \to f^*\sT_X\otimes \sI_B \to (f^*\sT_X\otimes \sI_B)_p
        \to 0.
    }
    
    \defn{
        We say that a curve $f:C\arrow X$ is $r$-free if for all effective divisors $D$ of
        degree $r\geq0$, $H^1(C, f^*\sT_X\otimes \sO_C(-D))=0$ and $f^*\sT_X \otimes
        \sO_C(-D)$ is generated by global sections. A $0$-free curve is called free
        whereas a $1$-free curve is called very free. 
        \label{rfreecurvedefn}
    }

    \rem{
        The condition of $r$-freeness makes formal the notion that the curve $C$ deforms
        in $X$ while keeping any general $r$ points fixed.
    }

    The following lemma follows immediately from Lemma \ref{appendixvfimpliesf}.

    \lem{
        If $f:C\to X$ is an $r$-free curve then $f$ is $r'$-free for all $r'\leq r$.
        \label{vfimpliesf}
    }
    
    In the case of $C=\bP^1$, Grothendieck's Theorem \cite{grothendieck} gives us that $f^*\sT_X =
    \oplus_{i=1}^n \sO_{\bP^1}(a_i)$ with $a_1\leq \ldots\leq a_n$. From Corollary
    \ref{vbp1} it follows that for $r\geq0$ the condition that $H^1(\bP^1,
    f^*\sT_X(-r-1))=0$ is equivalent to that of global generation of $f^*\sT_X(-r)$ (see also
    \cite{debarre} Remark $4.6$). We deduce that $f:\bP^1\to X$ is $r$-free if and only if
    $a_1\geq r$.

    \n
    We also give a relative version of the above definitions as have already been
    discussed in \cite{ksct}.

    \defn{
        Let $\cC\to Y$ be a flat proper family of curves where $Y$ is a variety over an
        algebraically closed field, and let $X$ be a smooth $Y$-variety. Let
        $B\subset\cC$ be a closed subscheme, finite and flat over $Y$ and $g:B\to X$ a
        $Y$-morphism. We say that a $Y$-morphism $f:\cC\to X$ is relatively free over $g$
        if $f|_B = g$, $f^*\sT_{X/Y}\otimes \sI_B$ is globally generated and $H^1(\cC,
        f^*\sT_{X/Y}\otimes \sI_B)=0$.
    }

    \prop{(\cite{ksct} Proposition $14$, \cite{kollar} II$.3.5-3.7$) Let $f:\cC\to X$ as
        above be a morphism which is free over $f|_B$ and assume that $\dim Y=1$ and
        that $\cC\to Y$ is surjective. Then
        \begin{enumerate}
            \item The connected component $\Hom_{[f]}(\cC/Y, X/Y, f|_B) \subset
                \Hom(\cC/Y, X/Y, f|_B)$ containing $f$ is smooth at $[f]$.
            \item The evaluation morphism $\cC\times \Hom_{[f]}(\cC/Y, X/Y, f|_B) \to X$
                dominates $X$.
            \item If $Z\subset X$ is any subset of codimension at least $2$, and $[f']\in
                \Hom_{[f]}(\cC/Y, X/Y, f|_B)$ a general point, then $(f')^{-1}(Z)\subset
                B$.
        \end{enumerate}
        \label{relativedeformationovercurve}
    }

    \rem{
        We should remark at this point that there do not exist complete intersection
        curves which are free on a general smooth hypersurface. For example, let $X$ be a
        degree $d$ smooth hypersurface in $\bP^n$. Assume $d\leq n$ since otherwise $X$
        will be of general type or Calabi-Yau and will not have any free curves. Let $Y_i$
        be $n-2$ suitably general hypersurfaces in $\bP^n$ all of degree $e$ and let $C =
        X \cap_{i=1}^{n-2} Y_i$ be the resulting curve. The degree of $C$ is $de^{n-2}$
        and we have a short exact sequence
        \eqn{
            0\to \sT_C\to \sT_X|_C\to \sN_{C/X} \to 0
        }
        where the normal bundle is 
        \eqn{
            \sN_{C/X}=\oplus_{i=1}^{n-2} \sO_{\bP^n}(Y_i)|_C = \oplus_{i=1}^{n-2}
            \sO_{\bP^n}(e)|_C. 
        }
        By adjunction, since $C$ is a complete intersection curve, we compute
        \eqn{
            \deg \sT_C = -\deg \omega_C = -d(-n-1+d+\sum_{i=1}^{n-2} e).
        }
        Even setting $e=1$ to make $\deg \sT_C$ as large as possible, and taking into
        account that $\deg \sN_{C/X} = e(n-2)$, we see that $\deg \sT_X|_C = \deg \sT_C +
        \deg \sN_{C/X}$ is not going to be positive for non-trivial values of $d$ and $n$.
        Positivity of the degree of $\sT_X|_C$ would be necessary for any ampleness
        conditions.
    }

    The following result gives that a general deformation of a free curve is an embedding.
    We will often use this implicitly throughout the remainder of this section.

    \theo{(\cite{kollar} II$.1.8$)
        Let $f:C\to X$ be a morphism from a smooth projective curve to a smooth variety
        over $k$. Let $B\subset C$ be a subscheme with ideal sheaf $\sI_B$ and $g=f|_B$.
        \begin{enumerate}
            \item If $h^1(C, f^*\sT_X\otimes \sI_B(-2p) \leq \dim X-2$ for every $p\in C$,
                then a general deformation of $f$ over $g$ is an immersion on $C\setminus B$.
                Furthermore, if $g$ is an immersion, then a general deformation of $f$ over
                $g$ is an immersion on $C$.
            \item If $h^1(C, f^*\sT_X\otimes \sI_B(-p-q) \leq \dim X-3$ for every $p,q\in C$,
                then a general deformation of $f$ over $g$ is an embedding on $C\setminus B$.
                Furthermore, if $g$ is an embedding, then a general deformation of $f$ over
                $g$ is an embedding on $C$.
        \end{enumerate}
    }

    Hence if the dimension of $X$ is at least $3$, a general deformation of a $2$-free
    morphism is an embedding into $X$. We will see (Theorem \ref{rfreeiffample}) that in
    fact this holds for any free morphism too. From \cite{kollar} $II.3.2$, if a family of
    curves mapping to a variety has a member which is free over $g$, then the locus of all
    such curves in this family is open.

    \n
    There is another type of positive curve one can consider for a smooth projective
    variety $X$, namely $f:C\to X$ such that $f^*\sT_X$ is ample. Note that such a curve
    automatically has $\sN_{C/X}$ ample as in Lemma \ref{defnamplecurve} since the
    quotient of an ample bundle is ample in the following short exact sequence
    \eqn{
        0\to \sT_C\to f^*\sT_X\to \sN_{C/X}\to 0.
    }
    We will prove that the existence of a curve of genus $g\geq1$ such that $f^*\sT_X$ is
    ample is in fact equivalent to the existence of a free curve of the same genus.

    \prop{
        Let $X$ be a smooth projective variety over an algebraically closed field $k$ and
        $f:C\to X$ a morphism from a smooth projective curve such that $f^*\sT_X$ is
        ample. Then for any closed subscheme $B\subset C$ with ideal sheaf $\sI_B$, there
        exists a morphism $f':C\to X$ which is free over $\sI_B$.
        \label{ampleimpliesfree}
    }
    \proof{
        If the characteristic of $k$ is positive, then the result follows from Proposition
        \ref{ampleisvgincharp} by pulling back by Frobenius as we will see as a special
        case in the following. Now, assume the characteristic of $k$ is $0$, denote by
        $H_X$ an ample divisor on $X$ and consider the flat proper models $\cX, \cC, \cB,
        H_\cX$ of $X, C, B, H_X$ respectively over the algebra $R$ which is $\bZ$ adjoined
        all the coefficients defining $C, X$ etc.\ and all the morphisms between them, as
        in \cite{kollar} II$.5.10$. Denote by $g_R$ the morphism $\cB\to\cX$. From the
        construction, the geometric generic fibre of $\cX$ is $X$ and after shrinking
        $\Spec R$ we can assume $\cC$ is smooth and the same for $\cX$ over the image of
        $f_R:\cC\to\cX$. Note also that after further suitably shrinking $\Spec R$ we can
        assume $f_R^*\sT_\cX$ is a locally free relatively ample bundle on $\cC$ over $R$
        and the same for $H_\cX$ on $\cX$. Now, for any closed fibre $\cX_p$ over a
        maximal $p$ in $R$, the reduction of $f_p:\cC_p\to \cX_p$ will also have
        $f_p^*\sT_{\cX_p}$ ample and from Proposition \ref{ampleisvgincharp} we have
        (after pulling back by Frobenius) that there exists a morphism $f_p':\cC_p\to
        \cX_p$ which is free over $\sI_{\cB_p}$. We can perform this operation for every
        closed point $p$ in $\Spec R$ and note that from the proof of Proposition
        \ref{ampleisvgincharp} there is a global bound $d$ on the intersection number
        $(f'_{p*}\cC_p) . H_{\cX_p}$ depending on the genus $g$ of $C$ and the length of
        $B$. Considering now the relative moduli space $\Hom_R(\cC, \cX, g_R)$ and its
        subset $\rho:\Hom_R^{\operatorname{free},\leq d}(\cC, \cX, g_R) \to \Spec R$ of
        Proposition \ref{relativedeformationovercurve} containing curves which are free
        over $g_R$ and of relative $H_\cX$-degree at most $d$, we see that $\rho$ has
        non-trivial fibres over the closed points, but these are dense in $\Spec R$. We
        also know that $\Hom_R^{\operatorname{free},\leq d}(\cC, \cX, g_R)$ is of finite
        type over $R$ so by Chevalley's Theorem, the image of $\rho$ is constructible and
        since it is dense, it must contain the generic point (see \cite{hartshorne} ex.\
        II$.3.18$) hence the morphism $\rho$ must be surjective. Hence there exists a
        suitably free curve over the generic fibre which means a morphism $C\to X$ which
        is free over $\sI_B$. We conclude that the curve $f_p':C_p\to X_p$ lifts to a
        curve which is free over $\sI_B$ in characteristic zero.
    }

    A closer look at the proof of Proposition \ref{ampleisvgincharp} shows that proving
    global generation and vanishing of first cohomology is just a matter of degrees, hence
    if $f:C\to X$ a curve such that $f^*\sT_X$ is ample, then from the above proposition
    there is a morphism $f':C\to X$ such that $f'$ is $r$-free for any $r\geq0$.
    Conversely from Lemma \ref{vfimpliesf}, if $C$ is $r$-free for some $r\geq0$ then
    it is free and from Proposition \ref{ggandh1impliesample} it follows that
    $f^*\sT_X$ is ample. The above results and discussion lead to the following.

    \theo{
        Let $X$ be a smooth projective variety and $C$ a smooth projective curve of genus
        $g\geq1$ over an algebraically closed field $k$. Then for any $r\geq0$, there
        exists an $f:C\to X$ which is $r$-free if and only if there exists a morphism
        $f':C\to X$ such that $f'^*\sT_X$ is ample.
        \label{rfreeiffample}
    }

    Note that the above theorem is not true if $C$ was a rational curve, namely there
    exist varieties which have a free rational curve (uniruled varieties) which do not
    have a very free rational curve (rationally connected varieties) - see Theorem
    \ref{rciffvfree}.

    \lem{
        Let $X$ be a smooth variety over an algebraically closed field $k$ and $f:C\to X$
        a free morphism from a smooth projective curve over $k$. Then the image of $C$
        misses only finitely many divisors in $X$.
    }
    \proof{
        Note first that deformations of $C$ all lie in the same class in the cone of
        curves $N_1(X)$ so for a given point $x\in X$, we can consider the open subset
        $U_x\subset X$ consisting of images of deformations of $f:C\to X$ which go through
        $x$. The complement is a proper closed subset of $X$ and so contains a finite
        number of irreducible closed subsets so a finite number of divisors.
    }
    \lem{
        Let $X$ be a smooth variety over an algebraically closed field $k$ with $D\subset
        X$ a divisor and $f:C\to X$ a free morphism from a smooth projective curve over
        $k$ and $p\in C$. Then there exists a deformation $f':C\to X$ with $f'(p)\notin
        D$.
        \label{defmissingadivisor}
    }
    \proof{
        By semicontinuity let $U\subset\Hom(C, X)$ be a connected open neighbourhood of
        $[f]$ such that $H^1(C, f_t^*\sT_X) = 0$ for all $[f_t]\in U$. From \cite{mori} it
        follows that the dimension of $U$ is $h^0(C, f^*\sT_X)$. Denote by $\sI_p$ the
        ideal sheaf on $C$ of the closed subscheme with unique point $p$. Since $f$ is
        free, we have $H^1(C, f_t^*\sT_X\otimes \sI_p) = 0$ for all $[f_t]\in U$ and so by
        fixing a point $x\in X$ such that $p\mapsto x$, we have
        \eqn{
            \dim(\Hom(C,X;p\mapsto x) \cap U) &=& h^0(C, f^*\sT_X\otimes \sI_p)\\ &=&
            h^0(C, f^*\sT_X) - \dim X \\ &=& \dim U - \dim X.
        }
        Next, denote by
        \eqn{
            V = \{[f_t] \in U \mid f_t(p)\in D\} = \bigcup_{x\in D}\{[f_t]\in U \mid f_t(p)=x\}
        }
        the subspace of all morphisms in $U$ which send $p$ to a point in the divisor $D$. It
        follows that
        \eqn{
            \codim(V, U) &\geq& \dim U - \dim V \\ &=& h^0(C,f^*\sT_X) - (h^0(C, f^*\sT_X) -
            \dim X + \dim X - 1) \\ &=& 1
        }
        and hence there exists an $[f']\in U\setminus V$ such that $f'(p)\notin D$.
    }
    \rem{
        Note that this does not hold for $C$-connected varieties since it may be that
        $H^1(C, f^*\sT_X\otimes \sI_p)\neq 0$ so when we fix a point in the source and target,
        the dimension of the space of morphisms does not necessarily decrease by $\dim
        X$. Consider for example $C\to C\times\bP^n$.
    }

    A similar proof leads to the following result, stating that a general deformation of a
    free curve will miss a codimension two and above subset.

    \lem{(\cite{kollar} II$.3.7$)
        Let $X$ be a smooth projective variety over an algebraically closed field $k$ and
        $f:C\to X$ a morphism from a smooth projective curve which is free over $g:B\to X$.
        Let $Z\subset X$ be a subscheme of codimension at least $2$. Then $f'(C\setminus
        B)\cap Z = \emptyset$ for a general deformation $f'$ of $f$ over $g$.
        \label{freecurvemisscodimtwo}
    }

    \prop{
        Let $X$ be a smooth variety over an algebraically closed field $k$ and $f:C\to X$
        a smooth projective curve which is free over $B\subset C$ a closed subscheme with
        ideal sheaf $\sI_B$. Let $g:X\dashrightarrow Y$ be a rational map to a smooth
        variety $Y$. Then it follows that $f':=g\circ f: C\dashrightarrow Y$ can be
        deformed to a morphism free over $B$ if either of the following two conditions
        holds
        \begin{enumerate}
            \item the characteristic of $k$ is $0$ or
            \item the characteristic of $k$ is positive and $g$ is generically smooth.
        \end{enumerate}
        \label{vfpushesdown}
    }
    \proof{
        First note that in any characteristic we can deform $f:C\to X$ so that it misses
        the codimension $2$ exceptional locus of $g$ (from Lemma
        \ref{freecurvemisscodimtwo}) so we can assume that the composition $g\circ f:
        C\dashrightarrow Y$ is in fact a morphism. We also have an exact sequence of
        sheaves on $X$
        \eqn{
            0\to \sT_{X/Y} \to \sT_X \to g^* \sT_Y
        }
        and applying $f^*$ and tensoring with $\sI_B$ we obtain the following exact
        sequence
        \eqnn{
            0\to f^*\sT_{X/Y}\otimes \sI_B\to f^*\sT_X\otimes \sI_B \to (g\circ f)^*\sT_Y\otimes
            \sI_B.
            \label{sse1}
        }
        If $k$ has characteristic zero then by generic smoothness (see \cite{hartshorne}
        III$.10$) we may assume that $g$ is a smooth morphism and the above sequences are
        exact on the right. We can now apply Lemma \ref{fullrkggh1} to conclude that
        $(g\circ f)^*\sT_Y\otimes \sI_B$ is globally generated and has $H^1(C, (g\circ
        f)^*\sT_Y \otimes \sI_B)=0$. Assume now that the characteristic of $k$ is positive
        and that $g:X\to Y$ is generically smooth. There exists an open $U\subset X$ such
        that $g|_U:U\to Y$ is smooth, and since $f$ is free, we can find a deformation
        whose image misses the complement of $U$ but for a finite number of points. Hence
        we have an exact sequence
        \eqn{
            f^*\sT_X \to (g\circ f)^*\sT_Y \to \sQ \to 0
        }
        where $\sQ$ is a torsion sheaf supported on this finite set of points. This
        implies that $f^*\sT_X \to (g\circ f)^*\sT_Y$ has full rank (i.e.\ its image has
        the same rank as the target). The result now follows from Lemma \ref{fullrkggh1}. 
    }

    \prop{(\cite{kollar} II$.3.5$)
        Let $C$ be a smooth proper curve, $p,q\in C$ and $X$ a smooth variety and
        $[f]\in\Hom(C,X,g)$ a smooth point. Using notation of the evaluation morphisms
        from the beginning of this section we have
        \begin{enumerate} 
            \item if $\phi(p,f)$ is surjective then $F$ is smooth at $(p,[f])$. The
                converse holds if $H^0(C, \sT_C\otimes \sI_B)\arrow \sT_C\otimes k(p)$ is
                surjective.
            \item If $\phi^{(2)}(p,q,f)$ is surjective then $F^{(2)}$ is smooth at
                $(p,q,[f])$. The converse holds if $H^0(C, \sT_C\otimes \sI_B)\arrow \sT_C\otimes
                k(p) \oplus \sT_C\otimes k(q)$ is surjective.
        \end{enumerate} 
        \label{kol35}
    }

    We summarise the known results for rationally connected varieties in the following
    theorem.
    \theo{
        Let $X$ be a smooth projective variety over an algebraically closed field $k$. If
        the characteristic of $k$ is zero then the following are equivalent
        \begin{enumerate}
            \item There exists a very free morphism $f:\bP^1\to X$.
            \item $X$ is $\bP^1$-connected.
            \item $X$ is genus $0$ connected.
            \item $X$ is genus $0$ chain connected.
        \end{enumerate}
        If the characteristic of $k$ is positive, then $(1)\implies (2)\iff (3)\implies
        (4)$. In any characteristic, if we furthermore assume separability in $(2), (3)$
        and $(4)$ then all statements are equivalent to $(1)$.
        \label{rciffvfree}
    }

    Similarly, if $k$ is algebraically closed of any characteristic, $X$ is separably
    uniruled if and only if there exists a free rational curve $f:\bP^1\to X$.

    \rem{
        An example of a smooth projective variety in positive characteristic $p$ which is
        rationally connected but does not have any free rational curves (hence no very
        free rational curves either) is given by the Fermat hypersurface $x_0^d + x_1^d +
        \ldots + x_N^d$ where $N\geq3$ and $d=p^r+1$ (see \cite{debarre} $4.4$).
        \label{nofreecurvescharp}
    }

    In the case of higher genus curves though there exist genus $g$ connected varieties
    which do not have a free or very free curve for all $g\geq1$, for example consider
    $E\times\bP^1$ where $E$ is an elliptic curve. As pointed out before Lemma
    \ref{allvarsaregconnforsomeg}, $E\times\bP^1$ is $E$-connected yet it is not possible
    that there exists a morphism $f:C\to E\times\bP^1$ from a curve $C$ such that
    $f^*\sT_{E\times\bP^1}$ is ample since this bundle is isomorphic to $\sO_C\oplus
    \sO_C(2)$ which has a non-ample quotient $\sO_C$.  One can however prove the following
    proposition.

    \prop{
        Let $X$ be a smooth variety over an algebraically closed field and $f:C\arrow X$
        a very free morphism for some smooth projective curve $C$. Then $X$ is separably
        $C$-connected.
        \label{vfreesogcon}
    }
    \proof{
        Let $[f]\in Y\subset \Hom(C,X)$ be an open and smooth neighbourhood with cycle map
        $u:C\times Y \arrow X$. We first show that the evaluation map
        \eqn{
            \phi^{(2)}(p,q,f): H^0(C, f^*\sT_X)\arrow f^*\sT_X\otimes k(p) \oplus
            f^*\sT_X\otimes k(q)
        }
        is surjective for $p\neq q$ general points in $C$. Consider the following exact
        sequences of sheaves
        \eqn{
            0 \to f^*\sT_X(-p-q) \to f^*\sT_X \to (f^*\sT_X\otimes k(p)) \oplus
            (f^*\sT_X\otimes k(q)) \to 0 \\
            0 \to f^*\sT_X(-p-q) \to f^*\sT_X(-p) \to f^*\sT_X(-p)\otimes k(q) \to 0
        }
        and note that by taking the long exact sequence in cohomology of the first, to show that
        $\phi^{(2)}(p,q,f)$ is surjective, we need to show that $H^1(C, f^*\sT_X(-p-q))=0$.
        Since $f$ is very free we have from the second sequence that $H^0(C,
        f^*\sT_X(-p))\to f^*\sT_X(-p)\otimes k(q)$ is surjective and also that $H^1(C,
        f^*\sT_X(-p))=0$ from which it follows that $H^1(C, f^*\sT_X(-p-q))=0$.
        
        \n
        Since $\phi^{(2)}(p,q,f)$ is surjective, it follows from Proposition \ref{kol35}
        that $u^{(2)}:C\times C\times Y\arrow X\times X$ is smooth at $(p,q,[f])$. We
        conclude that $X$ is separably $C$-connected and thus also separably connected by
        genus $g$ curves.
    }


\section{Proving uniruledness and rational connectedness}


    In this section we prove that the existence of a free curve of genus at least $1$ in
    characteristic zero implies the existence of a very free rational curve.

    \prop{
        Let $X$ be a smooth variety of $\dim X\geq2$ over an algebraically closed field
        $k$ and $f:C\to X$ a free morphism for $C$ a smooth projective curve of genus $g$.
        Then $X$ is uniruled.
        \label{fimpliesuniruled}
    }
    \proof{
        We can compactify $X$ and assume that it is projective (note that by Lemma
        \ref{freecurvemisscodimtwo} we can find a free deformation of $f$ which misses the
        exceptional locus of a birational map). From the Bend and Break Lemma
        \ref{bendandbreak}, if we show that $\dim_{[f]} \Hom(C,X; f|_{\{c\}})\geq1$, then
        there will be a rational curve in $X$ passing through $f(c)$. This condition is
        equivalent to $-K_X.C - g\dim X \geq 1$. Since $f$ is free, Riemann-Roch gives
        \eqn{
            h^0(C,f^*\sT_X) &=& \deg f^*\sT_X + (1-g)\dim X \\
                            &=& -K_X.C + (1-g)\dim X
        }
        and since $f^*\sT_X$ is globally generated, we have $h^0(C, f^*\sT_X)\geq d=\dim X$.
        In fact, consider $d$ linearly independent sections $f_1,\ldots,f_d\in
        H^0(C,f^*\sT_X)$ and consider the short exact sequence given by these sections
        \eqn{
            0\to\sO_C^{\oplus d} \to f^*\sT_X \to \sQ\to 0.
        }
        Here $\sQ$ is a torsion sheaf and by taking cohomology and noting that
        $H^1(C,f^*\sT_X)=0$ we obtain that $h^0(C,\sQ) \geq h^1(C,\sO_C^{\oplus d}) = gd$.
        Since $h^0(C, \sO_C^{\oplus d}) = d$, it must be that $h^0(C, f^*\sT_X) > d$.
        Hence we obtain $-K_X.C + (1-g)\dim X > \dim X$ from which $-K_X.C - g\dim X \geq
        1$ and so Bend and Break applies.
    }

    From Lemma \ref{vfimpliesf} we obtain that the same result holds if $f$ was very free
    or such that $f^*\sT_X$ was ample. Note also the above result follows as a
    combination of Theorem \ref{rfreeiffample}, Proposition \ref{vfreesogcon} and
    Proposition \ref{cconisuniruled}, but the above proof is simpler.

    \theo{
        Let $X$ be a smooth projective variety over an algebraically closed field $k$ and
        $f:C\to X$ a free morphism from a smooth projective curve of genus $g\geq1$. Then
        the tower of MRC fibrations terminates with a point.
        \label{mrctowerispoint}
    }
    \proof{
        Applying Theorem \ref{rfreeiffample} twice we can find a very free morphism
        $f':C\to X$. Replacing $f$ by $f'$, it follows from Proposition \ref{vfreesogcon}
        that $X$ is separably $C$-connected. Following the argument as in Proposition
        \ref{cconmrctopointiorcurve}, by showing that at each step of the tower of MRC
        fibrations $X\dashrightarrow R^1(X) \dashrightarrow\cdots \dashrightarrow R^n(X)$
        the target $R^i(X)$ is uniruled, it must be that $R^n(X)$ is either a point or a
        curve $C'$. In the latter case, call the dominant composition map $\pi: X
        \dashrightarrow C'$ where we can assume that $C'$ is a smooth projective curve. By
        invoking Lemma \ref{freecurvemisscodimtwo} it follows that we can assume that $C$
        misses the codimension two exceptional locus of $\pi$ and from Lemma
        \ref{defmissingadivisor}, for a point $p\in C$, we can deform $f$ so that the
        image of $p$ misses the inverse image under $\pi$ of $\pi(f(p))$. Hence there is
        an at least $1$ dimensional family of non-constant morphisms from $C$ to $C'$ and
        from de Franchis' Theorem (\cite{acgh} Theorem $8.27$) it follows that $C'$ has genus
        $0$ or $1$. Assume that $C'$ has genus $1$. We have $\sT_{C'}=\sO_{C'}$ and
        $f^*\pi^*\sO_{C'} \cong \sO_C$ and we thus obtain $\sO_C$ as a quotient of $f^*\sT_X$
        since
        \eqn{
            f^*\sT_X \to f^*\pi^*\sO_{C'} \to 0
        }
        is exact. Taking cohomology we see that since $f$ is free, one obtains a
        contradiction since $H^1(C, \sO_C) \neq 0$. Finally note that $C'=\bP^1$ is ruled
        out since we have assumed that the tower of MRC fibrations is maximal.
    }

    As in the discussion following after Proposition \ref{cconmrctopointiorcurve}, we
    obtain the following.

    \cor{
        In the above setup, if $k$ has characteristic zero then $X$ is rationally
        connected.
        \label{mrctowerispointcor}
    }

    We will see in the next section that a straightforward application of Mori's Bend and
    Break Lemma \ref{bendandbreak} in combination with the Graber-Harris-Starr Theorem
    \ref{ghs} leads to proving that every elliptically connected variety over an arbitrary
    algebraically closed field of characteristic zero is either rationally connected or a
    rationally connected fibration over an elliptic curve (see Theorem
    \ref{elconclassification}). Assuming the ampleness of the restriction of a foliation
    to a smooth curve in characteristic zero, results of this type have been demonstrated
    in the work of various people. A short summary follows.

    \theo{(\cite{bm} Theorem $0.1$)
        Let $X$ be a normal complex projective variety and $C\subset X$ a complete curve
        in the smooth locus of $X$. Assume that $\sF\subset \sT_X$ is a foliation regular
        along $C$ and such that $\sF|_C$ is ample. If $x\in C$ is any point, the leaf
        through $x$ is algebraic and if $x\in C$ is general then the closure of the leaf
        is also rationally connected.
        \label{bm}
    }
    This result was reproved in \cite{ksct} using a simpler Mori-type technique of
    reduction to characteristic $p$, proving uniruledness of the MRC quotient and deriving
    a contradiction using the Graber-Harris-Starr Theorem \cite{ghs}, in a manner similar
    to our results. See also \cite{ksc}.

    \theo{(\cite{bdpp04} Corollary $0.3$)
        Let $X/\bC$ be a projective manifold. If $K_X$ is not pseudo-effective then $X$ is
        uniruled.
    }

    Using this theorem, Peternell proved a weaker version of Mumford's conjecture on
    numerical characterisation of rationally connected varieties (Conjecture
    \ref{mumfordconj}) from which one can deduce the following theorem.

    \theo{(\cite{peternell06} $5.4$, $5.5$)
        Let $X/\bC$ be a projective manifold and $C\subset X$ a possibly singular curve. If
        $\sT_X|_C$ is ample then $X$ is rationally connected. If $\sT_X|_C$ is nef and
        $-K_X.C>0$ then $X$ is uniruled.
        \label{peternell}
    }

    Using either of Theorems \ref{peternell} or \ref{bm}, a smooth projective variety $X$
    over an algebraically closed field of characteristic zero with a free genus $g$ curve
    $f:C\to X$ such that $g\geq1$ is automatically rationally connected since from
    Proposition \ref{ggandh1impliesample} we have that $f^*\sT_X$ is ample. Similarly,
    from Proposition \ref{ampleimpliesfree} and Corollary \ref{mrctowerispointcor} we
    obtain the following.

    \theo{
        Let $X$ be a smooth projective variety over an algebraically closed field $k$ of
        characteristic zero and assume that there exists a smooth curve $f:C\to X$ of genus
        $g$ such that $f^*\sT_X$ is ample. Then $X$ is rationally connected.
        \label{ampleimpliesrc}
    }

    \rem{
        At this point we do not know whether the above statement holds in characteristic
        $p$ or whether, still in positive characteristic, assuming that we have a free
        curve $f:C\to X$ of genus $g\geq 1$ implies that $X$ is separably rationally
        connected or even rationally chain connected. It is tempting to hope that both
        statements are true though. Jason Starr informs us that his maximal free rational
        quotient (MFRC) \cite{starr06} gives a generically (on the source) smooth morphism
        $X\to R_f(X)$ over any algebraically closed field $k$, so if $X$ contained a free
        rational curve $f:\bP^1\to X$, then $\dim R_f(X) < \dim X$. Hence, if $f:C\to X$ a
        free curve of genus $g\geq1$ implied that we have a free rational curve $\bP^1\to
        X$ (we do not know how to show this), taking successive MFRC quotients and using
        Proposition \ref{vfpushesdown} would reduce the tower of MFRC quotients to a
        point. This does not mean that $X$ will necessarily be rationally connected, but
        since there is a free rational curve on $X$, it will at least be separably
        uniruled. Even though Bend and Break arguments give us the existence of many
        rational curves, the author does not know any techniques to construct free
        rational curves in positive characteristic.
        \label{conjectureremark}
    }



\section{Elliptically connected varieties}


    Let $k$ be an algebraically closed field of arbitrary characteristic. In this section
    we will study more carefully the cases of genus $1$ connected (elliptically
    connected), $E$-connected varieties and free morphisms $E\to X$ where $E$ is an
    elliptic curve.

    \n
    Denoting RC and EC to mean rationally and elliptically connected respectively, it
    follows from Lemma \ref{genusgconnisgconforbiggerg} that we have the following inclusions
    of sets of varieties
    \eqn{
        \{\text{rational}\} \subsetneq \{\text{unirational}\} \subseteq \{\text{RC}\}
        \subsetneq \{\text{EC}\} \subsetneq \{\text{uniruled}\}.
    }
    It is an open problem whether there exists a non-unirational rationally connected
    variety but it is widely expected these do exist. An example of an elliptically but
    not rationally connected variety is given by $E\times\bP^n$ as we saw in the
    discussion after Definition \ref{defncconn}. An example of an elliptically connected
    variety which is not $E$-connected for some genus $1$ curve is given at the end of
    this section. We begin with a Bend and Break lemma.

    \lem{
        Let $f:E\arrow X$ be a morphism from a smooth genus $1$ curve $E$ to a smooth
        projective variety $X$ over an algebraically closed field $k$ of dimension at
        least $2$ and let $e\in E$. Assume that $-K_X.f_*E>0$ (for example $-K_X$ ample).
        Then there exists a rational curve on $X$ through $f(e)$.
        \label{babell}
    }
    \proof{
        Mori's Bend and Break Lemma \ref{bendandbreak} tells us that the conclusion of
        this lemma holds if the condition $\dim_{[f]}\Hom(E, X;f|_{e})\geq 1$ holds. By
        \cite{debarre} $2.11$ this is equivalent to $-K_X.f_*E-\dim X\geq 1$. From the
        projection formula, the integer $-K_X.f_*E$ is equal to $f^*(-K_X).E$. Now we can
        consider $[n]:E\arrow E$ the multiplication by $n$ isogeny on the elliptic curve
        $E$ and precompose $E\arrow X$ by this isogeny thus arbitrarily increasing
        $-K_X.f_*E$ the degree of $E$ under the anti-canonical embedding.
    }

    \prop{
        Let $X$ be an elliptically connected smooth projective variety. Then 
        \begin{enumerate}
            \item if $X$ is a surface it is either rationally connected or ruled, in
                particular it is uniruled,
            \item if $\dim X\geq3$ then $X$ is uniruled.
        \end{enumerate}
        \label{uniruled}
    }
    \proof{
        Let $\cE\to Y$ be the family of genus $1$ curves which makes $X$ elliptically
        connected. For (i), assume that $X$ is a smooth elliptically connected surface.
        Pick $x$ a point on $X$ and $f:C\to X$ a smooth irreducible genus $1$ curve whose
        image passes through $x$. Since elliptic curves have trivial canonical bundle, the
        converse condition in Proposition \ref{kol35} $(1)$ is satisfied and it follows
        that $f^*\sT_X$ is globally generated at $x$. This holds for all $x\in C$ so
        $f^*\sT_X$ is a globally generated vector bundle. Now, we can assume $C$ is
        smoothly embedded in $X$ and consider the short exact sequence
        \eqn{
            \xymatrix{
                0 \to \sT_C \to \sT_X|_C \to \sN_{C/X} \to 0
            }
        }
        where $\sT_C = \sO_C$ is trivial and $\sN_{C/X}$ is the normal line bundle of $C$
        in $X$. Since $\sT_X|_C$ and $\sT_C$ are globally generated, it follows that $\deg
        \sN_{C/X} > 0$. From the adjunction formula $2g_C-2 = C.(C+K_X)$ and so $C.K_X =
        -C^2 = -\deg \sN_{C/X} < 0$ from which it follows that $K_X$ is not nef. From the
        classification of surfaces (see for example \cite{friedman} Theorem $10.4$), this
        implies that $X$ is either rational or ruled and hence uniruled. Alternatively,
        since $-C.K_X>0$ we can apply Bend and Break as in Lemma \ref{babell} and produce
        a rational curve through every point of $X$.

        \n
        For (ii), we apply Bend and Break as follows. Fix a general point $x\in X$. From
        Mumford's Rigidity Lemma, we know that no fibres of $\cE\arrow Y$ are contracted in
        $X$. Hence, since $\cE\times_Y\cE$ dominates $X\times X$ we know that $\dim Y \geq
        2\dim X-2$. Now, the subspace $Z$ in $Y$ of curves which go through $x$ has
        dimension at least $\dim Y - (\dim X-1)$ and so the subspace $Z'$ in $Z$ of curves
        of fixed $j$-invariant that go through $x$ has dimension $\dim Z'\geq \dim Y-(\dim
        X-1)-1$. Hence $\dim Z' \geq \dim X-2\geq 1$ since $X$ has dimension at least
        $3$. From Lemma \ref{bendandbreak} it now follows that there is a rational curve
        through $x$. After possibly an extension to an uncountable algebraically closed
        field this implies that $X$ is uniruled (see \cite{debarre} Remark $4.2(5)$). 
    }

    \theo{
        Let $X$ be a smooth projective variety over an algebraically closed field $k$ of
        characteristic zero. Then $X$ is elliptically connected if and only if it is
        rationally connected or a rationally connected fibration over an elliptic curve.
        \label{elconclassification}
    }
    \proof{
        Consider the MRC fibration $\pi: X \dashrightarrow R(X)$. From Corollary
        \ref{ghscor}, we know that $R(X)$ is not uniruled. On the other hand, we know from
        Lemma \ref{gconproperties} part $(1)$ that $R(X)$ is elliptically connected since
        $\pi$ is dominant. Since $R(X)$ is elliptically connected and not uniruled, it
        follows from Proposition \ref{uniruled} that it must be either of dimension $0$
        and thus $X$ is rationally connected, or of dimension $1$ and so an elliptic curve
        $E$ by the Riemann-Hurwitz Theorem. By resolving the indeterminacy locus of the
        rational map $X\dashrightarrow R(X)$ we find a birational model $X'$ of $X$ such
        that $X'\to E$ has rationally connected general fibre. Consider the commutative
        diagram
        \eqn{
            \xymatrix{
                X\ar[d]\ar[r] & X'\ar[d] \\
                \Alb X\ar[r] & E 
            }
        }
        where $X\to\Alb X$ is the Albanese morphism (which contracts all rational curves
        in $X$) and $\Alb X\to E$ comes from the universality of the Albanese variety. We
        obtain that $\Alb X$ is an elliptic curve $E'$ isogenous to $E$ and that the
        fibres of $X\to E'$ are rationally connected.

        \n
        Conversely, we have seen that a rationally connected variety is elliptically
        connected in Lemma \ref{genusgconnisgconforbiggerg}. If on the other hand $X$ is a
        rationally connected fibration over an elliptic curve $E$ then from Proposition
        \ref{rcfibrationovercurveiscconnected} we know that it is $E$-connected.
    }
    \rem{
        In case $k$ was of positive characteristic, using the same methods as in Lemma
        \ref{cconmrctopointiorcurve} we deduce that for an elliptically connected variety
        $X$, the tower of the MRC fibration $X \dashrightarrow R^1(X) \dashrightarrow
        \cdots \dashrightarrow R^n(X)$ terminates with $R^n(X)$ a point or a curve. 
    }

    We now give an example of a way of constructing elliptically connected varieties over
    an arbitrary field that are not rationally connected. Families of such objects were
    recently constructed by Bjorn Poonen \cite{poonen} as counterexamples to the
    sufficiency of the Brauer-Manin obstruction applied to \'etale covers and we outline
    the construction in the following paragraph.

    \n
    Let $k$ be any field and let $P_\infty(x), P_0(x)$ be two relatively prime and
    separable degree $4$ polynomials in $k[x]$. Consider the vector bundle $\sL=\sO(1,2)$ on
    $\bP^1\times\bP^1$ and define the section
    \eqn{
        s := u^2\tilde{P}_\infty(w,x) + v^2\tilde{P}_0(w,x) \in \Gamma(\bP^1\times\bP^1,
        \sL^{\otimes2})
    }
    where $\tilde{P}_\infty,\tilde{P}_0$ are the homogenisations and $(u,v)$ and $(w,x)$
    are the coordinates of each of the two copies of $\bP^1$ respectively. For $a\in
    k^*$, the equation $y^2-az^2=s$ defines a conic bundle $V\to\bP^1\times\bP^1$.
    Composing with the first projection $\bP^1\times\bP^1\to\bP^1$, the fibre in $V$ above
    $\infty\in\bP^1$ is the Ch\^atelet surface $V_\infty: y^2-ax^2=P_\infty(x)$. Now let
    $f:E\to\bP^1$ be a dominant map from an elliptic curve such that $f$ is \'etale over
    the branch locus of the first projection of the zero locus of $s$ to $\bP^1$ (i.e.\ the
    locus $Z\in\bP^1\times\bP^1$ over which the fibres in $V$ split into a nodal union of
    two lines). We have a pullback diagram
    \eqn{
        \xymatrix{
            X:=V\times_{\bP^1} C \ar[d]^\pi\ar[r] & V\ar[d]\\
            E\ar[r]^f & \bP^1
        }
    }
    where now $X\to E$ is a fibration in Ch\^atelet surfaces over an elliptic curve. In
    other words, after passing to an uncountable algebraically closed field from Lemma
    \ref{gconproperties} parts $(4)$ and $(6)$ and using Theorem
    \ref{elconclassification}, $X$ is an elliptically connected threefold which is not
    rationally connected.


\section{Towards a positive characteristic analogue}

    From Remark \ref{conjectureremark} and the work preceding it, we would like to
    demonstrate that the existence of a free higher genus curve implies the existence of a
    free rational curve in positive characteristic, something which holds in
    characteristic zero by combining Theorems \ref{mrctowerispointcor} and
    \ref{rciffvfree}. In this section we will prove small results in this direction.

    \lem{
        \label{freecurveintersectionnumber}
        Let $X$ be a smooth projective variety of dimension $n$ over an algebraically
        closed field $k$ with $f:C\to X$ a free morphism from a smooth projective curve of
        genus $g$. Then 
        \eqn{
            C.K_X \leq n - g(n+1).
        }
    }
    \proof{
        We have a short exact sequence
        \eqn{
            0\to \sT_C\to f^*\sT_X\to \sN_{C/X} \to 0.
        }
        Since $f$ is free, $\deg \sN_{C/X}>0$ and by taking cohomology of the above short
        exact sequence we know that $H^0(C, \sN_{C/X})\to H^1(C, \sT_C)$ is surjective and
        $H^1(C, \sN_{C/X})=0$. By Serre duality, $h^1(C, \sT_C) = h^0(C, \omega_C^{\otimes
        2}) = 4g-3$ and so $h^0(C, \sN_{C/X})\geq 4g-3$, where $g$ is the genus of $C$. By
        Riemann-Roch,
        \eqn{
            \deg \sN_{C/X} &=& h^0(C, \sN_{C/X}) - h^1(C, \sN_{C/X}) + (n-1)(g-1) \\
                         &\geq& (n+3)g - (n+2).
        }
        We now have that 
        \eqn{
            C.K_X &=& \deg f^*\omega_X \\ 
                  &=& \deg \left(f^*\wedge^n \sT_X^*\right) \\
                  &=& \deg \left(\sT_C^*\otimes\wedge^{n-1} \sN_{C/X}^*\right) \\
                  &=& 2g-2 - \deg \sN_{C/X}
        }
        and so $C.K_X\leq -g(n+1) + n$.
    }

    \prop{
        Let $X$ be a smooth projective surface over an algebraically closed field $k$ with
        $f:C\to X$ a very free morphism from a smooth projective curve $C$. It follows
        that $X$ is separably rationally connected.
    }
    \proof{
        If $C$ is of genus $0$ then $X$ is separably rationally connected by definition.
        In general we only need to assume that the curve is free and from the above Lemma
        \ref{freecurveintersectionnumber} we have that $K_X$ is not nef and from the
        classification of surfaces (\cite{friedman} Theorem $10.4$) this means that $X$ is
        either a rational or ruled surface. If $X$ was ruled it would admit a birational
        morphism to $\bP^1\times C$. From Proposition \ref{vfpushesdown} the free morphism
        $f:C\to X$ would give a free morphism $C\to \bP^1\times C$ which would mean $C$
        must be a rational curve and that $X$ was birational to $\bP^2$. In other words,
        the only possibility is that $X$ is a rational surface and the result follows.
    }

    \rem{
        Some remarks about the case of dimension three, where the minimal model program is
        incomplete in positive characteristic. From the main theorem in \cite{kollar91},
        assuming $X$ is smooth and that it admits a free morphism from a curve, we can
        contract extremal rays in the cone of curves in arbitrary characteristic, to
        obtain a Fano fibration over a curve, surface or point (i.e.\ $X$ is Fano). In the
        case where there exists a conic fibration $X\to Y$ where $Y$ is a smooth surface,
        Koll\'ar proves that if the characteristic of $k$ is not $2$ then the general
        fibre is smooth. From Proposition \ref{vfpushesdown} it follows that the
        composition morphism $C\to Y$ is free and so from the above proposition for the
        case of surfaces, $Y$ is a rational surface. Hence $X$ is a conic bundle over a
        rational surface so it is rational, hence separably rationally connected. If $X\to
        Y$ a Fano fibration over a curve, to the author's knowledge, it is not known
        whether the fibres of the del Pezzo surface fibration over $Y$ obtained in this
        way must be smooth. Assuming for the time being that they were, they would be
        separably rationally connected and from the deformation theory argument in Theorem
        \ref{mrctowerispoint} and de Franchis' Theorem (\cite{acgh} Theorem $8.27$), $Y$
        would be $\bP^1$. From the de Jong-Starr Theorem \ref{ghs} we would obtain
        sections $\bP^1\to X$ from which we could assemble combs with very free
        teeth to be smoothed to very free rational curves in $X$, showing that
        $X$ is separably rationally connected.
    }

    The following result is well known in the case of $\bP^1$ (see \cite{debarre}
    $4.18$) and easily extends to higher genus.

    \prop{
        \label{freecurveimpliesnoglobalforms}
        Let $f:C\to X$ be a very free morphism from a smooth projective curve $C$ to a
        smooth projective variety $X$ over an algebraically closed field $k$. Then for all
        positive integers $m,p$
        \eqn{
            H^0(X, (\Omega^p_X)^{\otimes m})=0.
        }
    }
    \proof{
        Since $f:C\to X$ is very free, from Proposition \ref{vfreesogcon} there is a
        variety $U$ such that $C\times U\to X$ makes $X$ separably $C$-connected. Being
        very free is an open property (\cite{kollar} II$.3.2$) so we can assume that the
        general morphism $f_u:C_u\to X$ for $u\in U$ is very free, and so $f_u^*\sT_X$ is
        ample from Proposition \ref{ggandh1impliesample} (and by definition of a very free
        curve in the genus $0$ case). We conclude that for a general point $x\in X$ there
        is a morphism $f_u:C_u\to X$ such that $f_u^*\sT_X$ is ample and whose image
        passes through $x$. Hence since $f_u^*\Omega^1_X$ is negative, any section of
        $(\Omega_X^p)^{\otimes m}$ must vanish on the image $f(C_u)$ hence on a dense open
        subset of $X$, and so on $X$.
    }

    \prop{
        Let $f:C\to X$ be a very free morphism from a smooth projective curve $C$ to a
        smooth projective variety $X$ over an algebraically closed field $k$. It follows
        that the Albanese variety $\Alb X$ is trivial.
    }
    \proof{
        Note that in characteristic zero, we have that 
        \eqn{
            \dim\Alb X = \dim \Pic^0 X = \dim H^1(X, \sO_X) = h^{0,1}. 
        }
        Hodge duality gives that $h^{1,0}=h^{0,1}$ but more generally over any
        algebraically closed field we have (see \cite{igusa}) that $\dim\Alb X \leq
        h^{1,0} = h^0(X, \Omega^1_X)$. The result follows from Proposition
        \ref{freecurveimpliesnoglobalforms}.
    }


\section{An example in positive characteristic}


    Let $X$ be the Fermat quintic surface $x_0^5+x_1^5+x_2^5+x_3^5=0$ in $\bP^3$ over an
    algebraically closed field of characteristic $p$. In \cite{shioda} it is proven that
    if $p\neq 5$ and $p$ is not congruent to $1$ modulo $5$, then $X$ is a unirational
    general type surface and if we quotient by the action of the group $G$ of $5$-th
    roots of unity $x_i\mapsto \zeta^i x_i$, then we obtain a Godeaux surface which is
    again unirational but has algebraic fundamental group $\pi_1^{\et}(X/G, \bar{y})
    \cong \bZ/5\bZ$. Note that in characteristic zero, the notions of rationally chain
    connected, rationally connected, freely rationally connected (see \cite{shen}) and
    separably rationally connected all coincide and it is known that each variety in
    this class is simply connected. In positive characteristic however these notions are
    in decreasing generality and can differ. A rationally chain connected variety always
    has finite fundamental group (see \cite{cl03}) whereas a freely rationally connected
    variety is simply connected (see \cite{shen}). Note that Shioda's example above gives
    a unirational and hence rationally connected variety over a characteristic $p$
    algebraically closed field which is not simply connected.  

    \n
    We show there is a smooth projective variety in characteristic $p$ which has infinite
    \'etale fundamental group but after a finite number of MRC quotients we terminate with
    a point. Let $C$ be a smooth $5$ to $1$ cover of $\bP^1$, with defining affine
    equation of the form $y^5=f(x)$ where $f$ is a general polynomial of high degree. We
    have an action of $G=\bZ/5\bZ$ on $C$ which we can extend to the product $X\times C$
    of the above Fermat quintic $X$ with $C$. Projecting from the quotient onto the second
    factor we have a morphism
    \begin{equation*}
        (X\times C)/G \to \bP^1
    \end{equation*}
    where we have identified $C/G$ with $\bP^1$. The general fibre of this morphism is
    isomorphic to $X$. We have a short exact sequence
    \begin{equation*}
        1\to \pi_1^{\et}(X,\bar{x})\times\pi_1^{\et}(C,\bar{c}) \to
        \pi_1^{\et}((X\times C)/G,\bar{z}) \to G\to1.
    \end{equation*}
    Hence we have constructed an example of a smooth projective variety over an
    algebraically closed field of characteristic $p$ whose fundamental group is infinite
    yet whose tower of MRC quotients terminates with a point.




%\bibliography{bibliography}
%\bibliographystyle{alpha}
%\nocite{*}

\begin{thebibliography}{alpha}

\bibitem[ACG11]{acgh}
Enrico Arbarello, Maurizio Cornalba, and Pillip~A. Griffiths.
\newblock {\em Geometry of algebraic curves. {V}olume {II}}, volume 268 of {\em
  Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of
  Mathematical Sciences]}.
\newblock Springer, Heidelberg, 2011.
\newblock With a contribution by Joseph Daniel Harris.

\bibitem[AK03]{ak}
Carolina Araujo and J{\'a}nos Koll{\'a}r.
\newblock Rational curves on varieties.
\newblock In {\em Higher dimensional varieties and rational points ({B}udapest,
  2001)}, volume~12 of {\em Bolyai Soc. Math. Stud.}, pages 13--68. Springer,
  Berlin, 2003.

\bibitem[BDPP04]{bdpp04} S\'ebastien Boucksom, Jean-Pierre Demailly, Mihai Paun, Thomas
Peternell, {The pseudo-effective cone of a compact Kahler manifold and varieties of
negative Kodaira dimension}, {http://arxiv.org/abs/math/0405285}, 2004.

\bibitem[BM01]{bm} Feodor A. Bogomolov and Michael L. McQuillan, {Rational curves on
foliated varieties}. IHES, Preprint, 2001.

\bibitem[Cam92]{campana92}
F.~Campana.
\newblock Connexit\'e rationnelle des vari\'et\'es de {F}ano.
\newblock {\em Ann. Sci. \'Ecole Norm. Sup. (4)}, 25(5):539--545, 1992.

\bibitem[CL03]{cl03}
Antoine Chambert-Loir.
\newblock A propos du groupe fondamental des vari\'et\'es rationnellement
  connexes.
\newblock 2003.

\bibitem[Deb01]{debarre}
Olivier Debarre.
\newblock {\em Higher-dimensional algebraic geometry}.
\newblock Universitext. Springer-Verlag, New York, 2001.

\bibitem[dJS03]{djs}
A.~J. de~Jong and J.~Starr.
\newblock Every rationally connected variety over the function field of a curve
  has a rational point.
\newblock {\em Amer. J. Math.}, 125(3):567--580, 2003.

\bibitem[Fri98]{friedman}
Robert Friedman.
\newblock {\em Algebraic surfaces and holomorphic vector bundles}.
\newblock Universitext. Springer-Verlag, New York, 1998.

\bibitem[Gro57]{grothendieck}
A.~Grothendieck.
\newblock Sur la classification des fibr\'es holomorphes sur la sph\`ere de
  {R}iemann.
\newblock {\em Amer. J. Math.}, 79:121--138, 1957.

\bibitem[EGA]{ega} A. Grothendieck, {\em \'Elements de G\'eom\'etrie Alg\'ebrique}, Math.
Publ. IHES {\bf 4, 8, 11, 17, 20, 24, 28, 32}, 1960-7.

\bibitem[SGA1]{sga1}
Alexander Grothendieck.
\newblock {\em Rev\^etements \'etales et groupe fondamental ({SGA} 1)}.
\newblock Documents Math\'ematiques (Paris) [Mathematical Documents (Paris)],
  3. Soci\'et\'e Math\'ematique de France, Paris, 2003.
\newblock S{\'e}minaire de g{\'e}om{\'e}trie alg{\'e}brique du Bois Marie
  1960--61. [Algebraic Geometry Seminar of Bois Marie 1960-61], Directed by A.
  Grothendieck, With two papers by M. Raynaud, Updated and annotated reprint of
  the 1971 original [Lecture Notes in Math., 224, Springer, Berlin; MR0354651
  (50 \#7129)].

\bibitem[GHS03]{ghs}
Tom Graber, Joe Harris, and Jason Starr.
\newblock Families of rationally connected varieties.
\newblock {\em J. Amer. Math. Soc.}, 16(1):57--67 (electronic), 2003.

\bibitem[Har66]{hartample}
Robin Hartshorne.
\newblock Ample vector bundles.
\newblock {\em Inst. Hautes \'Etudes Sci. Publ. Math.}, (29):63--94, 1966.

\bibitem[Har70]{hartamplesubvar}
Robin Hartshorne.
\newblock {\em Ample subvarieties of algebraic varieties}.
\newblock Notes written in collaboration with C. Musili. Lecture Notes in
  Mathematics, Vol. 156. Springer-Verlag, Berlin, 1970.

\bibitem[Har71]{hartcurveample}
Robin Hartshorne.
\newblock Ample vector bundles on curves.
\newblock {\em Nagoya Math. J.}, 43:73--89, 1971.

\bibitem[Har77]{hartshorne}
Robin Hartshorne.
\newblock {\em Algebraic geometry}.
\newblock Springer-Verlag, New York, 1977.
\newblock Graduate Texts in Mathematics, No. 52.

\bibitem[Igu55]{igusa}
Jun-ichi Igusa.
\newblock A fundamental inequality in the theory of {P}icard varieties.
\newblock {\em Proc. Nat. Acad. Sci. U.S.A.}, 41:317--320, 1955.

\bibitem[Kol91]{kollar91}
J{\'a}nos Koll{\'a}r.
\newblock Extremal rays on smooth threefolds.
\newblock {\em Ann. Sci. \'Ecole Norm. Sup. (4)}, 24(3):339--361, 1991.

\bibitem[Kol96]{kollar}
J{\'a}nos Koll{\'a}r.
\newblock {\em Rational curves on algebraic varieties}, volume~32 of {\em
  Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of
  Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd
  Series. A Series of Modern Surveys in Mathematics]}.
\newblock Springer-Verlag, Berlin, 1996.

\bibitem[KM98]{kollarmori}
J{\'a}nos Koll{\'a}r and Shigefumi Mori.
\newblock {\em Birational geometry of algebraic varieties}, volume 134 of {\em
  Cambridge Tracts in Mathematics}.
\newblock Cambridge University Press, Cambridge, 1998.
\newblock With the collaboration of C. H. Clemens and A. Corti, Translated from
  the 1998 Japanese original.

\bibitem[KMM92a]{kmm92a}
J{\'a}nos Koll{\'a}r, Yoichi Miyaoka, and Shigefumi Mori.
\newblock Rational connectedness and boundedness of {F}ano manifolds.
\newblock {\em J. Differential Geom.}, 36(3):765--779, 1992.

\bibitem[KMM92c]{kmm92c}
J{\'a}nos Koll{\'a}r, Yoichi Miyaoka, and Shigefumi Mori.
\newblock Rationally connected varieties.
\newblock {\em J. Algebraic Geom.}, 1(3):429--448, 1992.

\bibitem[KSC06]{ksc}
Stefan Kebekus and Luis Sol{\'a}~Conde.
\newblock Existence of rational curves on algebraic varieties, minimal rational
  tangents, and applications.
\newblock In {\em Global aspects of complex geometry}, pages 359--416.
  Springer, Berlin, 2006.

\bibitem[KSCT07]{ksct}
Stefan Kebekus, Luis Sol{\'a}~Conde, and Matei Toma.
\newblock Rationally connected foliations after {B}ogomolov and {M}c{Q}uillan.
\newblock {\em J. Algebraic Geom.}, 16(1):65--81, 2007.

\bibitem[Laz04]{lazarsfeld2}
Robert Lazarsfeld.
\newblock {\em Positivity in algebraic geometry. {II}}, volume~49 of {\em
  Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of
  Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd
  Series. A Series of Modern Surveys in Mathematics]}.
\newblock Springer-Verlag, Berlin, 2004.
\newblock Positivity for vector bundles, and multiplier ideals.

\bibitem[Mor79]{mori}
Shigefumi Mori.
\newblock Projective manifolds with ample tangent bundles.
\newblock {\em Ann. of Math. (2)}, 110(3):593--606, 1979.

\bibitem[MFK94]{git}
D.~Mumford, J.~Fogarty, and F.~Kirwan.
\newblock {\em Geometric invariant theory}, volume~34 of {\em Ergebnisse der
  Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related
  Areas (2)]}.
\newblock Springer-Verlag, Berlin, third edition, 1994.

\bibitem[Mum70]{mumford}
David Mumford.
\newblock {\em Abelian varieties}.
\newblock Tata Institute of Fundamental Research Studies in Mathematics, No. 5.
  Published for the Tata Institute of Fundamental Research, Bombay, 1970.

\bibitem[Occ06]{occhetta}
Gianluca Occhetta.
\newblock Extending rationally connected fibrations.
\newblock {\em Forum Math.}, 18(5):853--867, 2006.

\bibitem[Ott12]{ottem}
John~Christian Ottem.
\newblock Ample subvarieties and {$q$}-ample divisors.
\newblock {\em Adv. Math.}, 229(5):2868--2887, 2012.

\bibitem[Pet06]{peternell06}
Thomas Peternell.
\newblock Kodaira dimension of subvarieties. {II}.
\newblock {\em Internat. J. Math.}, 17(5):619--631, 2006.

\bibitem[Poo10]{poonen}
Bjorn Poonen.
\newblock Insufficiency of the {B}rauer-{M}anin obstruction applied to \'etale
  covers.
\newblock {\em Ann. of Math. (2)}, 171(3):2157--2169, 2010.

\bibitem[She10]{shen}
Mingmin Shen.
\newblock Foliations and rational connectedness in positive characteristic.
\newblock {\em J. Algebraic Geom.}, 19(3):531--553, 2010.

\bibitem[Shi74]{shioda}
Tetsuji Shioda.
\newblock An example of unirational surfaces in characteristic {$p$}.
\newblock {\em Math. Ann.}, 211:233--236, 1974.

\bibitem[Sta06]{starr06} J. Starr, {The maximal free rational quotient}, preprint,
http://arxiv.org/abs/math/0602640, 2006.

\bibitem[Uen75]{ueno75}
Kenji Ueno.
\newblock {\em Classification theory of algebraic varieties and compact complex
  spaces}.
\newblock Lecture Notes in Mathematics, Vol. 439. Springer-Verlag, Berlin,
  1975.
\newblock Notes written in collaboration with P. Cherenack.

\end{thebibliography}

\end{document}

1208.4055.extracted_bib
\begin{thebibliography}{alpha}

\bibitem[ACG11]{acgh}
Enrico Arbarello, Maurizio Cornalba, and Pillip~A. Griffiths.
\newblock {\em Geometry of algebraic curves. {V}olume {II}}, volume 268 of {\em
  Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of
  Mathematical Sciences]}.
\newblock Springer, Heidelberg, 2011.
\newblock With a contribution by Joseph Daniel Harris.

\bibitem[AK03]{ak}
Carolina Araujo and J{\'a}nos Koll{\'a}r.
\newblock Rational curves on varieties.
\newblock In {\em Higher dimensional varieties and rational points ({B}udapest,
  2001)}, volume~12 of {\em Bolyai Soc. Math. Stud.}, pages 13--68. Springer,
  Berlin, 2003.

\bibitem[BDPP04]{bdpp04} S\'ebastien Boucksom, Jean-Pierre Demailly, Mihai Paun, Thomas
Peternell, {The pseudo-effective cone of a compact Kahler manifold and varieties of
negative Kodaira dimension}, {http://arxiv.org/abs/math/0405285}, 2004.

\bibitem[BM01]{bm} Feodor A. Bogomolov and Michael L. McQuillan, {Rational curves on
foliated varieties}. IHES, Preprint, 2001.

\bibitem[Cam92]{campana92}
F.~Campana.
\newblock Connexit\'e rationnelle des vari\'et\'es de {F}ano.
\newblock {\em Ann. Sci. \'Ecole Norm. Sup. (4)}, 25(5):539--545, 1992.

\bibitem[CL03]{cl03}
Antoine Chambert-Loir.
\newblock A propos du groupe fondamental des vari\'et\'es rationnellement
  connexes.
\newblock 2003.

\bibitem[Deb01]{debarre}
Olivier Debarre.
\newblock {\em Higher-dimensional algebraic geometry}.
\newblock Universitext. Springer-Verlag, New York, 2001.

\bibitem[dJS03]{djs}
A.~J. de~Jong and J.~Starr.
\newblock Every rationally connected variety over the function field of a curve
  has a rational point.
\newblock {\em Amer. J. Math.}, 125(3):567--580, 2003.

\bibitem[Fri98]{friedman}
Robert Friedman.
\newblock {\em Algebraic surfaces and holomorphic vector bundles}.
\newblock Universitext. Springer-Verlag, New York, 1998.

\bibitem[Gro57]{grothendieck}
A.~Grothendieck.
\newblock Sur la classification des fibr\'es holomorphes sur la sph\`ere de
  {R}iemann.
\newblock {\em Amer. J. Math.}, 79:121--138, 1957.

\bibitem[EGA]{ega} A. Grothendieck, {\em \'Elements de G\'eom\'etrie Alg\'ebrique}, Math.
Publ. IHES {\bf 4, 8, 11, 17, 20, 24, 28, 32}, 1960-7.

\bibitem[SGA1]{sga1}
Alexander Grothendieck.
\newblock {\em Rev\^etements \'etales et groupe fondamental ({SGA} 1)}.
\newblock Documents Math\'ematiques (Paris) [Mathematical Documents (Paris)],
  3. Soci\'et\'e Math\'ematique de France, Paris, 2003.
\newblock S{\'e}minaire de g{\'e}om{\'e}trie alg{\'e}brique du Bois Marie
  1960--61. [Algebraic Geometry Seminar of Bois Marie 1960-61], Directed by A.
  Grothendieck, With two papers by M. Raynaud, Updated and annotated reprint of
  the 1971 original [Lecture Notes in Math., 224, Springer, Berlin; MR0354651
  (50 \#7129)].

\bibitem[GHS03]{ghs}
Tom Graber, Joe Harris, and Jason Starr.
\newblock Families of rationally connected varieties.
\newblock {\em J. Amer. Math. Soc.}, 16(1):57--67 (electronic), 2003.

\bibitem[Har66]{hartample}
Robin Hartshorne.
\newblock Ample vector bundles.
\newblock {\em Inst. Hautes \'Etudes Sci. Publ. Math.}, (29):63--94, 1966.

\bibitem[Har70]{hartamplesubvar}
Robin Hartshorne.
\newblock {\em Ample subvarieties of algebraic varieties}.
\newblock Notes written in collaboration with C. Musili. Lecture Notes in
  Mathematics, Vol. 156. Springer-Verlag, Berlin, 1970.

\bibitem[Har71]{hartcurveample}
Robin Hartshorne.
\newblock Ample vector bundles on curves.
\newblock {\em Nagoya Math. J.}, 43:73--89, 1971.

\bibitem[Har77]{hartshorne}
Robin Hartshorne.
\newblock {\em Algebraic geometry}.
\newblock Springer-Verlag, New York, 1977.
\newblock Graduate Texts in Mathematics, No. 52.

\bibitem[Igu55]{igusa}
Jun-ichi Igusa.
\newblock A fundamental inequality in the theory of {P}icard varieties.
\newblock {\em Proc. Nat. Acad. Sci. U.S.A.}, 41:317--320, 1955.

\bibitem[Kol91]{kollar91}
J{\'a}nos Koll{\'a}r.
\newblock Extremal rays on smooth threefolds.
\newblock {\em Ann. Sci. \'Ecole Norm. Sup. (4)}, 24(3):339--361, 1991.

\bibitem[Kol96]{kollar}
J{\'a}nos Koll{\'a}r.
\newblock {\em Rational curves on algebraic varieties}, volume~32 of {\em
  Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of
  Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd
  Series. A Series of Modern Surveys in Mathematics]}.
\newblock Springer-Verlag, Berlin, 1996.

\bibitem[KM98]{kollarmori}
J{\'a}nos Koll{\'a}r and Shigefumi Mori.
\newblock {\em Birational geometry of algebraic varieties}, volume 134 of {\em
  Cambridge Tracts in Mathematics}.
\newblock Cambridge University Press, Cambridge, 1998.
\newblock With the collaboration of C. H. Clemens and A. Corti, Translated from
  the 1998 Japanese original.

\bibitem[KMM92a]{kmm92a}
J{\'a}nos Koll{\'a}r, Yoichi Miyaoka, and Shigefumi Mori.
\newblock Rational connectedness and boundedness of {F}ano manifolds.
\newblock {\em J. Differential Geom.}, 36(3):765--779, 1992.

\bibitem[KMM92c]{kmm92c}
J{\'a}nos Koll{\'a}r, Yoichi Miyaoka, and Shigefumi Mori.
\newblock Rationally connected varieties.
\newblock {\em J. Algebraic Geom.}, 1(3):429--448, 1992.

\bibitem[KSC06]{ksc}
Stefan Kebekus and Luis Sol{\'a}~Conde.
\newblock Existence of rational curves on algebraic varieties, minimal rational
  tangents, and applications.
\newblock In {\em Global aspects of complex geometry}, pages 359--416.
  Springer, Berlin, 2006.

\bibitem[KSCT07]{ksct}
Stefan Kebekus, Luis Sol{\'a}~Conde, and Matei Toma.
\newblock Rationally connected foliations after {B}ogomolov and {M}c{Q}uillan.
\newblock {\em J. Algebraic Geom.}, 16(1):65--81, 2007.

\bibitem[Laz04]{lazarsfeld2}
Robert Lazarsfeld.
\newblock {\em Positivity in algebraic geometry. {II}}, volume~49 of {\em
  Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of
  Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd
  Series. A Series of Modern Surveys in Mathematics]}.
\newblock Springer-Verlag, Berlin, 2004.
\newblock Positivity for vector bundles, and multiplier ideals.

\bibitem[Mor79]{mori}
Shigefumi Mori.
\newblock Projective manifolds with ample tangent bundles.
\newblock {\em Ann. of Math. (2)}, 110(3):593--606, 1979.

\bibitem[MFK94]{git}
D.~Mumford, J.~Fogarty, and F.~Kirwan.
\newblock {\em Geometric invariant theory}, volume~34 of {\em Ergebnisse der
  Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related
  Areas (2)]}.
\newblock Springer-Verlag, Berlin, third edition, 1994.

\bibitem[Mum70]{mumford}
David Mumford.
\newblock {\em Abelian varieties}.
\newblock Tata Institute of Fundamental Research Studies in Mathematics, No. 5.
  Published for the Tata Institute of Fundamental Research, Bombay, 1970.

\bibitem[Occ06]{occhetta}
Gianluca Occhetta.
\newblock Extending rationally connected fibrations.
\newblock {\em Forum Math.}, 18(5):853--867, 2006.

\bibitem[Ott12]{ottem}
John~Christian Ottem.
\newblock Ample subvarieties and {$q$}-ample divisors.
\newblock {\em Adv. Math.}, 229(5):2868--2887, 2012.

\bibitem[Pet06]{peternell06}
Thomas Peternell.
\newblock Kodaira dimension of subvarieties. {II}.
\newblock {\em Internat. J. Math.}, 17(5):619--631, 2006.

\bibitem[Poo10]{poonen}
Bjorn Poonen.
\newblock Insufficiency of the {B}rauer-{M}anin obstruction applied to \'etale
  covers.
\newblock {\em Ann. of Math. (2)}, 171(3):2157--2169, 2010.

\bibitem[She10]{shen}
Mingmin Shen.
\newblock Foliations and rational connectedness in positive characteristic.
\newblock {\em J. Algebraic Geom.}, 19(3):531--553, 2010.

\bibitem[Shi74]{shioda}
Tetsuji Shioda.
\newblock An example of unirational surfaces in characteristic {$p$}.
\newblock {\em Math. Ann.}, 211:233--236, 1974.

\bibitem[Sta06]{starr06} J. Starr, {The maximal free rational quotient}, preprint,
http://arxiv.org/abs/math/0602640, 2006.

\bibitem[Uen75]{ueno75}
Kenji Ueno.
\newblock {\em Classification theory of algebraic varieties and compact complex
  spaces}.
\newblock Lecture Notes in Mathematics, Vol. 439. Springer-Verlag, Berlin,
  1975.
\newblock Notes written in collaboration with P. Cherenack.

\end{thebibliography}