On the syzygies and Hodge theory of nodal hypersurfaces
Alexandru Dimca$^1$
We give sharp lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining an even dimensional nodal hypersurface. This implies the validity of formulas due to M. Saito, L. Wotzlaw and the author for the graded pieces with respect to the Hodge filtration of the top cohomology of the hypersurface complement in many new cases.
1310.5344.tex
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\begin{document}
%\date{June 4, 2009}

\title [On the syzygies and Hodge theory of  nodal hypersurfaces]
{On the syzygies and Hodge theory of nodal hypersurfaces }




\author[Alexandru Dimca]{Alexandru Dimca$^1$}
\address{Univ. Nice Sophia Antipolis, CNRS,  LJAD, UMR 7351, 06100 Nice, France.}
\email{dimca@unice.fr}

\thanks{$^1$ Partially supported by Institut Universitaire de France.} 


\subjclass[2000]{Primary  14C30, 13D40; Secondary  32S35, 13D02}

\keywords{nodal hypersurfaces, syzygies, mixed Hodge structure, pole order filtration}

\begin{abstract} We give sharp lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining an even dimensional nodal hypersurface. 
This implies the validity of formulas due to M. Saito, L. Wotzlaw and the author for the graded pieces with respect to the Hodge filtration of the top cohomology of the hypersurface complement in many new cases.


\end{abstract}

\maketitle

%\tableofcontents

\section{Introduction} \label{sec:intro}


Let $S=\C[x_0,...,x_n]$ be the graded ring of polynomials in $x_0,,...,x_n$ with complex coefficients and denote by $S_r$ the vector space of homogeneous polynomials in $S$ of degree $r$. 
For any polynomial $f \in S_r$, we define the {\it Jacobian ideal} $J_f \subset S$ as the ideal spanned by the partial derivatives $f_0,...,f_n$ of $f$ with respect to $x_0,...,x_n$ 
and the corresponding graded {\it Milnor} (or {\it Jacobian}) {\it algebra} by
\begin{equation} 
\label{eq1}
M(f)=S/J_f.
\end{equation}

The Milnor algebra $M(f)$ can be seen (up to a twist in grading) as the top cohomology $H^{n+1}(K^*(f))$,
 where $K^*(f)$ is the Koszul complex of $f_0,...,f_n$ with the natural grading $|x_j|=|dx_j|=1$ defined by 
\begin{equation} 
\label{Koszul}
K^*(f) : \  \   \    \   \   \     0 \to \Omega^0 \to \Omega^1 \to ...   \to \Omega^{n+1}\to 0
\end{equation}
with all the arrows  given by the wedge product by $df=f_0dx_0+f_1dx_1+...+f_ndx_n$.


The homogeneous components of the next cohomology group, say $H^n(K^*(f))_{n+r}$, describe the syzygies
$$\sum_{j=0,n}a_jf_j=0$$
where $a_j \in S_r$, modulo the trivial syzygies generated by 
$$(f_j)f_i+(-f_i)f_j=0.$$

The following result was proved in \cite{DSt3}.

\begin{thm}
\label{thmA} Let $D:f=0$ be a nodal hypersurface of degree $d>2$ in $\PP^n$.

\smallskip

\noindent (i) If $n=2n_1+1$ is odd, then $H^n(K^*(f))_{m}=0$ for any $m \leq n_1d$.

\smallskip

\noindent (ii) If $n=2n_1$ is even, then $H^n(K^*(f))_{m}=0$ for any $m \leq n_1d-1$.


\end{thm}

Examples involving Chebyshev hypersurfaces shows that the bounds are best possible for $n$ even,
but not for $n$ odd, see \cite{DSt3}. The first purpose of this paper is to establish the following new bound in the case  $n$ odd, which is optimal, see Example \ref{ex1}.

\begin{thm}
\label{thmB} Let $D:f=0$ be a nodal hypersurface of degree $d>2$ in $\PP^n$.

If $n=2n_1+1$ is odd, then $H^n(K^*(f))_{m}=0$ for any $m \leq (n_1+1)d-\left[ \frac{d}{2} \right]-1$.

\end{thm}

The idea of proof of Theorem \ref{thmB} is similar to that used in the proof of 
Theorem \ref{thmA}, namely the interplay between Hodge filtrations, pole order filtrations
and some spectral sequences. In the case of Theorem \ref{thmA} it was enough to look at the cohomology of the complement $U=\PP^n \setminus D$, while in the case at hand we have to consider the eigenspaces $H^*(F)_{\lambda}$ of the monodromy action on the corresponding Milnor fiber $F: f(x)-1=0$, which is technically more complicated.

The second aim of this paper is to show that the information we have obtained on the syzygies
allows us to show that an algebraic description of some of the graded pieces $Gr_F^pH^n(U,\C)$
given in \cite{DSW} holds in many cases, see Theorem \ref{thmC}. For basic facts on mixed Hodge structures we refer \cite{PS}.
 








\section{Spectral sequences for Milnor fibers of homogeneous polynomials} \label{sec2}

For $k=1,...,d$, set $\lambda =\exp(-2 \pi ik/d)$ and let $L_k$ be the rank one local system on $U$ such that
\begin{equation} 
\label{eq2}
H^*(U,L_k)=H^*(F)_{\lambda},
\end{equation}
see for details \cite{D2}, p. 211 and note that here the eigenspaces are with respect to the local system monodromy $T$  acting on $H^*(F)$, \cite{DS3} . Let $j:U \to X=\PP^n$ be the inclusion, and let $\LL_k$ be the meromorphic extension of $L_k \otimes_{\C}\OO_U$. Then $\LL_k$ is a regular holonomic $\DD_Y$-module,
see \cite{Sa2}, section (4.8), and one clearly has
\begin{equation} 
\label{eq3}
\HH^*(X,DR(\LL_k))=\HH^*(X,Rj_*L_k)=H^*(F)_{\lambda}.
\end{equation}
The $\DD_Y$-module $\LL_k$ has a natural (increasing) pole order filtration $P_*$ such that 
$P_j\LL_k$ is isomorphic to $\OO_X((jd+k)$ for $j \in \N$ and $P_j\LL_k=0$ for $j<0$, see
 (3.1.3) in \cite{DS4}. Then the de Rham complex $DR(\LL_k)$ has an induced decreasing filtration
$P^*$ and this induces the {\it pole order filtration} $P^*$ on the eigenspaces $H^*(F)_{\lambda}$
of the Milnor fiber cohomology. One has the following fundamental inclusion
\begin{equation} 
\label{HP1}
F^sH^j(F)_{\lambda} \subset P^sH^j(F)_{\lambda},
\end{equation}
for any $s$ and any $j$, where $F^s$ denotes the canonical Hodge filtration on the cohomology of the smooth quasi-projective variety $F$, see (3.1.3), (4.4.7) and (4.4.8) in \cite{DS4}.  This extends the similar result for $H^*(U)$, see \cite{DD} and \cite{Sa1}.

On the other hand, for each $k=1,...,d$ there is a spectral sequence
\begin{equation} 
\label{sp2}
E^{p,q}_{1}(f,k)=H^{p+q}(K^*(f))_{qd+k} \Rightarrow H^{p+q-1}(F)_{\lambda}
\end{equation}
coming from the graded Gauss-Manin complex $C^*_f$ associated with $f$, see (4.4.4) and (4.5.3) in  \cite{DS4}. A similar spectral sequence is obtained by using the algebraic microlocal Gauss-Manin complex $\tilde C^*_f$ and the corresponding limit is the reduced cohomology of the Milnor fiber, see the   (4.5) in \cite{DS4}. In fact, one has
\begin{equation} 
\label{sp3}
H^{j+1}( C^*_f)_{k}=H^{j+1}( \tilde C^*_f)_{k}= H^{j}(F)_{\lambda}
\end{equation}
for any $k=1,...,d$ and $j>0$, see (4.2.3)  in \cite{DS4}.
These spectral sequences induce a filtration $P'$ on $H^*(F)_{\lambda}$
and one has $P^s=P'^{s+1}$, see (4.4.7) in  \cite{DS4}.


\begin{rk}
\label{rkMB}
Remark 4.11 in \cite{Sa2} gives a very explicit description of the pole order filtration on the top cohomology group $H^n(F)$. Assume that we have a (finite) family of monomials $(g_j(x))_{j \in J}$ in $S$ such that the cohomology classes $[\omega_j]$ of the differential forms $\omega_j=g_j(x)\Delta(dx_0 \wedge ...\wedge dx_n)$ for $j \in J$ yield a basis of the $\C$-vector space $H^n(F)$. Then $Gr_P^{n-q}H^n(F)_{\lambda}$ is spanned by the classes $[\omega_j]$    with $\deg \omega_j = qd+k.$ Note that the eigenspace $Gr_P^{n-q}H^n(F)_1$ is spanned exactly by the classes $[\omega_j]$ in $Gr_P^{n-q}H^n(F)$ having a maximal degree $\deg \omega_j =(q+1)d$. In particular, the pole order filtrations on $H^n(F)_1$ and on $H^n(U)$, the latter constructed using a generalization of Griffiths approach in  \cite{D1}, Chapter 6 or in  \cite{DSt2}
are the same, as it should be since we have a natural isomorphism $H^n(F)_1=H^n(U)$.
Note that the corresponding basis for $Gr_P^{n-q}H^n(U)$ is usually written as
$$\sigma_j=\omega_j/f^{q+1}.$$
Similarly, the limit term $E^{n-q+1,q}_{\infty}(f,k)$, which is isomorphic to $Gr_P^{n-q}H^n(F)_{\lambda}$, has a basis given with the above notation by
$$\eta_j=g_j(x)\cdot dx_0 \wedge ...\wedge dx_n.$$

\end{rk}


\section{Hodge filtration versus pole order filtration on Milnor fibers} \label{sec3}


If $D$ is a nodal hypersurface in $\PP^n$, then one has an equality
\begin{equation} 
\label{HP2}
F^sH^n(U)= P^sH^n(U),
\end{equation}
for any $s \geq n-m+1$, with $m=\alpha_D=\frac{n}{2}$, see Corollary (0.12) in M. Saito \cite{Sa1}, or  the formula (1.1.3) in \cite{DSW}. The purpose of this section is to prove the following
similar result for the associated Milnor fibers.

\begin{prop}
\label{propA}
Let $D:f=0$ be a nodal hypersurface of degree $d>2$ in $\PP^n$ with $n$ odd. Then 
$$F^sH^n(F)_{\lambda}= P^sH^n(F)_{\lambda},$$
for any $s \geq n-m+1$ and any $\lambda \in \mu_d$, $\lambda \ne 1$, with $m=\frac{n}{2}$.
\end{prop}

\proof
Consider the hypersurface $\tilde D$ defined in $\PP^{n+1}$ by the equation $\tilde f(x,t)=0$,
with $\tilde f(x,t)=f(x)-t^d$. Let $\tilde U=\PP^{n+1} \setminus \tilde D$ and let $H:t=0$ be the hyperplane at infinity in $\PP^{n+1}$ such that $\tilde U \cap H=U$. Consider the Gysin long exact sequence
\begin{equation} 
\label{G1}
...\to H^{n-1}(U) \to H^{n+1}(\tilde U) \to H^{n+1}(\tilde U \setminus U) \to H^n(U) \to...
\end{equation}
The group $\mu_d$ of $d$-roots of unity acts on $\PP^{n+1}$ via
$$ \be \cdot (x_0:...:x_n:t)=(x_0:...:x_n:\be t).$$
This action extends the  action of $\mu_d$ on $\C^{n+1}=\PP^{n+1} \setminus H$, which is used to define the local system monodromy  $T:F \to F$, namely
$$ \be \cdot (x_0,...,x_n)=(\be^{-1} x_0,...,\be^{-1}  x_n).$$

It follows that the Gysin exact sequence \eqref{G1} inherits a $\mu_d$-action, such that for any
$\lambda \in \mu_d$, $\lambda \ne 1$, one has the following isomorphism of eigenspaces
\begin{equation} 
\label{G2}
i^*: H^{n+1}(\tilde U)_{\lambda} \to H^{n+1}(\tilde U \setminus U)_{\lambda}.
\end{equation}
On the other hand, one has $\tilde U \setminus U=\C^{n+1} \setminus F$, and hence a new Gysin sequence
shows that one has an isomorphism
\begin{equation} 
\label{G3}
R:H^{n+1}(\tilde U \setminus U)_{\lambda} \to  H^n(F)_{\lambda},
\end{equation}
induced by a residue morphism $R$ which has Hodge type $(-1,-1)$. By composing the above two isomorphisms, we get isomorphisms
\begin{equation} 
\label{G4}
Ri^*: F^{s+1}H^{n+1}(\tilde U)_{\lambda} \to F^sH^n(F)_{\lambda},
\end{equation}
for any integer $s$ and any $\lambda \in \mu_d$, $\lambda \ne 1$.

Now we look at the corresponding $P^*$ filtrations and  show  that
\begin{equation} 
\label{G5}
\dim P^{s+1}H^{n+1}(\tilde U)_{\lambda} =\dim P^sH^n(F)_{\lambda},
\end{equation}
for any integer $s$ and any $\lambda \in \mu_d$, $\lambda \ne 1$.

The cohomology of the filtered algebraic microlocal Gauss-Manin complexes $\tilde C^*_f$ and $\tilde C^*_{\tilde f}$ are closely related, namely
$$H^{\ell+1}( \tilde C^*_{\tilde f},P'  )=H^{\ell}(  \tilde C^*_f ,P'  )\otimes H^1(\tilde C^*_{t^d},P')$$
see (4.9) in \cite{DS4}. Looking at the homogeneous components corresponding to $k=d$ and taking the $\lambda$-eigenspaces yields the equality \eqref{G5} in view of \eqref{sp3}.





Finally, any singularity of the hypersurface $\tilde D$ has type $A_{d-1}$, and hence the corresponding $\alpha_{\tilde D}$ is exactly $\tilde m=\frac{n}{2}+\frac{1}{d}$.
Using Corollary (0.12) in M. Saito \cite{Sa1}, or the formula (1.1.3) in \cite{DSW}, we see that
\begin{equation} 
\label{HP3}
F^sH^{n+1}(\tilde U)= P^sH^{n+1}(\tilde U),
\end{equation}
for any $s \geq n-\tilde m+2$. Using the formulas \eqref{G4} and \eqref{G5} and the inclusion \eqref{HP1} we complete the proof, as one may clearly replace $\tilde m$ by $m$ as soon as $d>2$ and $n$ is odd.
 
\endproof



\section{Proof of  Theorem \ref{thmB} and some examples} \label{sec4}

Assume from now on that $n=2n_1+1$. Then the equality of filtrations in Proposition \ref{propA}
holds for $s \geq n_1+2$. We show now that this equality may be extended as follows.

\begin{lem}
\label{L}
Let $D:f=0$ be a nodal hypersurface of degree $d>2$ in $\PP^{2n_1+1}$ and set $s=n_1+1$. Then 
$$F^sH^n(F)_{\lambda}= P^sH^n(F)_{\lambda},$$
for
$\lambda =\exp(-2 \pi ik/d)$ with 
$$0<k\leq k_0=d-\left[ \frac{d}{2} \right]-1.$$
More precisely, in these conditions one has
 $\dim Gr_F^sH^n(F)_{\lambda}= \dim Gr_P^sH^n(F)_{\lambda}= \dim M(f)_{n_1d+k-n-1}=\dim M(h)_{n_1d+k-n-1},$
 where $h$ is a homogeneous polynomial in $S$ of degree $d$ such that the hypersurface $D_h:h=0$
 is smooth.

\end{lem}

\proof In view of the inclusions in \eqref{HP1} and using Proposition \ref{propA}, it is enough to establish the inequality
$$\dim Gr_F^sH^n(F)_{\lambda} \geq \dim Gr_P^sH^n(F)_{\lambda}.$$
We know that
$$\dim Gr_P^sH^n(F)_{\lambda}=\dim E_{\infty}^{n_1+2,n_1}(f,k) \leq \dim E_1^{n_1+2,n_1}(f,k)=\dim H^{n+1}(K^*(f))_{n_1d+k}=$$
$$=\dim M(f)_{n_1d+k-n-1}= \dim M(h)_{n_1d+k-n-1}$$
where $ h \in S_d$ denotes a polynomial such that the associated hypersurface $D_h:h=0$ is smooth.
Indeed, the last equality follows from Corollary 2.2. $(i)$ in \cite{DSt3}, which is a direct consequence of Theorem \ref{thmA}.

On the other hand, using an argument similar to Proposition 4.1 in \cite{D3}, which goes back to Lemma 3.6. in \cite{Sa2}, it follows that for $\be \ne \pm 1$, the mixed Hodge structure induced on 
\begin{equation} 
\label{mhs1}
H^n(F)_{\be, \overline \be}=H^n(F)_{\be} \oplus H^n(F)_{ \overline \be}
\end{equation}
is pure of weight $n$. Therefore
$$\dim Gr_F^sH^n(F)_{\lambda}=h^{n_1+1,n_1}(H^n(F)_{\lambda})=h^{n_1+1,n_1}(H^n(\tilde D)_{\lambda})$$
in view of Corollary 1.2 in \cite{DL}. To compute the last equivariant Hodge number we use Proposition 5.2 in  \cite{DL}. The first term in the sum giving $h^{n_1+1,n_1}(H^n(\tilde D)_{\lambda})$ is the
corresponding number computed for the smooth hypersurface $\tilde D_h: \tilde h=h(x)-t^d=0$.
Using the standard identification going back to Griffiths \cite{Gr},
$$H^{n_1+1,n_1}(H^n(\tilde D_h))=H^{n+2}(K^*(\tilde h))_{(n_1+1)d},$$
and recalling that taking the $\lambda$-eigenspace means to look at forms of the form
$$g(x)t^{d-k-1}dx_0\wedge...\wedge dx_n\wedge dt,$$
where $g(x)$ is homogeneous of degree $n_1d+k-n-1$, we get
$$h^{n_1+1,n_1}(H^n(\tilde D_h)_{\lambda})= \dim M(h)_{n_1d+k-n-1}.$$ 
We show now that the other terms in the sum giving $h^{n_1+1,n_1}(H^n(\tilde D)_{\lambda})$ are trivial.
These terms are of two types:

(a) $h^{p,q}(H^{n+1}(\tilde D)_{\lambda})$, which are zero  
since a  $\be$-eigenspace  of the group $ H^{n+1}(\tilde D)$ under the $\mu_d$-action maybe nontrivial only for $\be=\pm 1$, see Example 6.3.24 in \cite{D1}, Theorem 1.1 in \cite{DL} and Theorem 4.1 in \cite{DSt3}. 

(b) $h^{p,q}(H^n(F(d))_{\lambda})$, where $F(d)$ is the affine Milnor fiber given by
$$g(y,t)=y_1^2+...y_n^2+t^d-1=0$$
in $\C^{n+1}$ with the corresponding $\mu_d$-action, i.e. 
$$\be (y_1,...,y_n,t)=(y_1,...,y_n,\be t)$$
replacing the monodromy action when eigenspaces are considered.
Since this is the Milnor fiber of an isolated weighted homogeneous singularity whose link is a rational homology sphere, we have $h^{p,q}(H^n(F(d))_{\lambda})=0$ for $p+q \ne n$. It follows that we have just one
such number to investigate, namely  $h^{n_1+1,n_1}(H^n(F(d))_{\lambda})$. Using the weights $wt(y_j)=2$ and $wt(t)=d$, we get using \cite{St}
$$h^{n_1+1,n_1}(H^n(F(d))=\sum_{j=1,2d-1}\dim M(g(y,t))_{j-d-2}.$$
The $\lambda$-eigenspace should come from the monomial $t^{d-k-1}$, of degree $2d-2k-2$.
Our condition on $k$ implies that $2d-2k-2>d-3$, hence this monomial is not giving a contribution to $h^{n_1+1,n_1}(H^n(F(d))$, i.e.
$h^{n_1+1,n_1}(H^n(F(d))_{\lambda})=0$. Moreover $2k<d$ in order to avoid the case $\lambda=-1$.

This shows that $\dim Gr_F^sH^n(F)_{\lambda}=\dim M(h)_{n_1d+k-n-1}$, completing the proof of Lemma \ref{L}.

\endproof

Now we give the proof of Theorem \ref{thmB}. Lemma \ref{L} implies that
$$E_1^{n_1+2,n_1}(f,k_0)=H^{n+1}(K^*(f))_{n_1d+k_0}=E_{\infty}^{n_1+2,n_1}(f,k_0).$$
Moreover, $E_1^{n_1+2+e,n_1-e}(f,k_0)=E_{\infty}^{n_1+1+e,n_1-e}(f,k_0)$ for all $e=1,2,...,n_1$
by similar (and simpler) computations based on Proposition \ref{propA}. It follows that all the differentials in the spectral sequence $E_r(f,k_0)$ starting from $E_r^{n_1+1,n_1}(f,k_0)$ are $0$.
Since $E_{\infty}^{n_1+1,n_1}(f,k_0)=0$ as well (the only eigenvalues of the monodromy on $H^{n-1}(F)$ are $\pm 1$), we get
$$E_1^{n_1+1,n_1}(f,k_0)=H^n(K^*(f))_{n_1d+k_0}=0$$
which completes the proof of Theorem \ref{thmB} . Indeed, recall that if the coordinates $x_0,...,x_n$ are choosen such that the hyperplane at infinity $H_0:x_0=0$ is transversal to $D$, then the multiplication by $x_0$ induces an injection
$H^n(K^*(f))_{s-1} \to H^n(K^*(f))_{s}$ for any $s$ (the dual statement for the homology is part of Corollary 11
in \cite{CD}).




Before proceeding, we recall the following notions, introduced in \cite{DSt2}.
\begin{definition}
\label{def}


For a hypersurface $D:f=0$ of degree $d$ with isolated singularities we introduce three integers, as follows:



\noindent (i) the {\it coincidence threshold} $ct(D)$ defined as
$$ct(D)=\max \{q~~:~~\dim M(f)_k=\dim M(h)_k \text{ for all } k \leq q\},$$
with $h$  a homogeneous polynomial in $S$ of degree $d=\deg f$ such that $D_h:h=0$ is a smooth hypersurface in $\PP^n$.



\noindent (ii) the {\it minimal degree of a nontrivial syzygy} $mdr(D)$ defined as
$$mdr(D)=\min \{q~~:~~ H^n(K^*(f))_{q+n}\ne 0\}$$
where $K^*(f)$ is the Koszul complex of $f_0,...,f_n$ with the natural grading defined in \cite{DSt2}.


\end{definition}

 Moreover
it is easy to see that one has 
\begin{equation} 
\label{REL}
ct(D)=mdr(D)+d-2.
\end{equation} 
In practice, for a given polynomial $f$, it is easy to compute $ct(D)$ using a number of computer algebra softwares.

\begin{cor}
\label{corA}
Let $D:f=0$ be a nodal hypersurface of degree $d>2$ in $\PP^n$.
If $n=2n_1+1$ is odd, then  
$$ct(D) \geq (n_1+2)d-\left[ \frac{d}{2} \right]-n-2.$$
\end{cor}

\begin{ex}
\label{ex1}
Let $\CC(3,d)$ be the Chebyshev surface of degree $d$ in $\PP^3$  as defined in \cite{DSt3}.
Then for $3 \leq d \leq 20$, numerical computation shows that one has
$$ct(\CC(3,d))=3d-\left[ \frac{d}{2} \right]-5,$$
i.e. we have equalities for these cases in Corollary \ref{corA}. It follows that in any such case the bound for the vanishing in Theorem \ref{thmB} is sharp, namely
$$H^n(K^*(f))_{m+1} \ne 0$$
for  $m =(n_1+1)d-\left[ \frac{d}{2} \right]-1.$
\end{ex}


\begin{rk} \label{sernesi}


Let $D:f=0$ be a degree $d$ hypersurface in $\PP^n$ having only isolated singularities. Let $\hat J$ be the saturation of the Jacobian ideal $J$ of $f$. Then 
the vector space $\hat J_d/J_d$ is naturally identified with the space of first order locally trivial deformations of $D$ in $\PP^n$ modulo those arising from the above $PGL(n+1)$-action, see E. Sernesi \cite{Se}.
 The dimension of the vector space $\hat J_d/J_d$ can be determined as follows.
$$\dim \hat J_d/J_d=\dim M(f)_{d}-\dim M(h)_d+\dim M(f)_{T-d}-\tau(D),$$
see G. Sticlaru \cite{Sti} for this formula and a number of interesting examples of rigid and non-rigid  hypersurfaces. Here $\tau(D)$ is the total Tjurina number of the hypersurface $D$, e.g. the number of nodes for a nodal hypersurface.

\end{rk} 

\section{Hodge theory of nodal hypersurfaces} \label{sec5}

Let $\I \subset \OO_{\PP^n}$ be the reduced ideal sheaf of the set of nodes $\NN=Sing D \subset \PP^n$. Set $I_k{(i)}=H^0(\PP^n,\I^i(k))\subset S_k$ and define $I^{(i)}=\oplus_kI_k{(i)}$,
a homogeneous ideal in the polynomial ring $S$.

For a degree $d$ nodal hypersurface $D$ in $\PP^n$, one of the main results in \cite{DSW} describe the graded pieces $Gr_F^pH^n(U)$ of the top cohomology of the complement $U=\PP^n \setminus D$ with respect to the Hodge filtration $F$ in terms of purely algebraic objects, namely one has
\begin{equation} 
\label{GR}
Gr_F^pH^n(U,\C)=(I^{(q-m+1)}/I^{(q-m)}J_f)_{(q+1)d-n-1}
\end{equation} 
for $q=n-p>m:=\left[ \frac{n}{2} \right]$ under a certain condition (B), see Theorem 2 in \cite{DSW}.

Recall that for a finite set of points $\NN \subset \PP^n$
we denote by 
$$\defect S_m(\NN)=|\NN| - \codim \{h \in S_m~~|~~ h(a)=0 \text{ for any } a \in \NN\},$$ 
the {\it defect (or superabundance) of the linear system of polynomials in $S_m$ vanishing at the points in $\NN$}, see \cite{D1}, p. 207. This positive integer is called the {\it failure of $\NN$ to impose independent conditions on homogeneous polynomials of degree $m$} in \cite{EGH}. 
There is a close relationship between defects $\defect S_m(\NN)$ and the syzygies described by $H^n(K^*(f))$, see \cite{DSt2} for nodal hypersurfaces and \cite{D4} for projective hypersurfaces with isolated singularities.


The discussion following the statement of Theorem 2 in \cite{DSW} shows that in fact (B) is equivalent to the condition
\begin{equation} 
\label{B}
(B'): \defect S_e(\NN)=0,
\end{equation} 
where $e=\left[ \frac{n}{2} \right](d-1)-p$ and $\NN $ is the set of nodes of $D$.

One has the following consequence of Theorems \ref{thmA} and \ref{thmB}.

\begin{thm}
\label{thmC}
Let $D:f=0$ be a nodal hypersurface of degree $d>2$ in $\PP^n$ and assume that
$q=n-p>m:=\left[ \frac{n}{2} \right]$.
\smallskip

\noindent (i) If $n=2n_1$ is even, then the isomorphism \eqref{GR} always hold.

\smallskip

\noindent (ii) If $n=2n_1+1$ is even, then the isomorphism \eqref{GR} holds if
$$p\leq n -\left[ \frac{n}{2} \right] -\left[ \frac{d}{2} \right] .  $$
In particular, the isomorphism \eqref{GR} holds for $d=3$.




\end{thm}

\proof When $n=2n_1$ is even, then $p<n_1$ and hence $e=n_1(d-1)-p=n_1d-n_1-p>n_1d-n.$
Use now Corollary 2.2 (ii) in \cite{DSt3} which says that
$\defect S_k(\NN)=0$ for $k\geq n_1d-n$.

\smallskip

When $n=2n_1+1$, we have $e=n_1(d-1)-p$.
On the other hand, one clearly has $\defect S_k(\NN)=0$ if and only if 
$k\geq T-ct(D)$ and Corollary \ref{corA} implies
$$T-ct(D) \leq 2(n_1+1)(d-2)-\left( (n_1+2)d-\left[ \frac{d}{2} \right]-n-2\right)=n_1d-n+\left[ \frac{d}{2} \right].$$
Hence $e \geq n_1d-n+\left[ \frac{d}{2} \right]$ as soon as $p\leq n -\left[ \frac{n}{2} \right] -\left[ \frac{d}{2} \right] .  $

\endproof

Note that Example 4.7 in \cite{DSW} shows that (B) may not be satisfied by a nodal surface, where $n=3$,  $d=4$ and $p=1$. This shows that the condition in Theorem \ref{thmC} (ii) is sharp,
as in the case at hand we get $1 \leq 3-1-2$, which fails just by $1$.



\begin{thebibliography}{00}





\bibitem{CD} A.~D.~R.~Choudary, A.~Dimca,  Koszul complexes and hypersurface singularities, 
        Proc. Amer. Math. Soc. 121(1994), 1009--1016. 


\bibitem{DD}
P.~Deligne and A.~Dimca,
Filtrations de Hodge et par l'ordre du p\^ole pour les hypersurfaces
singuli\`eres, Ann. Sci. Ecole Norm. Sup. (4) 23(1990), 645--656.




\bibitem{D1} A. Dimca, {\em Singularities and Topology of Hypersurfaces}, 
Universitext, Springer-Verlag, 1992.


\bibitem{D2} A. Dimca,
{\em Sheaves in Topology},  Universitext, Springer-Verlag, 2004.

\bibitem{D3} A. Dimca, Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements, Nagoya Math. J. 206 (2012), 75-97.

\bibitem{D4} A. Dimca, Syzygies of Jacobian ideals and defects of linear systems, 
Bull. Math. Soc. Sci.Math. Roumanie, Tome 56(104) No. 2, 2013, 191- 203.

\bibitem{DS3}
A. Dimca and M. Saito,
Some remarks on limit mixed Hodge structure and spectrum, arXiv:1210.3971,
to appear in An. \c St. Univ. Ovidius Constan\c ta. 


\bibitem{DS4} { A. Dimca,  M. Saito}, Graded  Koszul cohomology and spectrum of certain homogeneous polynomials, arXiv:1212.1081v2.




\bibitem{DSW} A. Dimca,  M. Saito, L. Wotzlaw, A generalization of Griffiths' theorem on rational integrals II,  Michigan Math. J. 58(2009),  603--625.

\bibitem{DL} A. Dimca, G. Lehrer, Hodge-Deligne equivariant polynomials and monodromy of hyperplane arrangements,  in: Configuration Spaces, Geometry, Combinatorics and Topology, Publications of Scuola Normale Superiore, vol. 14 (2012), 231-253.




\bibitem{DSt2} A. Dimca, G. Sticlaru, Koszul complexes and pole order filtrations,  arXiv:1108.3976, to appear in Proc. Edinburgh Math. Soc.

\bibitem{DSt3}  A. Dimca, G. Sticlaru, On the syzygies and Alexander polynomials of nodal hypersurfaces,  Math.\ Nachr.\ 285 (2012), 2120--2128.


 \bibitem{EGH} D. Eisenbud, M. Green, and J. Harris, Cayley Bacharach theorems and conjectures, Bull. Amer. Math. Soc. 33 (1996), 295--324.


\bibitem{Gr}
Ph.~Griffiths, On the period of certain rational integrals I,
II, Ann. Math. 90(1969), 460--541.



\bibitem{PS} C. Peters, J. Steenbrink, {\em Mixed Hodge Structures}, 
Ergeb. der Math. und ihrer Grenz. 3. Folge 52,
Springer, 2008.


\bibitem{Sa1}
M.~Saito, On $b$-function, spectrum and rational singularity,
Math. Ann. 295(1993), 51--74.

\bibitem{Sa2} M. Saito, Multiplier ideals, b-functions, and spectrum of a hypersurface singularity,
Compositio Math. 143 (2007), 1050--1068.

\bibitem{Se} E. Sernesi, The local cohomology of the jacobian ring, arXiv: 1306.373.

\bibitem{St} J. Steenbrink, Intersection form for quasi-homogeneous singularities, Compositio Math. 34(1977), 211-223.

\bibitem{Sti} G. Sticlaru, Invariants and rigidity of projective hypersurfaces,  arXiv:1309.7356.



\end{thebibliography}

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1310.5344.extracted_bib
\begin{thebibliography}{00}





\bibitem{CD} A.~D.~R.~Choudary, A.~Dimca,  Koszul complexes and hypersurface singularities, 
        Proc. Amer. Math. Soc. 121(1994), 1009--1016. 


\bibitem{DD}
P.~Deligne and A.~Dimca,
Filtrations de Hodge et par l'ordre du p\^ole pour les hypersurfaces
singuli\`eres, Ann. Sci. Ecole Norm. Sup. (4) 23(1990), 645--656.




\bibitem{D1} A. Dimca, {\em Singularities and Topology of Hypersurfaces}, 
Universitext, Springer-Verlag, 1992.


\bibitem{D2} A. Dimca,
{\em Sheaves in Topology},  Universitext, Springer-Verlag, 2004.

\bibitem{D3} A. Dimca, Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements, Nagoya Math. J. 206 (2012), 75-97.

\bibitem{D4} A. Dimca, Syzygies of Jacobian ideals and defects of linear systems, 
Bull. Math. Soc. Sci.Math. Roumanie, Tome 56(104) No. 2, 2013, 191- 203.

\bibitem{DS3}
A. Dimca and M. Saito,
Some remarks on limit mixed Hodge structure and spectrum, arXiv:1210.3971,
to appear in An. \c St. Univ. Ovidius Constan\c ta. 


\bibitem{DS4} { A. Dimca,  M. Saito}, Graded  Koszul cohomology and spectrum of certain homogeneous polynomials, arXiv:1212.1081v2.




\bibitem{DSW} A. Dimca,  M. Saito, L. Wotzlaw, A generalization of Griffiths' theorem on rational integrals II,  Michigan Math. J. 58(2009),  603--625.

\bibitem{DL} A. Dimca, G. Lehrer, Hodge-Deligne equivariant polynomials and monodromy of hyperplane arrangements,  in: Configuration Spaces, Geometry, Combinatorics and Topology, Publications of Scuola Normale Superiore, vol. 14 (2012), 231-253.




\bibitem{DSt2} A. Dimca, G. Sticlaru, Koszul complexes and pole order filtrations,  arXiv:1108.3976, to appear in Proc. Edinburgh Math. Soc.

\bibitem{DSt3}  A. Dimca, G. Sticlaru, On the syzygies and Alexander polynomials of nodal hypersurfaces,  Math.\ Nachr.\ 285 (2012), 2120--2128.


 \bibitem{EGH} D. Eisenbud, M. Green, and J. Harris, Cayley Bacharach theorems and conjectures, Bull. Amer. Math. Soc. 33 (1996), 295--324.


\bibitem{Gr}
Ph.~Griffiths, On the period of certain rational integrals I,
II, Ann. Math. 90(1969), 460--541.



\bibitem{PS} C. Peters, J. Steenbrink, {\em Mixed Hodge Structures}, 
Ergeb. der Math. und ihrer Grenz. 3. Folge 52,
Springer, 2008.


\bibitem{Sa1}
M.~Saito, On $b$-function, spectrum and rational singularity,
Math. Ann. 295(1993), 51--74.

\bibitem{Sa2} M. Saito, Multiplier ideals, b-functions, and spectrum of a hypersurface singularity,
Compositio Math. 143 (2007), 1050--1068.

\bibitem{Se} E. Sernesi, The local cohomology of the jacobian ring, arXiv: 1306.373.

\bibitem{St} J. Steenbrink, Intersection form for quasi-homogeneous singularities, Compositio Math. 34(1977), 211-223.

\bibitem{Sti} G. Sticlaru, Invariants and rigidity of projective hypersurfaces,  arXiv:1309.7356.



\end{thebibliography}