  In this paper we study $(i)$-curves with $i\in \{-1, 0, 1\}$ in the blown up
projective space $\mathbb{P}^r$ in general points. The notion of $(-1)$-curves
was analyzed in the early days of mirror symmetry by Kontsevich with the
motivation of counting curves on a Calabi-Yau threefold. In dimension two,
Nagata studied planar $(-1)$-curves in order to construct counterexample to
Hilbert's 14th problem.
  We introduce the notion of classes of $(0)$- and $(1)$-curves in
$\mathbb{P}^r$ with $s$ points blown up and we prove that their number is
finite if and only if the space is a Mori Dream Space. We further introduce a
bilinear form on a space of curves, and a unique symmetric Weyl-invariant
class, $F$, (that we will refer to as the anticanonical curve class). For Mori
Dream Spaces we prove that $(-1)$-curves can be defined arithmetically by the
linear and quadratic invariants determined by the bilinear form. Moreover,
$(0)$- and $(1)$-Weyl lines give the extremal rays for the cone of movable
curves in $\mathbb{P}^r$ with $r+3$ points blown up. As an application, we use
the technique of movable curves to reprove that if $F^2\leq 0$ then $Y$ is not
a Mori Dream Space and we propose to apply this technique to other spaces.
