  In this paper we analyze $(-1)$- curves 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. In dimension two, Nagata studied planar $(-1)$-curves
in order to construct counterexample to Hilbert's 14th problem.
  In this paper, we first analyze the Coxeter systems and the Weyl group action
on curves, to identify infinitely many $(-1)$-curves on the blown-up
$\mathbb{P}^r$ when the number of points is at least $r+5$. We introduce a
bilinear form on a space of curves, that extends the intersection product in
dimension $2$, and a unique symmetric Weyl-invariant class (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, while the set of $(0)$- and
$(1)$-Weyl lines determine the cone of effective divisors in blown-up
$\mathbb{P}^r$ with $r+3$ points.
