Effective cones of cycles on blowups of projective space

Abstract

In this paper we study the cones of higher codimension (pseudo)effective cycles on point blowups of projective space. We determine bounds on the number of points for which these cones are generated by the classes of linear cycles and for which these cones are finitely generated. Surprisingly, we discover that for (very) general points the higher codimension cones behave better than the cones of divisors. For example, for the blowup $X_r^n$ of ${\mathbb{P}}^n$, $n>4$ at $r$ very general points, the cone of divisors is not finitely generated as soon as $r>n+3$, whereas the cone of curves is generated by the classes of lines if $r\le 2^n$. In fact, if $X_r^n$ is a Mori dream space then all the effective cones of cycles on $X_r^n$ are finitely generated.