Effective cones of cycles on blow-ups of projective space

TL;DR

This paper investigates the structure of effective cones of higher codimension cycles on blow-ups of projective space, revealing conditions under which these cones are finitely generated and how they compare to divisor cones.

Contribution

It provides bounds on the number of points for finite generation of cones and shows that higher codimension cones often behave better than divisor cones on blow-ups.

Findings

Cones are finitely generated for certain point configurations.

Higher codimension cones are better behaved than divisor cones.

If the blow-up is a Mori Dream Space, all effective cones are finitely generated.

Abstract

In this paper, we study the cones of higher codimension (pseudo)effective cycles on point blow-ups 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 blow-up $X_r^n$ 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 $r \leq 2^n$. 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.