Birational geometry of blowups via Weyl chamber decompositions and actions on curves

Abstract

We study the birational geometry of $X^n_s$, the blow-up of $\mathbb{P}^n_\mathbb{C}$ at $s$ points in general position. We identify a set of subvarieties, which we call Weyl $r$-planes, that belong to an orbit for the action of the Weyl group on $r$-cycles. They satisfy the following properties: they appear as stable base locus of divisors; each Weyl $r$-plane is swept out by an $(n-r)$-moving curve class; moreover, if $s\ge n+3$, for any fixed $r$ all these curve classes belong to the same orbit for the Weyl action. For Mori dream spaces of type $X^n_s$, all such orbits are finite and they allow to reinterpret Mukai's description of the Mori chamber decomposition of the effective cone in terms of $(n-r)$-moving curve classes, unifying previous different approaches. If $X^n_s$ is not a Mori dream space, there are infinitely many Weyl $r$-planes. These yields the definition of the Weyl chamber decomposition of the pseudoeffective cone of divisors. We pose the question as to whether the nef chamber decomposition can be defined (in the negative part of $\overline{\mathrm{Eff}}(X^n_s)$) and, if this is the case, whether it coincides with the Weyl chamber decomposition. We conjecture that the answer is affirmative for $X^3_8$ and $X^5_9$.