The birational geometry of moduli spaces of sheaves on the projective plane

Abstract

We describe a close relation between wall crossings in the birational geometry of modulispaceofGiesekerstablesheaves MH (v)on P 2 andmini-wallcrossingsinthestability manifold Stab(D b (P 2 )). The general philosophy is that the birational geometry of MH (v) is closely related to the birational geometry of X.S o if−K X is ample, we hope −K MH (v) is ample as well. This turns out to be not quite right but is close. We first show that if v is a primitive topological type such that MH (v) is non-empty and irreducible, then MH (v) is smooth and −K MH (v) is big and nef. In particular, MH (v) is a Mori dream space. Therefore MH (v) provide a rich class of examples of the so called weak Fano varieties, whose classification theory is highly interesting in its own right. We achieve the statement by showing that the first Chern class