Stability conditions and birational geometry of projective surfaces

Abstract

We show that the minimal model program on any smooth projective surface is realized as a variation of the moduli spaces of Bridgeland stable objects in the derived category of coherent sheaves.Comment: 39 pages, to appear in Compositio Mathematic

A generalized Bogomolov-Gieseker inequality for the smooth quadric threefold

“…Examples of successful applications are found in the birational geometry of Hilbert schemes of points on smooth projective surfaces (see for example [ABCH13,BM13,MM13,YY14]). Toda shows that the minimal model program on any smooth projective surface is realized as a variation of moduli spaces of Bridgeland stable objects in [Tod12].…”

Wall-crossing implies Brill-Noether: Applications of stability conditions on surfaces

“…• Flips of secant varieties can be shown to arise naturally in the wall-crossing for moduli spaces of torsion sheaves on P 2 [Mar13]. • One can induce the minimal model programme of a surface X via wall-crossing in D b (X) [Tod12]; yet the moduli space becomes reducible if one tries to contract other curves of self-intersection less than -2 [Tra15]. • There is a a close relation between the location of the wall where a given ideal sheaf in Hilb n (P 2 ) gets destabilised and its Castelnuovo-Mumford regularity [CHP16].…”

Change of Polarization for Moduli of Sheaves on Surfaces as Bridgeland Wall-crossing

“…For instance, it has been shown that running a directed Minimal Model Program (MMP) for M H (v) when X is K3 [BM13], abelian [Yos12], Enriques [Nue14], or the projective plane [ABCH13, BMW14,CHW14], corresponds to varying a stability condition on X. Also, when v = [C x ] the moduli space M H (v) is canonically isomorphic to X and Toda [Tod12] has proven that any MMP for X can be obtained by varying stability conditions. By the boundedness results for walls on certain rays in Stab(X) obtained by Lo and Qin [LQ11] and generalized by Maciocia [Mac12], we know that moduli spaces of L-twisted H-Gieseker semistable sheaves are moduli spaces of Bridgeland semistable objects.…”

“…In Section 6 we treat the case of change of polarization for 1-dimensional sheaves. Finally, in Section 7 we use our methods to give a proof of the result of Toda [Tod12] that a contraction of a −1-curve can be interpreted as a Bridgeland wall-crossing.…”