| Description |
We study some birational geometric aspects of moduli spaces of semistable sheaves on surfaces. We observe that moduli spaces of semistable sheaves on a Del Pezzo surface are Mori-dream spaces and, following the techniques intro- duced by Arcara, Bertram, Coskun, and Huizenga, relate the Mori chambers to chambers in the Stability manifold introduced by Bridgeland. In the special case when X = P2, the wall-crossing phenomena for stability conditions (and therefore the wall-crossing in NE(X)) can be analyzed very closely. In fact, for the case of 1- dimensional sheaves with vanishing Euler-Poincare characteristic, we can describe several of the birational models, including the minimal. As we will study later, the wall-crossing phenomena in this case gives information about existence of flips of Secant varieties of Veronese surfaces. To do this we will need a generalization of a duality result of Maican for moduli spaces of 1-dimensional plane sheaves to any moduli space of Bridgeland semistable objects. In particular, we prove that Maican's result holds on arbitrary smooth complex surfaces. Finally, we show that the change of polarization for moduli spaces of sheaves on a smooth projective complex surface X, and the birational geometry of X itself, are a consequence of the wall-crossing phenomena on Stab(X). |
