Abstract:

Since Bridgeland introduced his definition of stability conditions in 2002, they have been used to solve various problems in algebraic geometry. Any Bridgeland stability condition gives us a moduli problem and associates to any Chern character a moduli space, which for surfaces is known to be an algebraic space. For the Chern character of a Gieseker semi stable sheaf on a surface, some special Bridgeland stability conditions (those in the so-called Gieseker chamber) give back the classical moduli space of semistable coherent sheaves constructed by Gieseker and Maruyama. Varying the stability conditions allows us to study the relationships between various moduli spaces of Bridgeland semi stable objects on surfaces. For example, one often obtains birational maps between moduli spaces using wall-crossing methods in the space of Bridgeland stability conditions, in particular the natural nef divisor on the moduli space define by. A. Bayer and E. Macrì. Thus far, these techniques have not been used to study moduli spaces associated to bielliptic surfaces. In this talk, we discuss how using wall-crossing techniques along with some other insights allows us to determine the nef cone of the moduli space of stable sheaves of Chern class (1, 0, -n), which corresponds to the Hilbert scheme of n points. Moreover, we determine the birational surgery that gives the first minimal model beyond the boundary of the nef cone.