| Description |
In this thesis we explore two enumerative problems from algebraic geometry, each with its own flavor. First, we investigate Le Potier's Strange Duality for moduli spaces of sheaves on surfaces. Strange duality is a conjectural perfect pairing between spaces of sections of the determinant line bundles (also called theta line bundles) on moduli spaces with discrete invariants satisfying a suitable orthogonality condition. Our approach, inspired by [MO07], is to construct zero dimensional Quot schemes whose points correspond to dual bases for the two spaces of sections. We restrict our study to the case when one of the moduli spaces is the Hilbert scheme of points on a del Pezzo surface. We compute the expected cardinality of these Quot schemes using multiple point formulas [MR10]. These numbers are shown to be equal to the Euler characteristics of the theta line bundles, as computed by the universal power series of [EGL01]. We also investigate conditions under which we can prove the existence of suitable Quot schemes. As part of this, we prove a result of independent interest: a general sheaf on $\PP^2$ of Euler characteristic at least 2 greater than its rank is globally generated. We also attach a paper about tropical geometry. In this paper, we consider the question of when points in tropical affine space uniquely determine a tropical hypersurface. We introduce a notion of multiplicity of points so that this question may be meaningful even if some of the points coincide. We give a geometric/combinatorial way and a tropical linear-algebraic way to approach this question. First, given a fixed hypersurface, we show how one can determine whether points on the hypersurface determine it by looking at a corresponding marking of the dual complex. With a regularity condition on the dual complex and when the number of points is minimal, we show that our condition is equivalent to the connectedness of an appropriate subcomplex. Second, we introduce notions of nonsingularity of tropical matrices and solutions to tropical linear equations that take into account our notion of multiplicity and prove a Cramer's Rule type theorem relating them. |
