Topics in the moduli theory of sheaves

PI计划调查曲线和曲面上各种滑轮空间的几何形状。特别地，她将研究一条曲线上平凡的n阶束的稳定束商空间，因为这条曲线被允许在Deligne-Mumford模空间中变化。稳定商空间是在与D.Oprea和R.Pandharipande合作的早期工作中构造的。在一个方向上，它可能有助于澄清曲线的模空间的重言环的特征，它允许适当的态射。在不同的方向上，稳定商的模空间的交理论与Grassmannians的Gromov-Witten理论和Grassmannians的完全交有密切的联系，如果完全理解这一点将是有趣的。PI还建议继续研究曲面上滑轮的模空间上的奇异对偶现象。近年来，有理曲面和K3曲面在这方面的研究取得了很大的进展，在数学和理论物理中都具有重要的意义。在数学上，他们的研究是由Mumford利用几何不变量理论构造光滑射影曲线上稳定丛的模空间开始的。后来，Donaldson证明了曲面上阶数为2的模空间的交理论是理解四维流形几何的一个有用的工具。在物理学中，向量丛的模空间在规范理论的基础研究中自然产生。PI计划进一步研究不同环境下的滑轮参数空间，并促进我们对这些重要几何结构的理解。

The PI plans to investigate the geometry of various spaces of sheaves on curves and surfaces. In particular, she will study a space of stable sheaf quotients of a trivial rank n sheaf on a curve, as the curve is allowed to vary in the Deligne-Mumford moduli space. The space of stable quotients was constructed earlier in joint work with D. Oprea and R. Pandharipande. In one direction, it may help to clarify features of the tautological ring of the moduli space of curves, to which it admits a proper morphism. In a different direction, the intersection theory of the moduli space of stable quotients has close connections with the Gromov-Witten theory of Grassmannians and complete intersections in Grassmannians, which would be interesting to understand completely. The PI also proposes to continue the study of strange duality phenomena on moduli spaces of sheaves on surfaces. Recently, there has been good progress in this area in the case of rational surfaces and K3 surfaces.Moduli spaces of sheaves on varieties are geometric objects of great importance in both mathematics and theoretical physics. Mathematically, their study was initiated by Mumford's construction of the moduli space of stable bundles on a smooth projective curve via geometric invariant theory. Later, Donaldson showed that the intersection theory of moduli spaces of rank 2 sheaves on surfaces is a useful tool for understanding the geometry of four-manifolds. In physics, moduli spaces of vector bundles arise naturally in the fundamental investigation of gauge theories. The PI plans to further the study of parameter spaces of sheaves in various settings, and advance our understanding of these important geometric structures.