Topics in the moduli theory of sheaves

PI计划研究曲线和曲面上不同空间的轮轴几何形状。特别是，她将研究曲线上平凡秩n的稳定层商的空间，因为曲线允许在delign - mumford模空间中变化。稳定商空间的构造较早，是与D. Oprea和R. Pandharipande共同完成的。在一个方向上，它可能有助于澄清曲线模空间的同义环的特征，它承认一个适当的态射。另一方面，稳定商模空间的交点理论与Grassmannians的Gromov-Witten理论以及Grassmannians中的完全交点有着密切的联系，完全理解它们会很有趣。论文还提出继续研究曲面上轴模空间上的奇异对偶现象。近年来，在有理曲面和K3曲面的情况下，这方面的研究取得了很好的进展。变轴上的模空间是数学和理论物理中非常重要的几何对象。在数学上，他们的研究始于Mumford利用几何不变理论构造光滑投影曲线上稳定束的模空间。后来，Donaldson证明了曲面上2阶轴模空间的交点理论是理解四流形几何的一个有用工具。在物理学中，向量束的模空间在规范理论的基础研究中自然出现。PI计划进一步研究各种设置下的滑轮参数空间，并推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.