Geometry of Linear Systems on Curves: Birational Geometry of Moduli Spaces

主要研究者建议研究曲线模空间的双有理模型。具体来说，他将继续他的调查之间的关系紧致的空间所产生的光滑抽象曲线的最小模型计划，几何不变理论，和变形理论的曲线奇点。此外，他建议研究曲线的模连同映射到射影空间;特别是，双有理模型的Kontsevich空间的稳定映射所产生的最小模型程序，也推广了Viscardi的建设模空间的稳定映射与域有奇点以外的节点。代数曲线理论主要涉及这样一个多项式的解与解集的几何之间的关系。为了充分理解这种关系，我们需要研究所有可能曲线的空间“曲线的模空间”及其几何。这反过来又要求我们研究这个空间的不同模型;这就是首席研究员建议做的。

The Principal Investigator proposes to work on birational models of moduli spaces of curves. Specifically, he will continue his investigation of the relationships among the compactifications of the space of smooth abstract curves arising from the Minimal Model Program, Geometric Invariant Theory, and the deformation theory of curve singularities. In addition, he proposes to study moduli of curves together with maps to projective space; specifically, birational models of the Kontsevich space of stable maps arising from the Minimal Model Program, and also generalizations of Viscardi's construction of moduli spaces of stable maps with domains having singularities other than nodes.An algebraic curve is, simply put, a polynomial in two variables; and the theory of algebraic curves is largely concerned with the relationship between the solutions of such a polynomial and the geometry of the set of solutions. To fully understand this relationship, we need to study the space of all possible curves 'the moduli space of curves' and its geometry. This in turn requires that we study different models of this space; and this is what the Principal Investigator proposes to do.