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

本课题拟研究曲线模空间的双民族模型。具体来说，他将继续研究由最小模型程序、几何不变理论和曲线奇点变形理论引起的光滑抽象曲线空间的紧化之间的关系。此外，他还提出将曲线的模与射影空间的映射一起研究；具体地说，由最小模型规划引起的稳定映射的Kontsevich空间的双空间模型，以及Viscardi关于稳定映射的模空间的构造的推广，该构造具有除节点以外的奇异域。简单地说，代数曲线是两个变量的多项式；而代数曲线理论主要关注的是这种多项式的解与解集的几何关系。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；这就是首席研究员的建议。

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.