The study of the birational geometry of various moduli spaces of curves with the help of the computer algebra system Macaulay

曲线的模空间Mg是给定亏格的代数曲线(Riemann曲面)的泛参数空间，因为它的点对应于亏格g的曲线的同构类.研究Mg的几何和拓扑是代数几何的一个中心问题.由Mumford提出的一个众所周知的原理是，亏格g≥2(有或没有标记点或水平结构)的所有模空间参数化曲线都是一般类型的簇，但有有限数量的例外出现在相对较小的亏格中，此时这些簇往往是单调的，甚至是单调的。当一个簇X通过一个普通点xЄX时，它通过一条有理曲线ƒ：P1(X，而当标准丛Kx具有最大截面数时，称X是一般类型的。从分类理论的角度来看，单一性是一般类型存在的对立面。本项目的目的是计算各种盖子的kodaira尺寸。它们包括泛Picard簇和模空间，它们分别由一条曲线和一个阶点组成的模空间组成。预计证据将依赖于麦考利系统辅助的合并性计算。

The moduli space of curves Mg is the universal parameter space for algebraic curves (Riemann surfaces) of given genus, in the sense that its points correspond to isomorphism classes of curves of genus g. The study of the geometry and topology of Mg is a central problem in algebraic geometry. A well-known principle, due to Mumford, asserts that all moduli spaces parameterizing curves of genus g ≥ 2 (with or without marked points or level structures), are varieties of general type, with a finite number of exceptions that occur in relatively small genus, when these varieties tend to be uniruled, or even unirational. A variety X is said to be uniruled, when through a general point x Є X there passes a rational curve ƒ : P1 ( X, whereas X is said to be of general type, when the canonical bundle Kx has the maximum number of sections. From the point of view of classification theory, uniruledness is the opposite of being of general type. The aim of this project is to compute the Kodaira dimension of various covers covers of Mg. These include universal Picard varieties and moduli spaces classifying pairs consisting of a curve together with a point of order l in the Jacobian variety respectively. The proofs are expected to rely on syzygy calculations assisted by the Macaulay system.