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

曲线Mg的模空间是给定格的代数曲线（黎曼曲面）的普适参数空间，因为它的点对应于g格曲线的同构类。Mg的几何和拓扑的研究是代数几何的中心问题。由Mumford提出的一个众所周知的原理断言，所有参数化格g≥2的曲线的模空间（有或没有标记的点或水平结构）都是一般类型的变种，在相对较小的格中有有限数量的例外，当这些变种趋向于无规则，甚至是无秩的时候。当通过一般点X Є X时，我们称变种X为无规型，而当规范束Kx具有最大的截面数时，我们称变种X为有理曲线f: P1 (X)。从分类理论的观点来看，普遍性是普遍性的对立面。这个项目的目的是计算各种封面封面的Kodaira维数。它们分别包括泛Picard变分和由一条曲线和雅可比变分中1阶点组成的模空间分类对。预计这些证明将依赖麦考利系统辅助的协同计算。

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.