Syzygies, experiments in algebraic geometry and unirationality questions for moduli spaces

一个代数簇M是单有理的，如果支配簇Pn为99 KM.在另一个极端，M是一般型的，如果标准丛KM是大的。在第一种情况下，很容易找到M上的点，在第二种情况下，通过M的一般点没有P1。对于模空间，例如曲线Mg的模，一个簇是否属于这些类型之一的问题尤为重要。如果模空间M是单有理的，那么我们原则上可以写出依赖于自由参数的支配族。在一般类型的情况下，任何一组参数都满足一个代数方程组。我们计划研究曲线的各种精化模空间，如曲线C的Mr g;d = f（C; gr d）g以及C上d次因子的r维线性系统gr d。在模空间是单有理的情况下，我们希望提供一个随机选择点的计算机代数代码，这将有助于进一步的实验研究。

An algebraic variety M is unirational, if dominant Pn 99K M. On the other extreme, M is of general type, if the canonical bundle KM is big. In the first case it is easy to find points on M, in the second case there is no P1 through a general point of M. The question whether a variety is of one of these types is especially important for moduli spaces, such as the moduli of curves Mg. If a moduli space M is unirational then we can in principle write down a dominant family depending on free parameters. In case of general type, any set of parameters satisfy a system of algebraic equations. We plan to investigate various refined moduli spaces of curves such as Mr g;d = f(C; gr d)g of curves C together with an r-dimensional linear system gr d of divisors of degree d on C. In case the moduli space is unirational we want to provide a computer algebra code which chooses points at random, which will be useful for further experimental investigations.