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

代数簇M是单调的，如果占优PN99K M。在另一个极端，如果标准丛Km大，则M是通用型的。在第一种情况下，很容易在M上找到点，在第二种情况下，不存在通过M的通用点的P1。对于模空间，例如曲线的模MG，一个变种是否属于这些类型中的一个的问题特别重要。如果模空间M是单调的，那么我们原则上可以写出一个依赖于自由参数的占优族。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；grd)g，以及C上d次因子的r维线性系统grd。如果模空间是单调的，我们想要提供一个随机取点的计算机代数代码，这将为进一步的实验研究提供有用的工具。

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.