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

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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.