×

Needles in a haystack: Special varieties via small fields. (English) Zbl 0994.14037

Eisenbud, David (ed.) et al., Computations in algebraic geometry with Macaulay 2. Berlin: Springer. Algorithms Comput. Math. 8, 251-279 (2002).
The authors illustrate how picking points over a finite field at random can help to investigate questions in algebraic geometry. In the first section they describe a Macaulay2 program that picks curves of genus \(g \leq 14\) at random. The authors interpret their approach as a ‘computer aided proof’ for the unirationality of the moduli space \({\mathcal M}_g\) for \(g \leq 13.\) In fact they contribute to the problem to find an explicit unirational parametrization of \({\mathcal M}_g\) for \(g \leq 13.\)
The second section is devoted to the application of ‘random curves’ to prove the consequences of Green’s conjecture on syzygies of canonical curves, and to compare these results with the corresponding statements for ‘Coble self-dual’ sets of \(2g-2\) points in \(\mathbb P^{g-2}.\)
In the final section the authors exploit their method to prove the existence of three components of the Hilbert scheme of Calabi-Yau threefolds of degree 17 in \(\mathbb P^6\) over the complex numbers. This is done by lifting to characteristic zero.
For the entire collection see [Zbl 0973.00017].

MSC:

14Q15 Computational aspects of higher-dimensional varieties
14G15 Finite ground fields in algebraic geometry
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14H10 Families, moduli of curves (algebraic)
PDFBibTeX XMLCite