Syzygies, finite length modules, and random curves

Summary

We apply the theory of Gröbner bases to the computation of free resolutions over a polynomial ring, the defining equations of a canonically embedded curve, and the unirationality of the moduli space of curves of a fixed small genus.

While a great deal of modern commutative algebra and algebraic geometry has taken a nonconstructive form, the theory of Gröbner bases provides an algorithmic approach. Algorithms currently implemented in computer algebra systems, such as Macaulay2 [Grayson and Stillman] and Singular [Decker et al. 2011], already exhibit the wide range of computational possibilities that arise from Gröbner bases.

In these lectures, we focus on certain applications of Gröbner bases to syzygies and curves. In Section 1, we use Gröbner bases to give an algorithmic proof of Hilbert’s syzygy theorem, which bounds the length of a free resolution over a polynomial ring. In Section 2, we prove Petri’s theorem about the defining equations for canonical embeddings of curves. We turn in Section 3 to the Hartshorne–Rao module of a curve, showing by example how a module M of finite length can be used to explicitly construct a curve whose Hartshorne– Rao module is M. Section 4 then applies this construction to the study of the unirationality of the moduli space 𝔐_g of curves of genus g.