Unirationality of the moduli spaces of curves of genus 11, 13 (and 12)
Given a field k, the moduli space \({\mathcal M}_ g(k)\) is an irreducible k- variety parametrizing the isomorphism classes of nonsingular curves of genus g defined over k. It is a classical problem whether \({\mathcal M}_ g(k)\) is unirational, or equivalently whether there exists a k-family of curves depending rationally on free parameters and containing up to isomorphism almost every curve of genus g. The answer is known to be yes if \(g\leq 10\), \(k={\mathbb{C}}\) (classical) or if \(g=12\), \(k={\mathbb{C}}\) (Sernesi, 1981), no if \(g\geq 23\) (Mumford, Harris, Eisenbud 1981-84). This paper extends the ''range of unirationality'' of \({\mathcal M}_ g(k)\) to \(g\leq 13\), k arbitrary (e.g. \(k=rational\) numbers). (The case \(g\leq 10\) is not discussed in detail in this paper, but can be done by the same method.) The proof has two parts: (a) one constructs a family of space curves having general moduli and some other good properties having to do mostly with the ''postulation''; this uses a combination of classical geometric techniques and elementary deformation theory, esp. some recent work of \textit{E. Sernesi} [Invent. Math. 75, 25-57 (1984; see the preceding review)]; (b) one represents the curves obtained as degeneracy loci of certain vector bundles on \({\mathbb{P}}^ 3\) and determines their cohomology; this allows one, using Horrocks' method for monads, to parametrize the bundles, hence the curves, by certain matrices of homogeneous forms; the latter are easily shown to form a rational variety, which proves the theorem.
unirationality of the moduli spaces of curves
scientific article; zbMATH DE number 3859296