Normal forms of hyperelliptic curves of genus 3

Abstract

The main motivation for the paper is to understand which hyperelliptic curves of genus 3 defined over a field K of characteristic $\ne$2 appear as the image of the Donagi–Livné–Smith construction. By results in Frey and Kani (Lecture Notes in Computer Science, vol. 7053, pp. 1–19. Springer, Heidelberg, 2012) this means that one has to determine the intersection W of a Hurwitz space defined by curves of genus 3 together with cover maps of degree 4 to $\mathbb {P}^1_K$ and a certain ramification type with the hyperelliptic locus in the moduli space of curves of genus 3. To achieve this aim we first study hyperelliptic curves of genus g as smooth curves C in $\mathbb {P}^1_K\times \mathbb {P}^1_K$ and prove that, under mild conditions on K, the curve C can be given by a “$(g+1,2)$-normal form”, namely by an affine equation in two variables of partial degrees $g+1$ and 2 and hence of total degree $\le$g + 3, which is smaller than the degree of Weierstraß normal forms. Such curves are naturally parameterized by a Hurwitz space $\overline{\mathcal{H}}_{g,g+1}$. We then specialize to $g=3$ and introduce Hurwitz spaces for 4-covers with special ramification types. The study of these spaces enables us to determine that W is irreducible of dimension 4. Moreover we find an explicitly given K-rational family of curves C in (4, 2)-normal form such that the isomorphism classes of its members are in W(K) and such that the image of the family in W is Zariski-dense. For these curves we describe the “inverse” of the Donagi–Livné–Smith construction.