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 $$\ne $$Â?2 appear as the image of the Donagi---Livne---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 $$\mathbb {P}^1_K$$PK1 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 $$\mathbb {P}^1_K\times \mathbb {P}^1_K$$PK1A—PK1 and prove that, under mild conditions on K, the curve C can be given by a "$$(g+1,2)$$(g+1,2)-normal form", namely by an affine equation in two variables of partial degrees $$g+1$$g+1 and 2 and hence of total degree $$\le $$≤g + 3, which is smaller than the degree of Weierstraβ normal forms. Such curves are naturally parameterized by a Hurwitz space $$\overline{\mathcal{H}}_{g,g+1}$$H¯g,g+1. We then specialize to $$g=3$$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---Livne---Smith construction.