Bielliptic curves of genus 3 in the hyperelliptic moduli

Abstract

Abstract. In this paper we study bielliptic curves of genus 3 defined over an algebraically closed field k and the intersection of the moduli space M b 3 of such curves with the hyperelliptic moduli H 3 . Such intersection S is an irreducible, 3-dimensional, rational algebraic variety. We determine the equation of this space in terms of the Gl(2, k)-invariants of binary octavics as defined in [27] and find a birational parametrization of S. We also compute all possible subloci of curves for all possible automorphism group G. Moreover, for every rational moduli point p ∈ S, such that |Aut (p)| > 4, we give explicitly a rational model of the corresponding curve over its field of moduli in terms of the Gl(2, k)-invariants. genus 3 hyperelliptic curves and dihedral invariants and absolute invariants

2-Weierstrass points of genus 3 hyperelliptic curves with extra involutions

“…Preserving the condition ε(X) = −X we can further modify X such that s 4 = 1. Then, we have the following lemma, which is proven in [9]. Lemma 2.…”

“…They describe such points using the dihedral invariants of such curves as defined in [11]. In [9] the automorphism groups of genus 3 hyperelliptic curves are characterized in terms of the dihedral invariants. Naturally one asks if the methods in [5] can be extended to genus 3 via methods described in [9].…”

“…We only focus on the groups G which determine a dimensional d > 0 family in H 3 . A complete list of such groups, signatures, and inclusions among the loci is given in [9]. The locus of curves C with the Klein viergrouppe V 4 ֒→ Aut (C) is a 3-dimensional locus in H 3 .…”

“…The locus of curves C with the Klein viergrouppe V 4 ֒→ Aut (C) is a 3-dimensional locus in H 3 . The appropriate invariants in this case are the dihedral invariants s 2 , s 3 , s 4 as defined in [9]. All the other cases can be described directly in terms of absolute invariants t 1 , .…”

“…The third section is also an introduction to genus 3 hyperelliptic curves with extra involutions and their dihedral invariants. Most of the material from that section can be found in [9]. The dihedral invariants s 2 , s 3 , s 4 which are defined in this section will be used in later sections to classify the Weierstrass points of genus 3 curves with extra automorphisms.…”