Bielliptic curves of genus 3 in the hyperelliptic moduli
In this paper we study bielliptic curves of genus 3 defined over an algebraically closed field \(k\) and the intersection of the moduli space \(\fancyscript{M}_3^b\) of such curves with the hyperelliptic moduli \(\fancyscript{H}_3\). Such intersection \(\fancyscript{S}\) 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 Shaska (Commun Algebra, to appear) and find a birational parametrization of \(\fancyscript{S}\). We also compute all possible subloci of curves for all possible automorphism group \(G\). Moreover, for every rational moduli point \(\mathfrak p \in \fancyscript{S}\), such that \(| \text{ Aut } (\mathfrak p ) | > 4\), 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 dihedral invariants absolute invariants 14h10 14h37 14q05