I‑AIMS Institute for Artificial Intelligence
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Bielliptic curves of genus 3 in hyperelliptic moduli

Tanush Shaska, F. Thompson

I-AIMS paper 2013-7 · Published in Appl. Algebra Engrg. Comm. Comput. 24 (2013), 387–412 · 2013

In this paper we study bielliptic curves of genus 3 defined over an algebraically closed field k and the intersection of the moduli space ℳ₃ᵇ of such curves with the hyperelliptic moduli ℋ₃. 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 and find a birational parametrization of 𝒮. We also compute all possible subloci of curves for all possible automorphism groups 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.

MSC classes: Primary 14H10; Secondary 14H37, 14Q05

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