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Some remarks on the hyperelliptic moduli of genus 3

Tanush Shaska

I-AIMS paper 2012-1 · Published in Comm. Algebra 42 (2014), 4110–4130 · 2014

In 1967, Shioda determined the ring of invariants of binary octavics and their syzygies using the symbolic method. We discover that the syzygies determined by Shioda are incorrect. In this paper, we compute the correct equations among the invariants of the binary octavics and give necessary and sufficient conditions for two genus 3 hyperelliptic curves to be isomorphic over an algebraically closed field k, char k ≠ 2, 3, 5, 7. For the first time, an explicit equation of the hyperelliptic moduli for genus 3 is computed in terms of absolute invariants.

MSC classes: Primary 14H10; Secondary 14H30, 13A50, 14H45

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