Some Remarks on the Hyperelliptic Moduli of Genus 3

Abstract

In 1967, Shioda [20] determined the ring of invariants of binary octavics and their syzygies using the symbolic method. We discover that the syzygies determined in [20] 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.

Key Words:

• Algebraic curves
• Binary forms
• Genus 3
• Invariants

2010 Mathematics Subject Classification:

• Primary 54C40, 14E20
• Secondary 46E25, 20C20

ACKNOWLEDGMENTS

The author thanks the anonymous referee for useful suggestions on improving this paper.

Notes

Communicated by P. Tiep.