Some Remarks on the Hyperelliptic Moduli of Genus 3

2014-04-23T18:20:12Z (GMT) by T. Shaska
<div><p>In 1967, Shioda [<a href="#CIT0020" target="_blank">20</a>] determined the ring of invariants of binary octavics and their syzygies using the symbolic method. We discover that the syzygies determined in [<a href="#CIT0020" target="_blank">20</a>] 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 <i>k</i>, char <i>k</i> ≠ 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.</p> </div>