A complete curve of genus 10 in the moduli space of curves of genus 3 and related questions

Abstract

We show the existence of a complete trigonal curve of geometric genus 10 in the moduli space $\mathcal{M}_3$ of smooth complex projective curves of genus $3$, and of a complete curve of geometric genus 102 through a general point of $\mathcal{M}_3$. We also show that the moduli space $\mathcal{A}_3$ of principally polarized complex abelian threefolds contains a complete curve whose normalization is a hyperelliptic curve of genus 4.

Disclosure

“[u1 , . . . , u6 ], which by a direct computation one can further check are non-zero. But then {f = 0} is a degree 8 hypersurface in C7 , which thus has degree of irrationality at most 8. Acknowledgments. The second-named author used the Gemini large language model to draft arguments for the generators and representation of Γ3 (1) (before finding appropriate references). The first-named author had a long conversation with ChatGPT 5.6 exploring possible curves in S invariant under”

PDF page 10

Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T18:53:43.027271Z

Structural counts