The Final Log Canonical Model of the Moduli Space of Stable Curves of Genus 4

Abstract

Abstract. We describe the GIT quotient of the linear system of (3, 3) curves on P 1 × P 1 as the final non-trivial log canonical model of M 4, isomorphic to M 4(α) for 8/17 < α ≤ 29/60. We describe singular curves parameterized by M 4(29/60), and show that the rational map M 4 M 4(29/60) contracts the Petri divisor, in addition to the boundary divisors ∆1 and ∆2. This answers a question of Farkas.

The geometry of the ball quotient model of the moduli space of genus four curves

“…[HL10]). Finally, Fedorchuk [Fed11] has constructed the final nontrivial step in the Hassett-Keel program for g = 4 by using GIT for (3, 3) curves on P 1 × P 1 . In this section we identify the GIT quotient…”

“…The value of α corresponding to our space M GIT 4 is intermediary between the slopes occurring in [HL10] and [Fed11] respectively. Proof.…”

“…However, to our knowledge, Theorem 3.1 is the first complete analysis for GIT stability on Chow 4,1 , and also the first description of the Hassett-Keel space M 4 5 9 . We also point out a related GIT computation (also motivated by Hassett-Keel program): GIT for genus 4 curves viewed as (3, 3) curves on a smooth quadric due to Fedorchuk [Fed11].…”

A family of divisors on M¯g,n and their log canonical models

“…While this problem is far from complete except small genera cases [16,19], we have understood many different compactifications of M g . For example, see [17,18,13,11,21,4].…”

“…Items (4),(5), and (6) are simple set-theoretical observations. Since ϕ A,B is a composition of smooth blow-ups, items (10),(11), and (12) are easily deduced. The rest of them come from Mumford's relation.…”