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

Abstract

Abstract. S. Kondo has constructed a ball quotient compactification for the moduli space of non-hyperelliptic genus four curves. In this paper, we show that this space essentially coincides with a GIT quotient of the Chow variety of canonically embedded genus four curves. More specifically, we give an explicit description of this GIT quotient, and show that the birational map from this space to Kondo's space is resolved by the blow-up of a single point. This provides a modular interpretation of the points in the boundary of Kondo's space. Connections with the slope nine space in the Hassett-Keel program are also discussed.

GIT Versus Baily-Borel Compactification for Quartic K3 Surfaces

“…However, this is a misleading point of view. We argue that the results in this paper have much more structure than those in the simpler geometric case discussed in [CMJL14, CMJL12] (for instance compare Corollary 1.4 to [CMJL12]). More generally, we believe that the Hassett-Keel-Looijenga program (for K3 type) has much more structure and is more regular than the Hassett-Keel program (for curves).…”

Chow Stability of Canonical Genus 4 Curves

“…We note thatC is a union ofC 1 andC 2 satisfying (a) eachC i is contained in 2 and L i,3 intersect at one point q i for each i = 1, 2, and (d) L 1,j and L 2,j meet at a point p j . From Section 11.3 in [4], it is induced thatC is Chow semistable.…”

“…[2], Theorem 3.1) classified Chow stable and semistable points in Chow 4,1 by using the GIT analysis for cubic threefolds. Our results are partial but we make a direct computation of the stability conditions on Chow 4,1 .…”

Log canonical models and variation of GIT for genus 4 canonical curves

“…, 8 17 ) M 4 ( 8 17 ) = { * } More specifically, i) the end point M 4 ( 8 17 + ǫ) is obtained via GIT for (3, 3) curves on P 1 × P 1 as discussed in [Fed12]; ii) the other end point M 4 ( 5 9 ) is obtained via GIT for the Chow variety of genus 4 canonical curves as discussed in [CMJL12]; iii) the remaining spaces M 4 (α) for α in the range 8 17 < α < 5 9 are obtained via appropriate Hilb m 4,1 quotients, with the exception of α = 23 44 . Thus in genus 4, the remaining unknown range for the Hassett-Keel program is the interval α ∈ ( 5 9 , 2 3 ).…”