Birational contraction of genus two tails in the moduli space of genus four curves I

Abstract

We show that for $\alpha \in (2/3,7/10)$, the log canonical model ${\bar{M}}_4(\alpha )$ of the pair $({\bar{M}}_4,\alpha \delta )$ is isomorphic to the moduli space ${\bar{M}}_4^{hs}$ of h-semistable curves, and that there is a birational morphism $\Xi :{\bar{M}}_4^{hs}\to {\bar{M}}_4(2/3)$ that contracts the locus of curves $C_1{\cup }_p{C}_2$ consisting of genus two curves meeting in a node $p$ such that $p$ is a Weierstrass point of $C_1$ or $C_2$. To obtain this morphism, we construct a compact moduli space ${\bar{M}}_{2,1}^{hs}$ of pointed genus two curves that have nodes, ordinary cusps and tacnodes as singularity, and prove that it is isomorphic to Rulla's flip constructed in his thesis.

References in corpus (6)

• Log minimal model program for the moduli space of stable curves: The first flip
• On Log Canonical Models of the Moduli Space of Stable Pointed Curves
• A geometric invariant theory construction of moduli spaces of stable maps
• Towards a classification of modular compactifications of the moduli space of curves
• Groebner techniques for low degree Hilbert stability
• Chow Stability of Curves of Genus 4 in P^3

Cited by in corpus (6)

• Log minimal model program for the moduli space of stable curves: The second flip
• GIT Constructions of Log Canonical Models of M_g
• Log canonical models and variation of GIT for genus four canonical curves
• The final log canonical model of the moduli space of stable curves of genus four
• An outline of the log minimal model program for the moduli space of curves
• Stability of genus five canonical curves