A Birational Contraction of Genus 2 Tails in the Moduli Space of Genus 4 Curves I

Abstract

We show that for a. (2 3, 7 10), the log canonical model M4(a) of the pair (M4, ad) is isomorphic to the moduli space M hs 4 of h-semistable curves constructed in [13] and that there is a birational morphism. : M hs 4. M4(2 3) which contracts the locus of curves C1. p C2 consisting of genus 2 curves meeting in a node p such that p is a Weierstrass point of C1 or C2. To obtain this morphism, we construct a compact moduli space M hs 2,1 of pointed genus 2 curves that have nodes, ordinary cusps, and tacnodes as singularities, and prove that it is isomorphic to Rulla's flip constructed in [23].