Log minimal model program for the moduli space of stable curves: the first flip

Abstract

Abstract. We give a geometric invariant theory (GIT) construction of the log canonical model M g (α) of the pairs (M g , αδ) for α ∈ (7/10 − ǫ, 7/10] for small ǫ ∈ Q + . We show that M g (7/10) is isomorphic to the GIT quotient of the Chow variety bicanonical curves; M g (7/10 − ǫ) is isomorphic to the GIT quotient of the asymptotically-linearized Hilbert scheme of bicanonical curves. In each case, we completely classify the (semi)stable curves and their orbit closures. Chow semistable curves have ordinary cusps and tacnodes as singularities but do not admit elliptic tails. Hilbert semistable curves satisfy further conditions, e.g., they do not contain elliptic bridges. We show that there is a small contraction Ψ : M g (7/10 + ǫ) → M g (7/10) that contracts the locus of elliptic bridges. Moreover, by using the GIT interpretation of the log canonical models, we construct a small contraction Ψ + : M g (7/10 − ǫ) → M g (7/10) that is the Mori flip of Ψ.

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

“…Hassett and Hyeon have explicitly constructed the log minimal models M g (α) for α ≥ 7 10 −ǫ (see [HH09,HH08]). Hyeon and Lee have also described the next stage of the program in the specific case of genus 4 (cf.…”

Good moduli spaces for Artin stacks

“…Hassett and Hyeon show in [14] for g 4 (the g = 3 case is handled in [17]) that a flip occurs at the next step in the log minimal model program at α = 7/10. Furthermore, they give modular interpretations for M g (7/10) and M g (7/10 − ) as the good moduli spaces (but not coarse moduli spaces) for the stack of Chow semi-stable curves (where curves are allowed as singularities nodes, cusps, and tacnodes do not admit elliptic tails) and Hilbert semi-stable curves (which are Chow semi-stable curves not admitting elliptic bridges), respectively.…”

Quartic del Pezzo surfaces over function fields of curves

“…The limit It is a fundamental result of Geometric Invariant Theory that orbit specializations are governed by specializations under one-parameter subgroups arising as automorphisms of objects with closed orbit (see [15,Proposition 4.2] whose action on U ss exchanges these non-closed G -orbits. Thus there is a single isomorphism class of del Pezzo surfaces in M with X 0 as a one-parameter limit.…”