Log minimal model program for the moduli space of stable curves: The first flip
By Brendan Hassett, Donghoon Hyeon. 2008
116 citations recorded by the source corpus.
We give a geometric invariant theory (GIT) construction of the log canonical model $\bar M_g(\alpha)$ of the pairs $(\bar M_g, \alpha \delta)$ for $\alpha \in (7/10 - \epsilon, 7/10]$ for small $\epsilon \in \mathbb Q_+$. We show that $\bar M_g(7/10)$ is isomorphic to the GIT quotient of the Chow variety bicanonical curves; $\bar M_g(7/10-\epsilon)$ 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 $\Psi: \bar M_g(7/10+\epsilon) \to \bar M_g(7/10)$ that contracts the locus of elliptic bridges. Moreover, by using the GIT interpretation of the log canonical models, we construct a small contraction $\Psi^+ : \bar M_g(7/10-\epsilon) \to \bar M_g(7/10)$ that is the Mori flip of $\Psi$.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/becfe3940017c3e85c9755ed8263b11b09610c05
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Log minimal model program for the moduli space of stable curves of genus three (2007, methodology): “This principle has been demonstrated in a series of work by Hassett and the authors [6, 8, 7, 9].”
- Good Moduli Spaces for Artin Stacks (2008, background): “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.”