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Second flip in the Hassett–Keel program: a local description

Part of: Curves Families, fibrations Algebraic groups

Published online by Cambridge University Press:  18 May 2017

Jarod Alper ,
Maksym Fedorchuk ,
David Ishii Smyth  and
Frederick van der Wyck
Jarod Alper
Affiliation:
Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia email jarod.alper@anu.edu.au
Maksym Fedorchuk
Affiliation:
Mathematics Department, Boston College, 140 Commonwealth Ave, Chestnut Hill, MA 02467, USA email maksym.fedorchuk@bc.edu
David Ishii Smyth
Affiliation:
Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia email david.smyth@anu.edu.au
Frederick van der Wyck
Affiliation:
Goldman Sachs International, 120 Fleet Street, London EC4A 2BE, UK email frederick.vanderwyck@gmail.com
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Abstract

This is the first of three papers in which we give a moduli interpretation of the second flip in the log minimal model program for $\overline{M}_{g}$, replacing the locus of curves with a genus $2$ Weierstrass tail by a locus of curves with a ramphoid cusp. In this paper, for $\unicode[STIX]{x1D6FC}\in (2/3-\unicode[STIX]{x1D716},2/3+\unicode[STIX]{x1D716})$, we introduce new $\unicode[STIX]{x1D6FC}$-stability conditions for curves and prove that they are deformation open. This yields algebraic stacks $\overline{{\mathcal{M}}}_{g}(\unicode[STIX]{x1D6FC})$ related by open immersions $\overline{{\mathcal{M}}}_{g}(2/3+\unicode[STIX]{x1D716}){\hookrightarrow}\overline{{\mathcal{M}}}_{g}(2/3){\hookleftarrow}\overline{{\mathcal{M}}}_{g}(2/3-\unicode[STIX]{x1D716})$. We prove that around a curve $C$ corresponding to a closed point in $\overline{{\mathcal{M}}}_{g}(2/3)$, these open immersions are locally modeled by variation of geometric invariant theory for the action of $\text{Aut}(C)$ on the first-order deformation space of $C$.

Keywords

moduli of curvesHassett–Keel program

MSC classification

Primary: 14D23: Stacks and moduli problems 14H10: Families, moduli (algebraic) 14L30: Group actions on varieties or schemes (quotients)
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 8 , August 2017 , pp. 1547 - 1583
Copyright
© The Authors 2017 

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