Second flip in the Hassett-Keel program: a local description. (English) Zbl 1403.14039
The Minimal Model Program (MMP, also called Mori’s program) aims at classifying algebraic varieties via divisorial contractions and flips. If one runs the MMP for a moduli space, the resulting birational models often have modular meanings that correspond to parameterized objects with altered geometric structures. Hassett and Keel initiated the (log) MMP for the Deligne-Mumford moduli space of stable curves \(\overline{\mathcal M}_g\), which has played a significant role in the theory of curve moduli.
The first divisorial contraction and the first flip in the MMP for \(\overline{\mathcal M}_g\) were carried out in [B. Hassett and D. Hyeon, Trans. Am. Math. Soc. 361, No. 8, 4471–4489 (2009; Zbl 1172.14018)] and [B. Hassett and D. Hyeon, Ann. Math. (2) 177, No. 3, 911–968 (2013; Zbl 1273.14034)] respectively.
The paper under review is the first of three impressive papers in which the authors describe the second flip in the MMP for \(\overline{\mathcal M}_g\). In this paper the authors introduce new stability conditions for curves and show that they are deformation open by using variation of geometric invariant theory for the action of the automorphism group on the first-order deformation space of the underlying curve. As a consequence, this provides an algebraic stack structure for the resulting moduli spaces.
The first divisorial contraction and the first flip in the MMP for \(\overline{\mathcal M}_g\) were carried out in [B. Hassett and D. Hyeon, Trans. Am. Math. Soc. 361, No. 8, 4471–4489 (2009; Zbl 1172.14018)] and [B. Hassett and D. Hyeon, Ann. Math. (2) 177, No. 3, 911–968 (2013; Zbl 1273.14034)] respectively.
The paper under review is the first of three impressive papers in which the authors describe the second flip in the MMP for \(\overline{\mathcal M}_g\). In this paper the authors introduce new stability conditions for curves and show that they are deformation open by using variation of geometric invariant theory for the action of the automorphism group on the first-order deformation space of the underlying curve. As a consequence, this provides an algebraic stack structure for the resulting moduli spaces.
Reviewer: Dawei Chen (Chestnut Hill)
MSC:
| 14D23 | Stacks and moduli problems |
| 14H10 | Families, moduli of curves (algebraic) |
| 14L30 | Group actions on varieties or schemes (quotients) |
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