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ON THE FIRST STEPS OF THE MINIMAL MODEL PROGRAM FOR THE MODULI SPACE OF STABLE POINTED CURVES

Part of: Birational geometry Families, fibrations Curves

Published online by Cambridge University Press:  22 April 2021

Giulio Codogni ,
Luca Tasin  and
Filippo Viviani Open the ORCID record for Filippo Viviani [Opens in a new window]
Giulio Codogni
Affiliation:
Dipartimento di Matematica, Università Tor Vergata, Via della Ricerca Scientica 1, 00133 Roma, Italy ([email protected])
Luca Tasin
Affiliation:
Dipartimento di Matematica F. Enriques, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy ([email protected])
Filippo Viviani
Affiliation:
Dipartimento di Matematica, Università Tor Vergata, Via della Ricerca Scientica 1, 00133 Roma, Italy ([email protected]) Dipartimento di Matematica e Fisica, Università Roma Tre, 00146 Roma, Italy ([email protected])
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Abstract

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The aim of this paper is to study all the natural first steps of the minimal model program for the moduli space of stable pointed curves. We prove that they admit a modular interpretation, and we study their geometric properties. As a particular case, we recover the first few Hassett–Keel log canonical models. As a by-product, we produce many birational morphisms from the moduli space of stable pointed curves to alternative modular projective compactifications of the moduli space of pointed curves.

Keywords

Minimal model programmoduli space of curvesmodular compactifications

MSC classification

Primary: 14H10: Families, moduli (algebraic)
Secondary: 14E30: Minimal model program (Mori theory, extremal rays) 14D23: Stacks and moduli problems 14H20: Singularities, local rings
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 22 , Issue 1 , January 2023 , pp. 145 - 211
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Abramovich, D., Olsson, M. and Vistoli, A., Tame stacks in positive characteristic, Ann. Inst. Fourier (Grenoble) 58(4) (2008), 1057–1091.CrossRefGoogle Scholar
Abramovich, D. and Vistoli, A., Compactifying the space of stable maps, J. Amer. Math. Soc. 15(1) (2002), 27–75.CrossRefGoogle Scholar
Alper, J., On the local quotient structure of Artin stacks, J. Pure Appl. Algebra 214(9) (2010), 1576–1591.CrossRefGoogle Scholar
Alper, J., Good moduli spaces for Artin stacks, Ann. Inst. Fourier (Grenoble) 63(6) (2013), 2349–2402.CrossRefGoogle Scholar
Alper, J., Adequate moduli spaces and geometrically reductive group schemes, Algebr. Geom. 1(4) (2014), 489–531.CrossRefGoogle Scholar
Alper, J., Fedorchuk, M. and Smyth, D. I., Singularities with ${G}_m$ -action and the log minimal model program for ${\overline{M}}_g$ , J. Reine Angew. Math. 721 (2016), 1–41.CrossRefGoogle Scholar
Alper, J., Fedorchuk, M. and Smyth, D. I., Second flip in the Hassett–Keel program: Projectivity, Int. Math. Res. Not. IMRN 24 (2017), 7375–7419.Google Scholar
Alper, J., Fedorchuk, M. and Smyth, D. I., Second flip in the Hassett-Keel program: Existence of good moduli spaces, Compos. Math. 153(8) (2017), 1584–1609.CrossRefGoogle Scholar
Alper, J., Fedorchuk, M., Smyth, D. I. and van der Wyck, F.,Second flip in the Hassett-Keel program: A local description, Compos. Math. 153(8) (2017), 1547–1583.CrossRefGoogle Scholar
Alper, J., Hall, J. and Rydh, D., A Luna étale slice theorem for algebraic stacks, Ann. of Math. (2) 191(3) (2020), 675–738.CrossRefGoogle Scholar
Alper, J., Halpern-Leistner, D. and Heinloth, J., ‘Existence of moduli spaces for algebraic stacks’, Preprint, 2018, http://arxiv.org/abs/1812.01128.Google Scholar
Alper, J. and Hyeon, D., GIT constructions of log canonical models of ${\overline{M}}_g$ , in Compact Moduli Spaces and Vector Bundles , Contemporary Mathematics, 564, pp. 87–106 (American Mathematical Society, Providence, RI, 2012).CrossRefGoogle Scholar
Alper, J., Smyth, D. I. and Van der Wyck, F., ‘Weakly proper moduli stacks of curves’, Preprint, 2011, http://arxiv.org/abs/1012.0538.Google Scholar
Ambro, F. and Kollar, J., ‘Minimal models of semi-log-canonical pairs’, in Moduli of K-stable varieties, 1–13, Springer INdAM Ser., 31, Springer, Cham, 2019.CrossRefGoogle Scholar
Arbarello, E., Cornalba, M. and Griffiths, P. A., Geometry of Algebraic Curves. Volume II, Grundlehren der Mathematischen Wissenschaften, 268 (Springer, Heidelberg, 2011). With a contribution by Harris, J. D..CrossRefGoogle Scholar
Bayer, A. and Macrì, E., Mmp for moduli of sheaves on k3s via wall-crossing: Nef and movable cones, Lagrangian fibrations, Invent. Math. 198(3) (2014), 505–590.CrossRefGoogle Scholar
Bertram, A. and Coskun, I., The birational geometry of the Hilbert scheme of points on surfaces, in Birational Geometry, Rational Curves, and Arithmetic, Simons Symposia, pp. 15–55 (Springer, Cham, Switzerland, 2013).Google Scholar
Birkar, C., Cascini, P., Hacon, C. and MKernan, J., Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23(2) (2010), 405–468.CrossRefGoogle Scholar
Casagrande, C., Codogni, G. and Fanelli, A., The blow-up of ${\mathbb{P}}^4$ at 8 points and its Fano model, via vector bundles on a degree 1 del Pezzo surface, Rev. Mat. Complut. 32(2) (2019), 475–529.CrossRefGoogle Scholar
Codogni, G., Tasin, L. and Viviani, F., ‘On some modular contractions of the moduli space of stable pointed curves’, Preprint, 2019, https://arxiv.org/abs/1904.13212, To be published on Algebra and Number Theory.Google Scholar
Coskun, I. and Huizenga, J., The birational geometry of the moduli spaces of sheaves on ${\mathbb{P}}^2$ , in Proceedings of the Gökova Geometry-Topology Conference 2014, Gökova Geometry/Topology Conference (GGT), pp. 114–155 (International Press, Boston, Gökova, Country, 2015).Google Scholar
Coskun, I. and Huizenga, J., The nef cone of the moduli space of sheaves and strong Bogomolov inequalities, Israel J. Math. 226(1) (2018), 205–236.CrossRefGoogle Scholar
Chiang-Hsieh, H.-J. and Lipman, J., A numerical criterion for simultaneous normalization, Duke Math. J. 133(2) (2006), 347–390.CrossRefGoogle Scholar
Debarre, O., Higher-Dimensional Algebraic Geometry, Universitext (Springer-Verlag, New York, 2001).CrossRefGoogle Scholar
Demazure, M., Pinkham, H. C. and Teissier, B. (eds.), Séminaire sur les Singularités des Surfaces, Held at the Centre de Mathématiques de l’École Polytechnique, Palaiseau, 1976–1977 , Lecture Notes in Mathematics, 777 (Springer, Berlin, 1980).Google Scholar
Fedorchuk, M. and Smyth, D. I., Alternate compactifications of moduli spaces of curves, in Handbook of Moduli, . Vol. I, Advanced Lectures in Mathematics, 24, pp. 331–413 (International Press, Somerville, MA, 2013).Google Scholar
Fringuelli, R. and Viviani, F., ‘On the Picard group scheme of the moduli stack of stable pointed curves’, Preprint, 2020, http://arxiv.org/abs/2005.06920.Google Scholar
Fujino, O., Applications of Kawamata’s positivity theorem, Proc. Japan Acad. Ser. A Math. Sci. 75(6) (1999), 75–79.CrossRefGoogle Scholar
Gibney, A., Keel, S. and Morrison, I., Towards the ample cone of ${\overline{M}}_{g,n}$ , J. Amer. Math. Soc. 15(2) (2002), 273–294.CrossRefGoogle Scholar
Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Publ. Math. Inst. Hautes Études Sci. 32 (1967), 361 pp.CrossRefGoogle Scholar
Hartshorne, R., Deformation Theory, Graduate Texts in Mathematics, 257 (Springer, New York, 2010).CrossRefGoogle Scholar
Hassett, B., Classical and minimal models of the moduli space of curves of genus two, in Geometric Methods in Algebra and Number Theory , Progress in Mathematics, 235, pp. 169–192 (Birkhäuser Boston, Boston, 2005).CrossRefGoogle Scholar
Hassett, B. and Hyeon, D., Log canonical models for the moduli space of curves: The first divisorial contraction, Trans. Amer. Math. Soc. 361(8) (2009), 4471–4489.CrossRefGoogle Scholar
Hassett, B. and Hyeon, D., Log minimal model program for the moduli space of stable curves: The first flip, Ann. of Math. (2) 177(3) (2013), 911–968.CrossRefGoogle Scholar
Huizenga, J., Birational geometry of moduli spaces of sheaves and Bridgeland stability, in Surveys on Recent Developments in Algebraic Geometry , Proceedings of Symposia in Pure Mathematics, 95, pp. 101–148 (American Mathematical Society, Providence, RI, 2017).Google Scholar
Hyeon, D. and Lee, Y., Stability of tri-canonical curves of genus two, Math. Ann. 337(2) (2007), 479–488.CrossRefGoogle Scholar
Hyeon, D. and Morrison, I., Stability of tails and 4-canonical models, Math. Res. Lett. 17(4) (2010), 721–729.CrossRefGoogle Scholar
Keel, S. and Mori, S., Quotients by groupoids, Ann. of Math. (2) 145(1) (1997), 193–213.CrossRefGoogle Scholar
Kempf, G. R., Instability in invariant theory, Ann. of Math. (2) 108(2) (1978), 299–316.CrossRefGoogle Scholar
Kollár, J., Projectivity of complete moduli, J. Differential Geom. 32(1) (1990), 235–268.CrossRefGoogle Scholar
Kollár, J., Exercises in the birational geometry of algebraic varieties, in Analytic and Algebraic Geometry , IAS/Park City Mathematics Series, 17, pp. 495–524 (American Mathematical Society, Providence, RI, 2010).CrossRefGoogle Scholar
Kollár, J. and Mori, S., Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, 134 (Cambridge University Press, Cambridge, UK, 1998). With the collaboration of Clemens, C. H. and Corti, A.; translated from the 1998 Japanese original.CrossRefGoogle Scholar
Li, C. and Zhao, X., Birational models of moduli spaces of coherent sheaves on the projective plane, Geom. Topol. 23(1) (2019), 347–426.CrossRefGoogle Scholar
Macrì, E. and Schmidt, B., Lectures on Bridgeland stability, in Moduli of Curves , Lecture Notes of the Unione Matematica Italiana, 21, pp. 139–211 (Springer, Cham, Switzerland, 2017).CrossRefGoogle Scholar
Matsuki, K., Introduction to the Mori Program, Universitext (Springer-Verlag, New York, 2002).CrossRefGoogle Scholar
Mori, S., Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116(1) (1982), 133–176.CrossRefGoogle Scholar
Moriwaki, A., The $\mathbb{Q}$ -Picard group of the moduli space of curves in positive characteristic, Internat. J. Math. 12(5) (2001), 519–534.CrossRefGoogle Scholar
Nuer, H., Projectivity and birational geometry of Bridgeland moduli spaces on an Enriques surface, Proc. Lond. Math. Soc. (3) 113(3) (2016), 345–386.CrossRefGoogle Scholar
Schubert, D., A new compactification of the moduli space of curves, Compos. Math. 78(3) (1991), 297–313.Google Scholar
Sernesi, E., Deformations of Algebraic Schemes, Grundlehren der Mathematischen Wissenschaften, 334 (Springer-Verlag, Berlin, 2006).Google Scholar
Smyth, D. I., Modular compactifications of the space of pointed elliptic curves I, Compos. Math. 147(3) (2011), 877–913.CrossRefGoogle Scholar
The Stacks Project Authors, Stacks Project, 2018, http://stacks.math.columbia.edu.Google Scholar
van Opstall, M. A. and Veliche, R., Maximally symmetric stable curves, Michigan Math. J. 55(3) (2007), 513–534.Google Scholar
Yoshioka, K., Bridgeland’s stability and the positive cone of the moduli spaces of stable objects on an abelian surface, in Development of Moduli Theory—Kyoto 2013 , Advanced Studies in Pure Mathematics, 69, pp. 473–537 (Mathematical Society of Japan, Tokyo, 2016).CrossRefGoogle Scholar