Moduli of stable maps in genus one and logarithmic geometry, I

Ranganathan, Dhruv¹ ; Santos-Parker, Keli² ; Wise, Jonathan³

¹ Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, United Kingdom
² Department of Mathematics, University of Colorado, Boulder, CO, United States, Medical School, University of Michigan, Ann Arbor, MI, United States
³ Department of Mathematics, University of Colorado, Boulder, CO, United States

Geometry & topology, Tome 23 (2019) no. 7, pp. 3315-3366.

Résumé

This is the first in a pair of papers developing a framework for the application of logarithmic structures in the study of singular curves of genus $1$. We construct a smooth and proper moduli space dominating the main component of Kontsevich’s space of stable genus $1$ maps to projective space. A variation on this theme furnishes a modular interpretation for Vakil and Zinger’s famous desingularization of the Kontsevich space of maps in genus $1$. Our methods also lead to smooth and proper moduli spaces of pointed genus $1$ quasimaps to projective space. Finally, we present an application to the log minimal model program for ${\overline{\mathcal{ℳ}}}_{1,n}$. We construct explicit factorizations of the rational maps among Smyth’s modular compactifications of pointed elliptic curves.

Détail

DOI : 10.2140/gt.2019.23.3315

Classification : 14N35, 14D23
Keywords: stable maps, quasimaps, elliptic singularities, logarithmic geometry

Bibliographie

[1] D Abramovich, C Cadman, B Fantechi, J Wise, Expanded degenerations and pairs, Comm. Algebra 41 (2013) 2346 | DOI

[2] D Abramovich, B Hassett, Stable varieties with a twist, from: "Classification of algebraic varieties" (editors C Faber, G van der Geer, E Looijenga), Eur. Math. Soc. (2011) 1 | DOI

[3] M Artin, Théorème de changement de base pour un morphisme propre, from: "Théorie des topos et cohomologie étale des schémas, Tome 3 (SGA )" (editors M Artin, A Grothendieck, J L Verdier), Lecture Notes in Math. 305, Springer (1973) 79

[4] P Belmans, A J De Jong, Others, The Stacks project, electronic reference (2017)

[5] R Cavalieri, M Chan, M Ulirsch, J Wise, A moduli stack of tropical curves, preprint (2017)

[6] I Ciocan-Fontanine, B Kim, Moduli stacks of stable toric quasimaps, Adv. Math. 225 (2010) 3022 | DOI

[7] Y Cooper, The geometry of stable quotients in genus one, Math. Ann. 361 (2015) 943 | DOI

[8] T Foster, D Ranganathan, M Talpo, M Ulirsch, Logarithmic Picard groups, chip firing, and the combinatorial rank, Math. Z. 291 (2019) 313 | DOI

[9] M Gross, B Siebert, Logarithmic Gromov–Witten invariants, J. Amer. Math. Soc. 26 (2013) 451 | DOI

[10] A Grothendieck, Eléments de géométrie algébrique, IV : Étude locale des schémas et des morphismes de schémas, I, Inst. Hautes Études Sci. Publ. Math. 20 (1964) 5

[11] A Grothendieck, Eléments de géométrie algébrique, IV : Étude locale des schémas et des morphismes de schémas, II, Inst. Hautes Études Sci. Publ. Math. 24 (1965) 5

[12] A Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique, IV : Les schémas de Hilbert, from: "Séminaire Bourbaki ", W A Benjamin (1966) 249

[13] J Hall, D Rydh, Coherent Tannaka duality and algebraicity of Hom-stacks, Algebra Number Theory 13 (2019) 1633 | DOI

[14] R Hartshorne, Algebraic geometry, 52, Springer (1977) | DOI

[15] D Holmes, Néron models of jacobians over base schemes of dimension greater than 1, J. Reine Angew. Math. 747 (2019) 109 | DOI

[16] Y Hu, J Li, Genus-one stable maps, local equations, and Vakil–Zinger’s desingularization, Math. Ann. 348 (2010) 929 | DOI

[17] T Kajiwara, Logarithmic compactifications of the generalized Jacobian variety, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 40 (1993) 473 | DOI

[18] F Kato, Exactness, integrality, and log modifications, preprint (1999)

[19] F Kato, Log smooth deformation and moduli of log smooth curves, Internat. J. Math. 11 (2000) 215 | DOI

[20] B Kim, Logarithmic stable maps, from: "New developments in algebraic geometry, integrable systems and mirror symmetry" (editors M H Saito, S Hosono, K Yoshioka), Adv. Stud. Pure Math. 59, Math. Soc. Japan (2010) 167 | DOI

[21] S L Kleiman, Relative duality for quasicoherent sheaves, Compositio Math. 41 (1980) 39

[22] J Kollár, S Mori, Birational geometry of algebraic varieties, 134, Cambridge Univ. Press (1998) | DOI

[23] R Lazarsfeld, Positivity in algebraic geometry, I : Classical setting — line bundles and linear series, 48, Springer (2004) | DOI

[24] C Manolache, Virtual pull-backs, J. Algebraic Geom. 21 (2012) 201 | DOI

[25] A Marian, D Oprea, R Pandharipande, The moduli space of stable quotients, Geom. Topol. 15 (2011) 1651 | DOI

[26] D Mumford, Abelian varieties, 5, Hindustan (1974)

[27] B Parker, Exploded manifolds, Adv. Math. 229 (2012) 3256 | DOI

[28] K S Parker, Semistable modular compactifications of moduli spaces of genus one curves, PhD thesis, University of Colorado at Boulder (2017)

[29] D Ranganathan, K Santos-Parker, J Wise, Moduli of stable maps in genus one and logarithmic geometry, II, Algebra Number Theory 13 (2019) 1765 | DOI

[30] D I Smyth, Modular compactifications of the space of pointed elliptic curves, I, Compos. Math. 147 (2011) 877 | DOI

[31] R Vakil, The enumerative geometry of rational and elliptic curves in projective space, J. Reine Angew. Math. 529 (2000) 101 | DOI

[32] R Vakil, A Zinger, A natural smooth compactification of the space of elliptic curves in projective space, Electron. Res. Announc. Amer. Math. Soc. 13 (2007) 53 | DOI

[33] R Vakil, A Zinger, A desingularization of the main component of the moduli space of genus-one stable maps into Pn, Geom. Topol. 12 (2008) 1 | DOI

[34] M Viscardi, Alternate compactifications of the moduli space of genus one maps, Manuscripta Math. 139 (2012) 201 | DOI

[35] A Zinger, The reduced genus 1 Gromov–Witten invariants of Calabi–Yau hypersurfaces, J. Amer. Math. Soc. 22 (2009) 691 | DOI

[36] A Zinger, A sharp compactness theorem for genus-one pseudo-holomorphic maps, Geom. Topol. 13 (2009) 2427 | DOI