Reduced invariants from cuspidal maps

Battistella, Luca; Carocci, Francesca; Manolache, Cristina

doi:10.1090/tran/8141

Reduced invariants from cuspidal maps
HTML articles powered by AMS MathViewer

by Luca Battistella, Francesca Carocci and Cristina Manolache PDF

Trans. Amer. Math. Soc. 373 (2020), 6713-6756 Request permission

Abstract:

We consider genus one enumerative invariants arising from the Smyth-Viscardi moduli space of stable maps from curves with nodes and cusps. We prove that these invariants are equal to the reduced genus one invariants of the quintic threefold, providing a modular interpretation of the latter.

References

Dan Abramovich, Martin Olsson, and Angelo Vistoli, Twisted stable maps to tame Artin stacks, J. Algebraic Geom. 20 (2011), no. 3, 399–477. MR 2786662, DOI 10.1090/S1056-3911-2010-00569-3
K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45–88. MR 1437495, DOI 10.1007/s002220050136
Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822, DOI 10.1007/978-3-642-51438-8
Pierrick Bousseau, Paires stables, invariants de Pandharipande-Thomas et invariants bps, Mèmoire de Master 2, École Normale Supérieure - Paris XI, 2014.
Huai-Liang Chang and Jun Li, Gromov-Witten invariants of stable maps with fields, Int. Math. Res. Not. IMRN 18 (2012), 4163–4217. MR 2975379, DOI 10.1093/imrn/rnr186
Huai-Liang Chang and Jun Li, An algebraic proof of the hyperplane property of the genus one GW-invariants of quintics, J. Differential Geom. 100 (2015), no. 2, 251–299. MR 3343833
Brian Conrad, Arithmetic moduli of generalized elliptic curves, J. Inst. Math. Jussieu 6 (2007), no. 2, 209–278. MR 2311664, DOI 10.1017/S1474748006000089
Kevin Costello, Higher genus Gromov-Witten invariants as genus zero invariants of symmetric products, Ann. of Math. (2) 164 (2006), no. 2, 561–601. MR 2247968, DOI 10.4007/annals.2006.164.561
Olivier Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001. MR 1841091, DOI 10.1007/978-1-4757-5406-3
B. Fantechi, L. Göttsche, and D. van Straten, Euler number of the compactified Jacobian and multiplicity of rational curves, J. Algebraic Geom. 8 (1999), no. 1, 115–133. MR 1658220
William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
Rajesh Gopakumar and Cumrun Vafa, M-theory and topological strings–I, arXiv:hep-th/9809187, 1998.
Rajesh Gopakumar and Cumrun Vafa, M-theory and topological strings–II, arXiv:hep-th/9812127, 1998.
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
Brendan Hassett and Donghoon Hyeon, Log canonical models for the moduli space of curves: the first divisorial contraction, Trans. Amer. Math. Soc. 361 (2009), no. 8, 4471–4489. MR 2500894, DOI 10.1090/S0002-9947-09-04819-3
Yi Hu and Jun Li, Genus-one stable maps, local equations, and Vakil-Zinger’s desingularization, Math. Ann. 348 (2010), no. 4, 929–963. MR 2721647, DOI 10.1007/s00208-010-0504-8
Bumsig Kim, Andrew Kresch, and Tony Pantev, Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee, J. Pure Appl. Algebra 179 (2003), no. 1-2, 127–136. MR 1958379, DOI 10.1016/S0022-4049(02)00293-1
Young-Hoon Kiem and Jun Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26 (2013), no. 4, 1025–1050. MR 3073883, DOI 10.1090/S0894-0347-2013-00768-7
Andrew Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999), no. 3, 495–536. MR 1719823, DOI 10.1007/s002220050351
Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné; Oxford Science Publications. MR 1917232
Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 39, Springer-Verlag, Berlin, 2000 (French). MR 1771927, DOI 10.1007/978-3-540-24899-6
Jun Li and Gang Tian, Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, Topics in symplectic $4$-manifolds (Irvine, CA, 1996) First Int. Press Lect. Ser., I, Int. Press, Cambridge, MA, 1998, pp. 47–83. MR 1635695
Jun Li and Aleksey Zinger, Gromov-Witten invariants of a quintic threefold and a rigidity conjecture, Pacific J. Math. 233 (2007), no. 2, 417–480. MR 2366383, DOI 10.2140/pjm.2007.233.417
Jun Li and Aleksey Zinger, On the genus-one Gromov-Witten invariants of complete intersections, J. Differential Geom. 82 (2009), no. 3, 641–690. MR 2534991
Cristina Manolache, Virtual pull-backs, J. Algebraic Geom. 21 (2012), no. 2, 201–245. MR 2877433, DOI 10.1090/S1056-3911-2011-00606-1
Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
R. Pandharipande, Hodge integrals and degenerate contributions, Comm. Math. Phys. 208 (1999), no. 2, 489–506. MR 1729095, DOI 10.1007/s002200050766
Dhruv Ranganathan, Keli Santos-Parker, and Jonathan Wise, Moduli of stable maps in genus one and logarithmic geometry, I, Geom. Topol. 23 (2019), no. 7, 3315–3366. MR 4046967, DOI 10.2140/gt.2019.23.3315
Jean-Pierre Serre, Local algebra, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. Translated from the French by CheeWhye Chin and revised by the author. MR 1771925, DOI 10.1007/978-3-662-04203-8
David Ishii Smyth, Modular compactifications of the space of pointed elliptic curves I, Compos. Math. 147 (2011), no. 3, 877–913. MR 2801404, DOI 10.1112/S0010437X10005014
David Ishii Smyth, Modular compactifications of the space of pointed elliptic curves II, Compos. Math. 147 (2011), no. 6, 1843–1884. MR 2862065, DOI 10.1112/S0010437X11005549
David Ishii Smyth, Towards a classification of modular compactifications of $\scr {M}_{g,n}$, Invent. Math. 192 (2013), no. 2, 459–503. MR 3044128, DOI 10.1007/s00222-012-0416-1
The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu, 2020.
Ravi Vakil, The enumerative geometry of rational and elliptic curves in projective space, J. Reine Angew. Math. 529 (2000), 101–153. MR 1799935, DOI 10.1515/crll.2000.094
Michael Viscardi, Alternate compactifications of the moduli space of genus one maps, Manuscripta Math. 139 (2012), no. 1-2, 201–236. MR 2959678, DOI 10.1007/s00229-011-0513-2
Ravi Vakil and Aleksey Zinger, A natural smooth compactification of the space of elliptic curves in projective space, Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 53–59. MR 2320682, DOI 10.1090/S1079-6762-07-00174-6
Ravi Vakil and Aleksey Zinger, A desingularization of the main component of the moduli space of genus-one stable maps into $\Bbb P^n$, Geom. Topol. 12 (2008), no. 1, 1–95. MR 2377245, DOI 10.2140/gt.2008.12.1
Jie Wang, Deformations of pairs $(X,L)$ when $X$ is singular, Proc. Amer. Math. Soc. 140 (2012), no. 9, 2953–2966. MR 2917069, DOI 10.1090/S0002-9939-2011-11230-9
Aleksey Zinger, On the structure of certain natural cones over moduli spaces of genus-one holomorphic maps, Adv. Math. 214 (2007), no. 2, 878–933. MR 2349722, DOI 10.1016/j.aim.2007.03.009
Aleksey Zinger, Standard versus reduced genus-one Gromov-Witten invariants, Geom. Topol. 12 (2008), no. 2, 1203–1241. MR 2403808, DOI 10.2140/gt.2008.12.1203
Aleksey Zinger, Reduced genus-one Gromov-Witten invariants, J. Differential Geom. 83 (2009), no. 2, 407–460. MR 2577474
Aleksey Zinger, The reduced genus 1 Gromov-Witten invariants of Calabi-Yau hypersurfaces, J. Amer. Math. Soc. 22 (2009), no. 3, 691–737. MR 2505298, DOI 10.1090/S0894-0347-08-00625-5
Aleksey Zinger, A sharp compactness theorem for genus-one pseudo-holomorphic maps, Geom. Topol. 13 (2009), no. 5, 2427–2522. MR 2529940, DOI 10.2140/gt.2009.13.2427