×

Boundedness of the space of stable logarithmic maps. (English) Zbl 1453.14081

Summary: We prove that the moduli space of stable logarithmic maps from logarithmic curves to a fixed target logarithmic scheme is a proper algebraic stack when the target scheme is projective with fine and saturated logarithmic structure. This was previously known only with further restrictions on the logarithmic structure of the target.

MSC:

14H10 Families, moduli of curves (algebraic)
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14D23 Stacks and moduli problems
14A20 Generalizations (algebraic spaces, stacks)
14A21 Logarithmic algebraic geometry, log schemes
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abramovich, D., Cadman, C., Fantechi, B., Wise, J.: Expanded degenerations and pairs. Comm. Algebra 41, 2346-2386 (2013)Zbl 1326.14020 MR 3225278 · Zbl 1326.14020
[2] Abramovich, D., Cadman, C., Wise, J.: Relative and orbifold Gromov-Witten invariants.arXiv:1004.0981(2010) · Zbl 1493.14093
[3] Abramovich, D., Caporaso, L., Payne, S.: The tropicalization of the moduli space of curves. Ann. Sci. École Norm. Sup. (4) 48, 765-809 (2015)Zbl 06502666 MR 3377065 · Zbl 1410.14049
[4] Abramovich, D., Chen, Q.: Stable logarithmic maps to Deligne-Faltings pairs II. Asian J. Math. 18, 465-488 (2014)Zbl 1321.14025 MR 3257836 [ACG+13] Abramovich, D., Chen, Q., Gillam, D., Huang, Y., Olsson, M., Satriano, M., Sun, S.: Logarithmic geometry and moduli, In: Handbook of Moduli. Vol. I, Adv. Lect. Math. 24, Int. Press, Somerville, MA, 1-61 (2013)Zbl 1322.14023 MR 3184161 · Zbl 1322.14023
[5] Abramovich, D., Chen, Q., Gillam, W. D., Marcus, S.: The evaluation space of logarithmic stable maps.arXiv:1012.5416(2010) [ACM+16] Abramovich, D., Chen, Q., Marcus, S., Ulirsch, M., Wise, J.: Skeletons and fans of logarithmic structures. In: Nonarchimedean and Tropical Geometry, M. Baker and S. Payne (eds.), Simons Symposia, Springer, 287-336 (2016)Zbl 06654330 · Zbl 1364.14047
[6] Abramovich, D., Karu, K.: Weak semistable reduction in characteristic 0. Invent. Math. 139, 241-273 (2000)Zbl 0958.14006 MR 1738451 · Zbl 0958.14006
[7] Abramovich, D., Marcus, S., Wise, J.: Comparison theorems for Gromov-Witten invariants of smooth pairs and of degenerations. Ann. Inst. Fourier (Grenoble) 64, 1611- 1667 (2014)Zbl 1317.14123 MR 3329675 · Zbl 1317.14123
[8] Abramovich, D., Matsuki, K., Rashid, S.: A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension. Tohoku Math. J. (2) 51, 489- 537 (1999)Zbl 0991.14020 MR 1725624 · Zbl 0991.14020
[9] Abramovich, D., Temkin, M.: Functorial factorization of birational maps for qe schemes in characteristic 0, part 2: proof of the main result. Manuscript in preparation (2013) · Zbl 1442.14042
[10] Abramovich, D., Wise, J.: Invariance in logarithmic Gromov-Witten theory. arXiv:1306.1222(2013) · Zbl 1420.14124
[11] Cavalieri, R., Chan, M., Ulirsch, M., Wise, J.: A moduli stack of tropical curves. arXiv:1704.03806(2017) · Zbl 1444.14005
[12] Cavalieri, R., Marcus, S., Wise, J.: Polynomial families of tautological classes on Mrtg,n. J. Pure Appl. Algebra 216, 950-981 (2012)Zbl 1273.14053 MR 2864866 · Zbl 1273.14053
[13] Chen, Q.: Stable logarithmic maps to Deligne-Faltings pairs I. Ann. of Math. (2) 180, 455-521 (2014)Zbl 1311.14028 MR 3224717 · Zbl 1311.14028
[14] Cox, D. A., Little, J. B., Schenck, H. K.: Toric Varieties. Grad. Stud. Math. 124, Amer. Math. Soc., Providence, RI (2011)Zbl 1223.14001 MR 2810322 · Zbl 1223.14001
[15] Fulton, W.: Intersection Theory. 2nd ed., Ergeb. Math. Grenzgeb. 2, Springer, Berlin (1998)Zbl 0885.14002 MR 1644323 · Zbl 0885.14002
[16] Gillam, W. D.: Logarithmic stacks and minimality. Int. J. Math. 23, 1250069, 38 pp. (2012)Zbl 1248.18008 MR 2945649 · Zbl 1248.18008
[17] Gross, M., Siebert, B.: Logarithmic Gromov-Witten invariants. J. Amer. Math. Soc. 26, 451-510 (2013)Zbl 1281.14044 MR 3011419 · Zbl 1281.14044
[18] Grothendieck, A., Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes. Inst. Hautes Études Sci. Publ. Math. 8, 222 pp. (1961)Zbl 0118.36206 MR 0217084 Boundedness of the space of stable logarithmic maps2809
[19] Kato, F.: Exactness, integrality, and log modifications.arXiv.org:math/9906124(1999)
[20] Kato, F.: Log smooth deformation and moduli of log smooth curves. Int. J. Math. 11, 215-232 (2000)Zbl 1100.14502 MR 1754621 · Zbl 1100.14502
[21] Kato, K.: Toric singularities. Amer. J. Math. 116, 1073-1099 (1994)Zbl 0832.14002 MR 1296725 · Zbl 0832.14002
[22] Kempf, G., Knudsen, F. F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings. I. Lecture Notes in Math. 339, Springer, Berlin (1973)Zbl 0271.14017 MR 0335518 · Zbl 0271.14017
[23] Milne, J. S.: Étale Cohomology. Princeton Math. Ser. 33, Princeton Univ. Press, Princeton, NJ (1980)Zbl 0433.14012 MR 559531 · Zbl 0433.14012
[24] Olsson, M. C.: Logarithmic geometry and algebraic stacks. Ann. Sci. École Norm. Sup. (4) 36, 747-791 (2003)Zbl 1069.14022 MR 2032986 · Zbl 1069.14022
[25] Olsson, M. C.: The logarithmic cotangent complex. Math. Ann. 333, 859-931 (2005) Zbl 1095.14016 MR 2195148 · Zbl 1095.14016
[26] Olsson, M. C.: (Log) twisted curves. Compos. Math. 143, 476-494 (2007) Zbl 1138.14017 MR 2309994 · Zbl 1138.14017
[27] Thuillier, A.: Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels. Manuscripta Math. 123, 381-451 (2007)Zbl 1134.14018 MR 2320738 · Zbl 1134.14018
[28] Ulirsch, M.: Functorial tropicalization of logarithmic schemes: the case of constant coefficients. Proc. London Math. Soc. 114, 1081-1113 (2017) · Zbl 1419.14088
[29] Wise, J.: Uniqueness of minimal morphisms of logarithmic schemes.arXiv: 1601.02968(2016) · Zbl 1343.14020
[30] Wise, J.: Moduli of morphisms of logarithmic schemes. Algebra Number Theory 10, 695-735 (2016)Zbl 1343.14020 MR 3519093 · Zbl 1343.14020
[31] Włodarczyk, J.: Toroidal varieties and the weak factorization theorem. Invent. Math. 154, 223-331 (2003)Zbl 1130.14014 MR 2013783 · Zbl 1130.14014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.