Geometry & Topology Volume 27, issue 3 (2023)

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A smooth compactification of the space of genus two curves in projective space: via logarithmic geometry and Gorenstein curves

Luca Battistella and Francesca Carocci

Geometry & Topology 27 (2023) 1203–1272

DOI: 10.2140/gt.2023.27.1203

Bibliography

1	D Abramovich, Q Chen, S Marcus, M Ulirsch, J Wise, Skeletons and fans of logarithmic structures, from: "Nonarchimedean and tropical geometry" (editors M Baker, S Payne), Springer (2016) 287 MR3702314
2	D Abramovich, Q Chen, S Marcus, J Wise, Boundedness of the space of stable logarithmic maps, J. Eur. Math. Soc. 19 (2017) 2783 MR3692887
3	D Abramovich, A Corti, A Vistoli, Twisted bundles and admissible covers, Comm. Algebra 31 (2003) 3547 MR2007376
4	D Abramovich, K Karu, Weak semistable reduction in characteristic 0, Invent. Math. 139 (2000) 241 MR1738451
5	D Abramovich, J Wise, Birational invariance in logarithmic Gromov–Witten theory, Compos. Math. 154 (2018) 595 MR3778185
6	J Alper, M Fedorchuk, D I Smyth, Singularities with 𝔾_m–action and the log minimal model program for ℳ_g, J. Reine Angew. Math. 721 (2016) 1 MR3574876
7	J Alper, M Fedorchuk, D I Smyth, F van der Wyck, Second flip in the Hassett–Keel program : a local description, Compos. Math. 153 (2017) 1547 MR3705268
8	O Amini, M Baker, E Brugallé, J Rabinoff, Lifting harmonic morphisms, II : Tropical curves and metrized complexes, Algebra Number Theory 9 (2015) 267 MR3320845
9	M Bainbridge, D Chen, Q Gendron, S Grushevsky, M Möller, The moduli space of multi-scale differentials, preprint (2019) arXiv:1910.13492
10	L Battistella, Alternative compactifications in low genus Gromov–Witten theory, PhD thesis, Imperial College London (2018)
11	L Battistella, Modular compactifications of ℳ_2,n with Gorenstein curves, Algebra Number Theory 16 (2022) 1547 MR4496075
12	L Battistella, F Carocci, A geographical study of ℳ₂(ℙ²,4)^main, Adv. Geom. 22 (2022) 463 MR4497183
13	L Battistella, N Nabijou, D Ranganathan, Curve counting in genus one: elliptic singularities and relative geometry, Algebr. Geom. 8 (2021) 637 MR4371543
14	D Bayer, D Eisenbud, Ribbons and their canonical embeddings, Trans. Amer. Math. Soc. 347 (1995) 719 MR1273472
15	K Behrend, B Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997) 45 MR1437495
16	P Belmans, A J de Jong, others, The Stacks project, electronic reference (2005–)
17	N Borne, A Vistoli, Parabolic sheaves on logarithmic schemes, Adv. Math. 231 (2012) 1327 MR2964607
18	S Bozlee, Contractions of subcurves of families of log curves, Comm. Algebra 49 (2021) 4616 MR4304662
19	S Bozlee, B Kuo, A Neff, A classification of modular compactifications of the space of pointed elliptic curves by Gorenstein curves, Algebra Number Theory 17 (2023) 127 MR4549282
20	F Catanese, Pluricanonical Gorenstein curves, from: "Enumerative geometry and classical algebraic geometry" (editors P Le Barz, Y Hervier), Progr. Math. 24, Birkhäuser (1982) 51 MR685764
21	R Cavalieri, M Chan, M Ulirsch, J Wise, A moduli stack of tropical curves, Forum Math. Sigma 8 (2020) MR4091085
22	R Cavalieri, H Markwig, D Ranganathan, Tropicalizing the space of admissible covers, Math. Ann. 364 (2016) 1275 MR3466867
23	M Chan, Tropical hyperelliptic curves, J. Algebraic Combin. 37 (2013) 331 MR3011346
24	H L Chang, S Guo, J Li, BCOV’s Feynman rule of quintic 3–folds, preprint (2018) arXiv:1810.00394
25	I Ciocan-Fontanine, B Kim, Moduli stacks of stable toric quasimaps, Adv. Math. 225 (2010) 3022 MR2729000
26	T Coates, C Manolache, A splitting of the virtual class for genus one stable maps, preprint (2018) arXiv:1809.04162
27	M A Cueto, H Markwig, Tropical geometry of genus two curves, J. Algebra 517 (2019) 457 MR3869282
28	O Debarre, Higher-dimensional algebraic geometry, Springer (2001) MR1841091
29	J M Drézet, Paramétrisation des courbes multiples primitives, Adv. Geom. 7 (2007) 559 MR2360903
30	D Ferrand, Conducteur, descente et pincement, Bull. Soc. Math. France 131 (2003) 553 MR2044495
31	L Y Fong, Rational ribbons and deformation of hyperelliptic curves, J. Algebraic Geom. 2 (1993) 295 MR1203687
32	W D Gillam, Logarithmic stacks and minimality, Internat. J. Math. 23 (2012) MR2945649
33	T Graber, J Kock, R Pandharipande, Descendant invariants and characteristic numbers, Amer. J. Math. 124 (2002) 611 MR1902891
34	M Gross, B Siebert, Logarithmic Gromov–Witten invariants, J. Amer. Math. Soc. 26 (2013) 451 MR3011419
35	S Guo, F Janda, Y Ruan, A mirror theorem for genus two Gromov–Witten invariants of quintic threefolds, preprint (2017) arXiv:1709.07392
36	C Haase, G Musiker, J Yu, Linear systems on tropical curves, Math. Z. 270 (2012) 1111 MR2892941
37	J Harris, D Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982) 23 MR664324
38	Y Hu, J Li, Genus-one stable maps, local equations, and Vakil–Zinger’s desingularization, Math. Ann. 348 (2010) 929 MR2721647
39	Y Hu, J Li, Derived resolution property for stacks, Euler classes and applications, Math. Res. Lett. 18 (2011) 677 MR2831834
40	Y Hu, J Li, J Niu, Genus two stable maps, local equations and modular resolutions, preprint (2018) arXiv:1201.2427v3
41	Y Hu, J Niu, A theory of stacks with twisted fields and resolution of moduli of genus two stable maps, preprint (2020) arXiv:2005.03384
42	D Johnson, A Polishchuk, Birational models of ℳ_2,2 arising as moduli of curves with nonspecial divisors, Adv. Geom. 21 (2021) 23 MR4203283
43	F Kato, Log smooth deformation and moduli of log smooth curves, Internat. J. Math. 11 (2000) 215 MR1754621
44	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 MR2683209
45	B Kim, A Kresch, T Pantev, Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee, J. Pure Appl. Algebra 179 (2003) 127 MR1958379
46	J Kollár, S Mori, Birational geometry of algebraic varieties, 134, Cambridge Univ. Press (1998) MR1658959
47	M Kontsevich, Enumeration of rational curves via torus actions, from: "The moduli space of curves" (editors R Dijkgraaf, C Faber, G van der Geer), Progr. Math. 129, Birkhäuser (1995) 335 MR1363062
48	S J Kovács, Rational singularities, preprint (2017) arXiv:1703.02269
49	A G Kuznetsov, Hyperplane sections and derived categories, Izv. Ross. Akad. Nauk Ser. Mat. 70 (2006) 23 MR2238172
50	Y P Lee, Quantum Lefschetz hyperplane theorem, Invent. Math. 145 (2001) 121 MR1839288
51	J Li, A Zinger, On the genus-one Gromov–Witten invariants of complete intersections, J. Differential Geom. 82 (2009) 641 MR2534991
52	C Manolache, Virtual push-forwards, Geom. Topol. 16 (2012) 2003 MR2975298
53	S Marcus, J Wise, Logarithmic compactification of the Abel–Jacobi section, Proc. Lond. Math. Soc. 121 (2020) 1207 MR4133706
54	H Matsumura, Commutative ring theory, 8, Cambridge Univ. Press (1989) MR1011461
55	S Mochizuki, The geometry of the compactification of the Hurwitz scheme, Publ. Res. Inst. Math. Sci. 31 (1995) 355 MR1355945
56	S Molcho, Universal stacky semistable reduction, Israel J. Math. 242 (2021) 55 MR4282076
57	M Möller, M Ulirsch, A Werner, Realizability of tropical canonical divisors, J. Eur. Math. Soc. 23 (2021) 185 MR4186466
58	W Nizioł, Toric singularities: log-blow-ups and global resolutions, J. Algebraic Geom. 15 (2006) 1 MR2177194
59	M C Olsson, Logarithmic geometry and algebraic stacks, Ann. Sci. École Norm. Sup. 36 (2003) 747 MR2032986
60	M C Olsson, (Log) twisted curves, Compos. Math. 143 (2007) 476 MR2309994
61	K S Parker, Semistable modular compactifications of moduli spaces of genus one curves, PhD thesis, University of Colorado at Boulder (2017) MR3697678
62	A Popa, The genus one Gromov–Witten invariants of Calabi–Yau complete intersections, Trans. Amer. Math. Soc. 365 (2013) 1149 MR3003261
63	D Ranganathan, K Santos-Parker, J Wise, Moduli of stable maps in genus one and logarithmic geometry, I, Geom. Topol. 23 (2019) 3315 MR4046967
64	D Ranganathan, K Santos-Parker, J Wise, Moduli of stable maps in genus one and logarithmic geometry, II, Algebra Number Theory 13 (2019) 1765 MR4017534
65	M Reid, Nonnormal del Pezzo surfaces, Publ. Res. Inst. Math. Sci. 30 (1994) 695 MR1311389
66	J P Serre, Algebraic groups and class fields, 117, Springer (1988) MR918564
67	D I Smyth, Modular compactifications of the space of pointed elliptic curves, I, Compos. Math. 147 (2011) 877 MR2801404
68	D I Smyth, Modular compactifications of the space of pointed elliptic curves, II, Compos. Math. 147 (2011) 1843 MR2862065
69	D E Speyer, Tropical geometry, PhD thesis, University of California, Berkeley (2005) MR2707751
70	J Stevens, On the classification of reducible curve singularities, from: "Algebraic geometry and singularities" (editors A Campillo López, L Narváez Macarro), Progr. Math. 134, Birkhäuser (1996) 383 MR1395193
71	B Teissier, The hunting of invariants in the geometry of discriminants, from: "Real and complex singularities" (editor P Holm), Sijthoff and Noordhoff (1977) 565 MR0568901
72	M Ulirsch, Functorial tropicalization of logarithmic schemes: the case of constant coefficients, Proc. Lond. Math. Soc. 114 (2017) 1081 MR3661346
73	M Ulirsch, Non-Archimedean geometry of Artin fans, Adv. Math. 345 (2019) 346 MR3899966
74	R Vakil, The enumerative geometry of rational and elliptic curves in projective space, J. Reine Angew. Math. 529 (2000) 101 MR1799935
75	R Vakil, Murphy’s law in algebraic geometry : badly-behaved deformation spaces, Invent. Math. 164 (2006) 569 MR2227692
76	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 MR2377245
77	M Viscardi, Alternate compactifications of the moduli space of genus one maps, Manuscripta Math. 139 (2012) 201 MR2959678
78	J Wang, Deformations of pairs (X,L) when X is singular, Proc. Amer. Math. Soc. 140 (2012) 2953 MR2917069
79	P M H Wilson, On blowing up conductor ideals, Math. Proc. Cambridge Philos. Soc. 83 (1978) 445 MR463171
80	F D W van der Wyck, Moduli of singular curves and crimping, PhD thesis, Harvard University (2010) MR2941435
81	A Zinger, Enumeration of genus-two curves with a fixed complex structure in ℙ² and ℙ³, J. Differential Geom. 65 (2003) 341 MR2064428
82	A Zinger, Standard versus reduced genus-one Gromov–Witten invariants, Geom. Topol. 12 (2008) 1203 MR2403808
83	A Zinger, The reduced genus 1 Gromov–Witten invariants of Calabi–Yau hypersurfaces, J. Amer. Math. Soc. 22 (2009) 691 MR2505298
84	A Zinger, A sharp compactness theorem for genus-one pseudo-holomorphic maps, Geom. Topol. 13 (2009) 2427 MR2529940