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Mirror symmetry and the classification of orbifold del Pezzo surfaces. (English) Zbl 1360.14106

This paper explores mirror symmetry for del Pezzo surfaces with cyclic quotient singularities and state a number of conjectures that together allow one to classify a broad class of such surfaces. The conjectures relate mutation-equivalence classes of Fano polygons with \(\mathbb Q\)-Gorenstein deformation classes of del Pezzo surfaces. As evidence, the authors show that their conjectures hold true in the smooth case and the case of simplest residual singularity \(\frac{1}{3}(1,1)\).

MSC:

14J33 Mirror symmetry (algebro-geometric aspects)
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14J26 Rational and ruled surfaces
14J45 Fano varieties
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[1] Abramovich, Dan; Graber, Tom; Vistoli, Angelo, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math., 130, 5, 1337-1398 (2008) · Zbl 1193.14070 · doi:10.1353/ajm.0.0017
[2] Abramovich, D.; Vistoli, A., Twisted stable maps and quantum cohomology of stacks. Intersection theory and moduli, ICTP Lect. Notes, XIX, 97-138 (electronic) (2004), Abdus Salam Int. Cent. Theoret. Phys., Trieste · Zbl 1095.14054
[3] Akhtar, Mohammad; Coates, Tom; Galkin, Sergey; Kasprzyk, Alexander M., Minkowski polynomials and mutations, SIGMA Symmetry Integrability Geom. Methods Appl., 8, Paper 094, 17 pp. (2012) · Zbl 1280.52014 · doi:10.3842/SIGMA.2012.094
[4] [AK14] Mohammad Akhtar and Alexander M. Kasprzyk, Singularity content, \newblockarXiv: 1401.5458 [math.AG], 2014.
[5] Altmann, Klaus; Hausen, J{\"u}rgen, Polyhedral divisors and algebraic torus actions, Math. Ann., 334, 3, 557-607 (2006) · Zbl 1193.14060 · doi:10.1007/s00208-005-0705-8
[6] Altmann, Klaus; Hausen, J{\`“u}rgen; S{\'”u}ss, Hendrik, Gluing affine torus actions via divisorial fans, Transform. Groups, 13, 2, 215-242 (2008) · Zbl 1159.14025 · doi:10.1007/s00031-008-9011-3
[7] Altmann, Klaus; Ilten, Nathan Owen; Petersen, Lars; S{\`“u}{\ss }, Hendrik; Vollmert, Robert, The geometry of \(T\)-varieties. Contributions to algebraic geometry, EMS Ser. Congr. Rep., 17-69 (2012), Eur. Math. Soc., Z\'”urich · Zbl 1316.14001 · doi:10.4171/114-1/2
[8] Chen, Weimin; Ruan, Yongbin, Orbifold Gromov-Witten theory. Orbifolds in mathematics and physics, Madison, WI, 2001, Contemp. Math. 310, 25-85 (2002), Amer. Math. Soc., Providence, RI · Zbl 1091.53058 · doi:10.1090/conm/310/05398
[9] Chen, Weimin; Ruan, Yongbin, A new cohomology theory of orbifold, Comm. Math. Phys., 248, 1, 1-31 (2004) · Zbl 1063.53091 · doi:10.1007/s00220-004-1089-4
[10] Ciocan-Fontanine, Ionu{\c{t}}; Kim, Bumsig, Wall-crossing in genus zero quasimap theory and mirror maps, Algebr. Geom., 1, 4, 400-448 (2014) · Zbl 1322.14083 · doi:10.14231/AG-2014-019
[11] [QC105] Tom Coates, Alessio Corti, Sergey Galkin, and Alexander M. Kasprzyk, \newblockQuantum periods for \(3\)-dimensional Fano manifolds, \newblockarXiv:1303.3288 [math.AG], 2013. · Zbl 1348.14105
[12] [CCIT] Tom Coates, Alessio Corti, Hiroshi Iritani, and Hsian-Hua Tseng, \newblockA mirror theorem for toric stacks, \newblockarXiv:1310.4163 [math.AG], 2013. · Zbl 1330.14093
[13] [CCIT2] Tom Coates, Alessio Corti, Hiroshi Iritani, and Hsian-Hua Tseng, \newblockSome applications of the mirror theorem for toric stacks, \newblockarXiv:1401.2611 [math.AG], 2014. · Zbl 1330.14093
[14] [corti-heuberger14] Alessio Corti and Liana Heuberger, Del Pezzo surfaces with \(\frac13(1,1)\) points, \newblockarXiv:1505.02092 [math.AG], 2015. · Zbl 1375.14056
[15] [FY14] Kento Fujita and Kazunori Yasutake, \newblockClassification of log del Pezzo surfaces of index three, \newblockarXiv:1401.1283 [math.AG], 2014.
[16] [GU] Sergey Galkin and Alexandr Usnich, Mutations of Potentials, \newblock preprint IPMU 10-0100, 2010.
[17] Greb, Daniel; Kebekus, Stefan; Kov{\'a}cs, S{\'a}ndor J., Extension theorems for differential forms and Bogomolov-Sommese vanishing on log canonical varieties, Compos. Math., 146, 1, 193-219 (2010) · Zbl 1194.14056 · doi:10.1112/S0010437X09004321
[18] Greb, Daniel; Kebekus, Stefan; Kov{\'a}cs, S{\'a}ndor J.; Peternell, Thomas, Differential forms on log canonical spaces, Publ. Math. Inst. Hautes \'Etudes Sci., 114, 87-169 (2011) · Zbl 1258.14021 · doi:10.1007/s10240-011-0036-0
[19] [GHK11] Mark Gross, Paul Hacking, and Se\'an Keel, Mirror symmetry for log Calabi-Yau surfaces I, \newblockarXiv:1106.4977 [math.AG], 2011. · Zbl 1351.14024
[20] [GHK12] Mark Gross, Paul Hacking, and Se\'an Keel, Moduli of surfaces with an anti-canonical cycle, \newblockarXiv:1211.6367 [math.AG], 2012. · Zbl 1330.14062
[21] Gross, Mark; Siebert, Bernd, From real affine geometry to complex geometry, Ann. of Math. (2), 174, 3, 1301-1428 (2011) · Zbl 1266.53074 · doi:10.4007/annals.2011.174.3.1
[22] Ilten, Nathan Owen, Mutations of Laurent polynomials and flat families with toric fibers, SIGMA Symmetry Integrability Geom. Methods Appl., 8, Paper 047, 7 pp. (2012) · Zbl 1276.14073 · doi:10.3842/SIGMA.2012.047
[23] [KNP14] Alexander M. Kasprzyk, Benjamin Nill, and Thomas Prince, Minimality and mutation-equivalence of polygons, \newblockarXiv:1501.05335 [math.AG], 2015. · Zbl 1394.14034
[24] [KT14] Alexander M. Kasprzyk and Ketil Tveiten, Maximally mutable Laurent polynomials, \newblock in preparation.
[25] Koll{\'a}r, J{\'a}nos, Flips, flops, minimal models, etc. Surveys in differential geometry, Cambridge, MA, 1990, 113-199 (1991), Lehigh Univ., Bethlehem, PA · Zbl 0755.14003
[26] Koll{\'a}r, J.; Shepherd-Barron, N. I., Threefolds and deformations of surface singularities, Invent. Math., 91, 2, 299-338 (1988) · Zbl 0642.14008 · doi:10.1007/BF01389370
[27] [OP14] Alessandro Oneto and Andrea Petracci, On the quantum periods of del Pezzo surfaces with \(\frac13(1,1)\) singularities, \newblock\hrefhttp://arxiv.org/abs/1507.08589arXiv:1507.08589 [math.AG], 2015. · Zbl 1395.14032
[28] [Tveiten] Ketil Tveiten, \newblock Period integrals and mutation, \newblock\hrefhttp://arxiv.org/abs/1501.05095arXiv:1501.05095 [math.AG], 2015.
[29] Wahl, Jonathan M., Elliptic deformations of minimally elliptic singularities, Math. Ann., 253, 3, 241-262 (1980) · Zbl 0431.14012 · doi:10.1007/BF0322000
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