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A support theorem for Hilbert schemes of planar curves. (English) Zbl 1303.14019

In the paper under review, the authors consider a family of integral plane curves and prove that if the relative Hilbert scheme of points is smooth, then the pushforward of the constant sheaf on the relative Hilbert scheme contains no summands other than the expected ones.
As a corollary, they show that the perverse filtration on the cohomology of the compactified Jacobian of an integral plane curve encodes the cohomology of all Hilbert schemes of points on the curve. An interesting consequence for the enumerative geometry of Calabi-Yau three-folds is also mentioned.

MSC:

14C05 Parametrization (Chow and Hilbert schemes)
14H40 Jacobians, Prym varieties
14D15 Formal methods and deformations in algebraic geometry
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
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References:

[1] Altman, A., Iarrobino, A., Kleiman, S.: Irreducibility of the compactified Jacobian. In: Real and Complex Singularities (Oslo, 1976), Sijthoff and Noordhoff, 1-12 (1977) · Zbl 0415.14014
[2] Altman, A., Kleiman, S.: Compactifying the Picard scheme. Adv. Math. 35, 50-112 (1980) · Zbl 0427.14015 · doi:10.1016/0001-8708(80)90043-2
[3] Behrend, K.: Donaldson-Thomas invariants via microlocal geometry. Ann. of Math. 170, 1307-1338 (2009) · Zbl 1191.14050 · doi:10.4007/annals.2009.170.1307
[4] Beilinson, A. A., Bernstein, J., Deligne, P.: Faisceaux pervers. Astérisque 100, 5-171 (1982) · Zbl 0536.14011
[5] Brianc\?on, J., Granger, M., Speder, J.-P.: Sur le schéma de Hilbert d’une courbe plane. Ann. Sci. École Norm. Sup. 14, 1-25 (1981) · Zbl 0463.14001
[6] Bryan, J., Pandharipande, R.: BPS states in Calabi-Yau 3-folds. Geom. Topol. 5, 287-318 (2001) · Zbl 1063.14068 · doi:10.2140/gt.2001.5.287
[7] Cattani, E., Kaplan, A.: Polarized mixed Hodge structures and the local monodromy of a variation of Hodge structure. Invent. Math. 67, 101-115 (1982) · Zbl 0516.14005 · doi:10.1007/BF01393374
[8] Diaz, S., Harris, J.: Ideals associated to deformations of singular plane curves. Trans. Amer. Math. Soc. 309, 433-468 (1988) · Zbl 0707.14022 · doi:10.2307/2000919
[9] Fantechi, B., Göttsche, L., van Straten, D.: Euler number of the compactified Jacobian and multiplicity of rational curves. J. Algebr. Geom. 8, 115-133 (1999) · Zbl 0951.14017
[10] Gopakumar, R., Vafa, C.: M-theory and topological strings I & II. hep-th/9809187, hep- th/9812187 · Zbl 0922.32015
[11] Greuel, G.-M., Lossen, C., Shustin, E.: Introduction to Singularities and Deformations. Springer (2007) · Zbl 1125.32013 · doi:10.1007/3-540-28419-2
[12] Hosono, S., Saito, M.-H., Takahashi, A.: Relative Lefschetz action and BPS state counting. Int. Math. Res. Notices 2001, 783-816 · Zbl 1060.14017 · doi:10.1155/S107379280100040X
[13] Khovanov, M., Rozansky, L.: Matrix factorizations and link homology, I, II. Fund. Math. 199, 1-91 (2008); Geom. Topol. 12, 1387-1425 (2008) · Zbl 1145.57009 · doi:10.4064/fm199-1-1
[14] Kool, M., Shende, V., Thomas, R.: A short proof of the Göttsche conjecture. Geom. Topol. 15, 397-406 (2011) · Zbl 1210.14011 · doi:10.2140/gt.2011.15.397
[15] Kool, M., Thomas, R.: Reduced classes and curve counting on surfaces I, II. · Zbl 1322.14086 · doi:10.14231/AG-2014-018
[16] Laumon, G.: Fibres de Springer et jacobiennes compactifiées. In: Algebraic Geometry and Number Theory, Progr. Math. 253, Birkhäuser Boston, 515-563 (2006) · Zbl 1129.14301
[17] Macdonald, I. G.: The Poincaré polynomial of a symmetric product. Math. Proc. Cam- bridge Philos. Soc. 58, 563-568 (1962) · Zbl 0121.39601
[18] Maulik, D., Yun, Z.: Macdonald formula for curves with planar singularities. · Zbl 1304.14036 · doi:10.1515/crelle-2012-0093
[19] Ng\hat o, B. C.: Le lemme fondamental pour les alg‘ebres de Lie. Publ. Math. IHES 111 (2010) · Zbl 1200.22011 · doi:10.1007/s10240-010-0026-7
[20] Oblomkov, A., Rasmussen, J., Shende, V.: The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link. · Zbl 1256.14025 · doi:10.1215/00127094-1593281
[21] Oblomkov, A., Shende, V.: The Hilbert scheme of a plane curve singularity and the HOM- FLY polynomial of its link. Duke Math. J. 161, 1277-1303 (2012) · Zbl 1256.14025 · doi:10.1215/00127094-1593281
[22] Pandharipande, R., Thomas, R. P.: Stable pairs and BPS invariants. J. Amer. Math. Soc. 23, 267-297 (2010) · Zbl 1250.14035 · doi:10.1090/S0894-0347-09-00646-8
[23] Shende, V.: Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation. Compos. Math. 148, 531-547 (2012) · Zbl 1312.14015 · doi:10.1112/S0010437X11007378
[24] Tessier, B.: Résolution simultanée-I. Familles de courbes. In: Séminaire sur les singu- larités des surfaces, Lecture Notes in Math. 777, Springer, 71-81 (1980) · Zbl 0415.00010 · doi:10.1007/BFb0085872
[25] Varagnolo, M., Vasserot, E.: Finite-dimensional representations of DAHA and affine Springer fibers: The spherical case. Duke Math. J. 147, 439-540 (2009) · Zbl 1237.20008 · doi:10.1215/00127094-2009-016
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