Abstract
Let \(X^{2n}\subseteq \mathbb {P}^N\) be a smooth projective variety. Consider the intersection cohomology complex of the local system \(R^{2n-1}\pi {_*}\mathbb {Q}\), where \(\pi \) denotes the projection from the universal hyperplane family of \(X^{2n}\) to \({(\mathbb {P}^N)}^{\vee }\). We investigate the cohomology of the intersection cohomology complex \(IC(R^{2n-1}\pi {_*}\mathbb {Q})\) over the points of a Severi variety, parametrizing nodal hypersurfaces, whose nodes impose independent conditions on the very ample linear system giving the embedding in \(\mathbb {P}^N\).
Dedicated to Ciro Ciliberto on his seventieth birthday.
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References
G. Albanese, Sui sistemi continui di curve piane algebriche, in Collected Papers of G. Albanese, with a biography of Albanese in English by Ciro Ciliberto and Edoardo Sernesi, ed. by Ciliberto, P. Ribenboim and Sernesi. Queen’s Papers in Pure and Applied Mathematics, vol. 103 (Queen’s University, Kingston, 1996), xii+408 pp
A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Analysis and Topology on Singular Spaces, I (Luminy, 1981), Astérisque, vol. 100 (Société Mathématique de France, Paris, 1982), pp. 5–171
E. Cattani, A. Kaplan, W. Schmidt, \(L^2\) and intersection cohomologies of polarizable variation of Hodge structures. Invent. Math. 87, 217–252 (1987)
L. Chiantini, E. Sernesi, Nodal curves on surfaces of general type. Math. Ann. 307, 41–56 (1997)
M.A. de Cataldo, L. Migliorini, The Hodge theory of algebraic maps. Ann. Sci. École Norm. Sup. 4, 38(5), 693–750 (2005)
M.A. de Cataldo, L. Migliorini, The decomposition theorem, perverse sheaves and the topology of algebraic maps. Bull. Amer. Math. Soc. (N.S.) 46(4), 535–633 (2009)
M.A. de Cataldo, L. Migliorini, On singularities of primitive cohomology classes. PAMS 137, 3593–3600 (2009)
P. Deligne, Equations differentielles a points singuliers reguliers. LNM 163 (1970)
V. Di Gennaro, D. Franco, Factoriality and Néron-Severi groups. Commun. Contemp. Math. 10(5), 745–764 (2008)
I.A. Chel’tsov, The factoriality of nodal threefolds and connectedness of the set of log canonical singularities. Math. Sb. 197(3), 87–116 (2006). Translation in Sb. Math. 197(3–4), 387–414 (2006)
S. Cynk, Defect of a nodal hypersurface. Manuscripta Math. 104(3), 325–331 (2001)
S. Cynk, Defect formula for nodal complete intersection threefolds. Internat. J. Math. 30(4), 1950020, 14 pp (2019)
V. Di Gennaro, D. Franco, On the existence of a Gysin morphism for the Blow-up of an ordinary singularity. Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 63(1), 75–86, (2017). https://doi.org/10.1007/s11565-016-0253-z
V. Di Gennaro, D. Franco, Néron-Severi group of a general hypersurface. Commun. Contemp. Math. 19(01), Article ID 1650004, 15 p. (2017). https://doi.org/10.1142/S0219199716500048
V. Di Gennaro, D. Franco, On the topology of a resolution of isolated singularities. J. Singul. 16, 195–211 (2017). https://doi.org/10.5427/jsing.2017.16j
V. Di Gennaro, D. Franco, On a resolution of singularities with two strata. Results Math. 74(3), 74:115 (2019). https://doi.org/10.1007/s00025-019-1040-9
V. Di Gennaro, D. Franco, On the topology of a resolution of isolated singularities, II. J. Singul. 20, 95–102 (2020). https://doi.org/10.5427/jsing.2020.20e
A. Dimca, Singularity and Topology of Hypersurfaces (Springer Universitext, New York, 1992)
A. Dimca, Sheaves in Topology (Springer Universitext, 2004)
Ph. Ellia, D. Franco, On codimension two subvarieties of \(\mathbb{P}^5\) and \(\mathbb{P}^6\). J. Algebraic Geom. 11(3), 513–533 (2002)
Ph. Ellia, D. Franco, L. Gruson, Smooth divisors of projective hypersurfaces. Comment. Math. Helv. 83(2), 371–385 (2008)
D. Franco, Explicit decomposition theorem for special Schubert varieties. Forum Math. 32(2), 447–470 (2020)
W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete; 3.Folge, Bd. 2 (Springer, 1984)
C.G. Gibson, K. Wirthmuller, A.A. du Plessis, E.J.N. Looijenga, Topological stability of smooth map**s. LNM 552 (1976)
M. Green, P. Griffiths, Algebraic cycles and singularities of normal functions, in Algebraic Cycles and Motives, LMS, vol. 343 (Cambridge University Press, Cambridge, 2007), pp. 206–263
P. Griffiths, J. Harris, Principles of Algebraic Geometry (Wiley, New York, 1978)
J. Harris, Algebraic Geometry - A Firs Course, vol. 133 (Springer GTM, 1992)
J. Harris, On the Severi problem. Invent. Math. 84, 445–461 (1986)
N. Katz, Pinceaux de Lefschetz: théorème d’existence, Séminaire de Géométrie Algébrique du Bois Marie - 1967-69 - Groupes de monodromie en géométrie algébrique - SGA7 II (1973), pp. 212–253
M. Kerr, G. Pearlstein, An exponential history of functions with logarithmic growth. Topol. Stratif. Spaces MSRI Publ . 58, 281–374 (2011)
S. Kleiman, Geometry on grassmannians and applications to splitting bundles and smoothing cycles. Publ. Math. Inst. Hautes Études Sci. 36, 281–297 (1969)
E.J.N. Looijenga, Isolated Singular Points on Complete Intersections. London Mathematical Society Lecture Note Series 77 (Cambridge University Press, Cambridge, 1984)
J. Milnor, Singular Points of Complex Hypersurfaces. Annals of Mathematics Studies (1968)
V. Navarro Aznar, Sur la théorie de Hodge des variétés algébriques à singularités isolées. Astérisque 130, 272–307 (1985)
M. Saito, Mixed Hodge modules. Publ. RIMS, Kyoto Univ. 26, 221–333 (1990)
Ch. Schnell, Computing cohomology of local systems, https://www.math.stonybrook.edu/~cschnell/pdf/notes/locsys.pdf
E. Sernesi, On the existence of certain families of curves. Invent. Math. 75, 25–57 (1984)
F. Severi, Vorlesungen über algebraische Geometrie (Teuner, Leipzig, 1921)
R.P. Thomas, Nodes and the Hodge conjecture. J. Alg. Geom. 14, 177–185 (2005)
C. Voisin, Hodge Theory and Complex Algebraic Geometry, I. Cambridge Studies in Advanced Mathematics 76 (Cambridge University Press, Cambridge, 2002)
J. Wahl, Deformations of plane curves with nodes and cusps. Amer. J. Math. 96, 529–577 (1974)
G. Williamson, Hodge theory of the decomposition theorem [after M.A. de Cataldo and L. Migliorini]. Séminaire BOURBAKI, 2015–2016, n. 1115, p. 31
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We thank the Referee for his valuable suggestions which allowed us to substantially improve the presentation.
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Di Gennaro, V., Franco, D. (2023). Intersection Cohomology and Severi Varieties. In: Dedieu, T., Flamini, F., Fontanari, C., Galati, C., Pardini, R. (eds) The Art of Doing Algebraic Geometry. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-11938-5_6
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