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Castelnuovo bounds for higher-dimensional varieties

Part of: Differential topology Analytic and descriptive geometry Special varieties Projective and enumerative geometry Surfaces and higher-dimensional varieties Complex manifolds (Co)homology theory

Published online by Cambridge University Press:  09 July 2012

F. L. Zak
F. L. Zak*
Affiliation:
CEMI (Central Economics and Mathematics Institute of the Russian Academy of Sciences), Nakhimovskiĭ av. 47, Moscow 117418, Russia (email: zak@mccme.ru)
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Abstract

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We give bounds for the Betti numbers of projective algebraic varieties in terms of their classes (degrees of dual varieties of successive hyperplane sections). We also give bounds for classes in terms of ramification volumes (mixed ramification degrees), sectional genus and, eventually, in terms of dimension, codimension and degree. For varieties whose degree is large with respect to codimension, we give sharp bounds for the above invariants and classify the varieties on the boundary, thus obtaining a generalization of Castelnuovo’s theory for curves to varieties of higher dimension.

Keywords

Castelnuovo bounddual varietyclassramificationBetti numbervariety of minimal degreeLefschetz theory

MSC classification

Secondary: 14F25: Classical real and complex (co)homology 14F45: Topological properties 14J40: $n$-folds ($n>4$) 14M99: None of the above, but in this section 14N15: Classical problems, Schubert calculus 14P25: Topology of real algebraic varieties 32Q55: Topological aspects of complex manifolds 51N15: Projective analytic geometry 51N35: Questions of classical algebraic geometry 57R19: Algebraic topology on manifolds 57R40: Embeddings

Information

Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 4 , July 2012 , pp. 1085 - 1132
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[Ale1914]Alexander, J. W., Sur les cycles des surfaces algébriques et sur une définition topologique de l’invariant de Zeuthen–Segre, Rend. R. Accad. Lincei Cl. Fis. Mat. Nat. (2) 23 (1914), 55–62.Google Scholar
[AA1938]Alexandrov, A. D., To the theory of mixed volumes of convex bodies. Part IV: mixed discriminants and mixed volumes, Mat. Sb. 3(45) (1938), no. 2, 227–251; Reprinted in: Selected works, Geometry and Applications, vol. 1, eds O. A. Ladyzhenskaya, Yu. G. Reshetnyak, V. A. Alexandrov, Yu. D. Burago, S. S. Kutateladze and N. N. Ural’tseva (Nauka, Novosibirsk, 2006), 116–143 (in Russian); English translation: Selected works, Selected Scientific Papers, vol. 1, Part 1, eds Yu. G. Reshetnyak and S. S. Kutateladze (Gordon and Breach, Amsterdam, 1996), 119–144.Google Scholar
[BBS1989]Beltrametti, M., Biancofiore, A. and Sommese, A. J., Projective n-folds of log-general type. I, Trans. Amer. Math. Soc. 314 (1989), 825–849.Google Scholar
[BFJ2009]Boucksom, S., Favre, C. and Jonsson, M., Differentiability of volumes of divisors and a problem of Teissier, J. Algebraic Geom. 18 (2009), 279–308.CrossRefGoogle Scholar
[Bro1937]Bronowski, J., Curves whose grade is not positive in the theory of the base, J. Lond. Math. Soc. 13 (1937), 86–90.Google Scholar
[Cas1889]Castelnuovo, G., Ricerche di geometria sulle curve algebriche, Atti Accad. Sci. Torino 24 (1889), 346–373; Reprinted in: Memorie Scelte, vol. 1 (Nicola Zanichelli Editore, Bologna, 1937), 21–44.Google Scholar
[Cas1893]Castelnuovo, G., Sui multipli di una serie lineare di gruppi di punti appartenente ad una curva algebrica, Rend. Circ. Mat. Palermo 7 (1893), 89–110; Reprinted in: Memorie Scelte, vol. 1 (Nicola Zanichelli Editore, Bologna, 1937), 95–113.CrossRefGoogle Scholar
[Cil1987]Ciliberto, C., Hilbert functions of finite sets of points and the genus of a curve in a projective space, in Space curves, proceedings of the conference in Rocca di Papa, 1985, Lecture Notes in Mathematics, vol. 1266 (Springer, Berlin–Heidelberg–New York, 1987), 24–73.Google Scholar
[DK1973]Deligne, P. and Katz, N., Groupes de monodromie en géométrie algébrique. II, in Séminaire de géométrie algébrique du Bois-Marie 1967–1969 (SGA 7 II), Lecture Notes in Mathematics, vol. 340 (Springer, Berlin–New York, 1973).Google Scholar
[Dem1993]Demailly, J.-P., A numerical criterion for very ample line bundles, J. Differential Geom. 37 (1993), 323–374.CrossRefGoogle Scholar
[DG2001]Di Gennaro, V., Self-intersection of the canonical bundle of a projective variety, Comm. Algebra 29 (2001), 141–156.CrossRefGoogle Scholar
[DN2006]Dinh, T.-C. and Nguyên, V.-A., The mixed Hodge–Riemann bilinear relations for compact Kähler manifolds, Geom. Funct. Anal. 16 (2006), 838–849.CrossRefGoogle Scholar
[Ein1982]Ein, L., The ramification divisor for branched coverings of  Pn, Math. Ann. 261 (1982), 483–485.CrossRefGoogle Scholar
[EH1987]Eisenbud, D. and Harris, J., On varieties of minimal degree (a centennial account), in Algebraic geometry, Bowdoin (Brunswick, Maine, 1985), Proceedings of the Symposia in Pure Mathematics, vol. 46, Part 1 (American Mathematical Society, Providence, RI, 1987), 3–13.CrossRefGoogle Scholar
[Ful1998]Fulton, W., Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Bd. 2, second edition (Springer, Berlin, 1998).CrossRefGoogle Scholar
[GH1978]Griffiths, Ph. and Harris, J., Principles of algebraic geometry (John Wiley–Interscience, London–New York, 1978).Google Scholar
[Gro1990]Gromov, M., Convex sets and Kähler manifolds, in Advances in differential geometry and topology (World Scientific, Teaneck, NJ, 1990), 1–38.Google Scholar
[Gr1958]Grothendieck, A., Sur une note de Mattuck–Tate, J. Reine Angew. Math. 200 (1958), 208–215.CrossRefGoogle Scholar
[Hal1882]Halphen, G.-H., Mémoire sur la classification des courbes gauches algébriques, J. Éc. Polytech. 52 (1882), 1–200; Also in: Oeuvres, vol. 3 (Gauthier-Villars, Paris, 1921), 261–455.Google Scholar
[Har1981]Harris, J., A bound on the geometric genus of projective varieties, Ann. Sc. Norm. Super. Pisa 8 (1981), 35–68.Google Scholar
[HE1982]Harris, J., with collaboration of D. Eisenbud, Curves in projective space, Séminaire de Mathématiques Supérieures, vol. 85 (Presses de l’Université de Montréal, Montreal, 1982).Google Scholar
[Har1992]Harris, J., Algebraic geometry. A first course, Graduate Texts in Mathematics, vol. 133 (Springer, New York, 1992; corrected third printing, 1995).Google Scholar
[Hir1966]Hirzebruch, F., Topological methods in algebraic geometry (Springer, New York, 1966).Google Scholar
[Hod1937]Hodge, W. V. D., Note on the theory of the base for curves on an algebraic surface, J. Lond. Math. Soc. 12 (1937), 58–63.CrossRefGoogle Scholar
[Ili1998]Ilic, B., Geometric properties of the double-point divisor, Trans. Amer. Math. Soc. 350 (1998), 1643–1661.CrossRefGoogle Scholar
[KP2004]Katz, N. M. and Pandharipande, R., Inequalities related to Lefschetz pencils and integrals of Chern classes, in Geometric aspects of Dwork theory (Walter de Gruyter, Berlin, 2004), 805–818.CrossRefGoogle Scholar
[KKh2012]Kaveh, K. and Khovanskiǐ, A. G., Newton–Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, arXiv.org:0904.3350 v.2 [math.AG], Ann. of Math. (2), to appear.Google Scholar
[KhZ]Kharlamov, V. and Zak, F. L., to appear.Google Scholar
[Kho1979]Khovanskiǐ, A. G., The geometry of convex polyhedra and algebraic geometry, Uspekhi Mat. Nauk 34 (1979), 160–161 (in Russian).Google Scholar
[Kle1986]Kleiman, S., Tangency and duality, in Proceedings of the 1984 Vancouver conference in algebraic geometry, Canadian Mathematical Society Conference Proceedings, vol. 6, eds Carrell, J., Geramita, A. and Russell, R. (American Mathematical Society, Providence, RI, 1986), 163–225.Google Scholar
[KK1989]Kulikov, V. S. and Kurchanov, P. V., Complex algebraic varieties, periods of integrals and Hodge structures, in Current problems in mathematics. Fundamental directions, Itogi Nauki i Tekhniki, vol. 36 (VINITI, Moscow, 1989), 5–231 (in Russian); English translation in: Algebraic geometry, III, Encyclopaedia of Mathematical Sciences, vol. 36 (Springer, Berlin, 1998).Google Scholar
[Lam1981]Lamotke, I., The topology of complex projective varieties after S. Lefschetz, Topology 20 (1981), 15–51.CrossRefGoogle Scholar
[LV2006]Laszlo, Y. and Viterbo, C., Estimates of characteristic classes of real algebraic varieties, Topology 45 (2006), 261–280.CrossRefGoogle Scholar
[Laz2004]Lazarsfeld, R. K., Positivity in algebraic geometry I–II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vols. 48–49 (Springer, Berlin, 2004).CrossRefGoogle Scholar
[LT1981]Tráng, Lê Dũng and Teissier, B., Variétés polaires locales et classes de Chern des variétés singulières, Ann. of Math. (2) 114 (1981), 457–491.CrossRefGoogle Scholar
[Lef1924]Lefschetz, S., L’Analysis situs et la géométrie algébrique (Gauthier-Villars, Paris, 1924).Google Scholar
[Mil1964]Milnor, J., On the Betti numbers of real varieties, Proc. Amer. Math. Soc. 15 (1964), 275–280; Reprinted in: Collected papers I, geometry (Publish or Perish, Houston, 1994), 133–140.CrossRefGoogle Scholar
[Moi1967]Moishezon, B. G., Algebraic homology classes on algebraic varieties, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 225–268 (in Russian); English translation: Math. USSR Izvestija 1, 209–251.Google Scholar
[Ole1951]Oleinik, O. A., Estimates of the Betti numbers of real algebraic hypersurfaces, Mat. Sb. 28 (1951), 635–640 (in Russian).Google Scholar
[Pie1978]Piene, R., Polar classes of singular varieties, Ann. Sci. Éc. Norm. Supér (4) 11 (1978), 247–276.CrossRefGoogle Scholar
[SeB1937]Segre, B., Intorno ad un teorema di Hodge sulla teoria della base per le curve di una superficie algebrica, Ann. Mat. Pura Appl. 16 (1937), 157–163.CrossRefGoogle Scholar
[Seg1895/96]Segre, C., Intorno ad un carattere delle superficie e delle varietà superiori algebriche, Atti. R. Accad. Torino XXXI (1895/96), 341–357; Also in: Opere, vol. I (Cremonese Editore, Roma, 1957), 312–326.Google Scholar
[SR1949]Semple, J. G. and Roth, L., Introduction to algebraic geometry (Oxford University Press, Oxford, 1949).Google Scholar
[Sev1902]Severi, F., Sulle intersezioni delle varietà algebriche e sopra i loro caratteri e singolarità proiettive, Mem. R. Accad. Sci. Torino (2) 52 (1902), 61–118; Reprinted in: Memorie Scelte, vol. 1 (Edizioni Cremonese, Firenze, 1950), 41–115.Google Scholar
[Tei1979]Teissier, B., Du théorème de l’index de Hodge aux inégalités isopérimétriques, C. R. Acad. Sci. Paris 288 (1979), A287–A289.Google Scholar
[Tev2005]Tevelev, E., Projective duality and homogeneous spaces, in Invariant theory and algebraic transformation groups IV, Encyclopaedia of Mathematical Sciences, vol. 133 (Springer, Berlin–Heidelberg–New York, 2005).Google Scholar
[Tho1965]Thom, R., Sur l’homologie des variétés algébriques réelles, in Differential and combinatorial topology, A Symposium in Honor of Marston Morse, ed. Cairns, S. S. (Princeton University Press, Princeton, NJ, 1965), 255–265.CrossRefGoogle Scholar
[Tim1999]Timorin, V. A., An analogue of the Hodge–Riemann relations for simple polytopes, Uspekhi Mat. Nauk 54 (1999), 113–162 (in Russian); English translation: Russian Math. Surveys 54 (1999), 381–426.Google Scholar
[Tod1937]Todd, J. A., The arithmetical invariants of algebraic loci, Proc. Lond. Math. Soc. 43 (1937), 190–225.Google Scholar
[Voi2002]Voisin, C., Hodge theory and complex algebraic geometry I, Cambridge Studies in Advanced Mathematics, vol. 76 (Cambridge University Press, Cambridge, 2002).CrossRefGoogle Scholar
[Zak1973]Zak, F. L., Surfaces with zero Lefschetz cycles, Mat. Zametki 13 (1973), 869–880; English translation: Math. Notes 13 (1973), 520–525.Google Scholar
[Zak1993]Zak, F. L., Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, vol. 127 (American Mathematical Society, Providence, RI, 1993).Google Scholar
[Zak1999]Zak, F. L., Projective invariants of quadratic embeddings, Math. Ann. 313 (1999), 507–545.CrossRefGoogle Scholar
[Zak2004]Zak, F. L., Determinants of projective varieties and their degrees, in Algebraic transformation groups and algebraic varieties, Encyclopedia of Mathematical Sciences, Subseries Invariant Theory and Algebraic Transformation Groups, vol. 132(3) (Springer, 2004), 207–238.CrossRefGoogle Scholar
[Zak2012]Zak, F. L., Asymptotic behaviour of numerical invariants of algebraic varieties, J. Eur. Math. Soc. 14 (2012), 255–271.CrossRefGoogle Scholar