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An analogue of Max Noether’s theorem. (English) Zbl 0589.14034
It is a general hope that basic properties of complete intersections remain true for dependency loci of sections of an ample vector bundle. Along this line the author proves the following generalisation of Max Noether’s theorem [see P. Deligne, Sém. Géom. algébr. 1967- 1968, SGA 7 II, Lect. Notes Math. 340, exposé XIX, 328-340 (1973; Zbl 0269.14019)]: if E is a (-2)-regular rank r vector bundle on $$P^ n$$, (n$$\geq 3$$, $$r\geq n-2)$$, $$T\subset H^ 0(E)$$ is a generic $$(r+3-n)$$- subspace and X is the locus of those $$x\in P^ n$$ where the map $$T\to E(x)$$ has rank at most $$r+2-n$$, then Pic(X) is either $${\mathbb{Z}}$$, generated by a hyperplane section or $${\mathbb{Z}}\oplus {\mathbb{Z}}$$ (according to whether $$r+3-n=1$$ or $$r+3-n\geq 2)$$ unless E is one of the following ”exceptional” vector bundles $$E={\mathcal O}_{{\mathbb{P}}^ 3}(2)$$, $${\mathcal O}_{{\mathbb{P}}^ 3}(3)$$, $${\mathcal O}_{{\mathbb{P}}^ 4}(2)\oplus {\mathcal O}_{{\mathbb{P}}^ 4}(2)$$ (these are precisely the exceptions which occur in the classical Max Noether theorem 1) As an application the author computes Pic(X) for certain generic weighted complete intersection surfaces, thus generalizing a result of J. Steenbrink [Algebr. geom., Proc. internat. Conf., La Rábida/Spain 1981, Lect. Notes Math. 961, 302-313 (1982; Zbl 0507.14025)].
Reviewer: A.Buium

##### MSC:
 14C22 Picard groups 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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##### References:
 [1] P. Deligne and N. Katz, Groupes de monodromie en géométrie algébrique. II , Lecture Notes in Mathematics, vol. 340, Springer-Verlag, Berlin, 1973. · Zbl 0258.00005 [2] G. Evans and P. Griffith, The syzygy problem , Ann. of Math. (2) 114 (1981), no. 2, 323-333. JSTOR: · Zbl 0497.13013 · doi:10.2307/1971296 · links.jstor.org [3] W. Fulton and R. Lazarsfeld, On the connectedness of degeneracy loci and special divisors , Acta Math. 146 (1981), no. 3-4, 271-283. · Zbl 0469.14018 · doi:10.1007/BF02392466 [4] W. Fulton and R. Lazarsfeld, Connectivity and its applications in algebraic geometry , Algebraic geometry (Chicago, Ill., 1980), Lecture Notes in Math., vol. 862, Springer, Berlin, 1981, pp. 26-92. · Zbl 0484.14005 [5] P. A. Griffiths, Hermitian differential geometry, Chern classes, and positive vector bundles , Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 185-251. · Zbl 0201.24001 [6] J. Carlson, M. Green, P. Griffiths, and J. Harris, Infinitesimal variation of Hodge structures , Comp. Math. · Zbl 0531.14006 [7] P. Griffiths and J. Harris, Infinitesimal variation of Hodge structures II , · Zbl 0576.14008 [8] R. Hartshorne, Ample subvarieties of algebraic varieties , Notes written in collaboration with C. Musili. Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin, 1970. · Zbl 0208.48901 · doi:10.1007/BFb0067839 · eudml:203415 [9] J. Le Potier, Annulation de la cohomolgie à valeurs dans un fibré vectoriel holomorphe positif de rang quelconque , Math. Ann. 218 (1975), no. 1, 35-53. · Zbl 0313.32037 · doi:10.1007/BF01350066 · eudml:162783 [10] D. Mumford, Lectures on curves on an algebraic surface , With a section by G. M. Bergman. Annals of Math. Studies, No. 59, Princeton University Press, Princeton, N.J., 1966. · Zbl 0187.42701 [11] S. Mori, On a generalization of complete intersections , J. Math. Kyoto Univ. 15 (1975), no. 3, 619-646. · Zbl 0332.14019 [12] C. Okonek, M. Schneider, and H. Spindler, Vector bundles on complex projective spaces , Progress in Mathematics, vol. 3, Birkhäuser Boston, Mass., 1980. · Zbl 0438.32016 [13] F. Oort and J. Steenbrink, The local Torelli problem for algebraic curves , Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn, 1980, pp. 157-204. · Zbl 0444.14007 [14] A. Sommese, Submanifolds of Abelian varieties , Math. Ann. 233 (1978), no. 3, 229-256. · Zbl 0381.14007 · doi:10.1007/BF01405353 · eudml:163110 [15] J. Steenbrink, On the Picard group of certain smooth surfaces in weighted projective spaces , Algebraic geometry (La Rábida, 1981), Lecture Notes in Math., vol. 961, Springer, Berlin, 1982, pp. 302-313. · Zbl 0507.14025 [16] I. Dogachev, Weighted projective spaces , Lecture Notes in Math., vol. 956, 1982.
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