×

zbMATH — the first resource for mathematics

The Grothendieck-Lefschetz theorem for normal projective varieties. (English) Zbl 1123.14004
The authors prove that if \(X\) is a normal projective variety in characteristic zero, \(L\) a base-point free ample line bundle on \(X\), and \(Y\) a general member of \(| L|\), then the restriction map of divisor class groups \(\text{Cl}(X)\rightarrow \text{Cl}(Y)\) is an isomorphism provided that \(\text{dim}\,X\geq 4\).

MSC:
14C22 Picard groups
14C20 Divisors, linear systems, invertible sheaves
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] L. Barbieri-Viale, A. Rosenschon, and M. Saito, Deligne’s conjecture on 1-motives, Ann. of Math. (2) 158 (2003), no. 2, 593 – 633. · Zbl 1124.14014 · doi:10.4007/annals.2003.158.593 · doi.org
[2] Luca Barbieri-Viale and Vasudevan Srinivas, Albanese and Picard 1-motives, Mém. Soc. Math. Fr. (N.S.) 87 (2001), vi+104 (English, with English and French summaries). · Zbl 1085.14011
[3] J. Biswas and V. Srinivas, Roitman’s theorem for singular projective varieties, Compositio Math. 119 (1999), no. 2, 213 – 237. · Zbl 0969.14005 · doi:10.1023/A:1001793226084 · doi.org
[4] Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5 – 57 (French). Pierre Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5 – 77 (French).
[5] A. Grothendieck, J. Dieudonne, Elements de Geometrie Algebrique I, III, Publ. Math. IHES. 4 (1960), 11 (1961).
[6] Hélène Esnault and Eckart Viehweg, Lectures on vanishing theorems, DMV Seminar, vol. 20, Birkhäuser Verlag, Basel, 1992. · Zbl 0779.14003
[7] William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. · Zbl 0885.14002
[8] Mark Goresky and Robert MacPherson, Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 14, Springer-Verlag, Berlin, 1988. · Zbl 0639.14012
[9] Alexander Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (\?\?\? 2), North-Holland Publishing Co., Amsterdam; Masson & Cie, Éditeur, Paris, 1968 (French). Augmenté d’un exposé par Michèle Raynaud; Séminaire de Géométrie Algébrique du Bois-Marie, 1962; Advanced Studies in Pure Mathematics, Vol. 2. · Zbl 0159.50402
[10] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001
[11] Robin Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. Notes written in collaboration with C. Musili. · Zbl 0208.48901
[12] Serge Lang, Abelian varieties, Springer-Verlag, New York-Berlin, 1983. Reprint of the 1959 original. · Zbl 0516.14031
[13] V. B. Mehta and V. Srinivas, A Characterisation of Rational Singularities, Asian J. Math, Vol. 1, No. 2, 249-271. · Zbl 0920.13020
[14] Niranjan Ramachandran, One-motives and a conjecture of Deligne, J. Algebraic Geom. 13 (2004), no. 1, 29 – 80. · Zbl 1058.14034
[15] Jean-Louis Verdier, Stratifications de Whitney et théorème de Bertini-Sard, Invent. Math. 36 (1976), 295 – 312 (French). · Zbl 0333.32010 · doi:10.1007/BF01390015 · doi.org
[16] André Weil, Sur les critères d’équivalence en géométrie algébrique, Math. Ann. 128 (1954), 95 – 127 (French). · Zbl 0057.13002 · doi:10.1007/BF01360127 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.