×

zbMATH — the first resource for mathematics

Vanishing theorems, a theorem of Severi, and the equations defining projective varieties. (English) Zbl 0762.14012
The authors present a vanishing theorem about the cohomology of a smooth complex projective variety under a condition in terms of the codimension of the variety and the degrees of the generating hypersurfaces. They call it a “variant” of a theorem of F. Severi [Rend. Circ. Mat. Palermo 17, 73-103 (1903; JFM 34.0700.01)]. — They point out a surprising number of applications to questions involving the equations defining projective varieties like Cohen-Macaulayness, complete intersection, Castelnuovo-Mumford regularity, Hodge type and projectively normal embeddings.
The idea of the proof is to consider a blowing-up along the variety and to apply the Kawamata-Viehweg vanishing theorem. See also Y. Kawamata, Math. Ann. 261, 43–46 (1982; Zbl 0476.14007); E. Viehweg, J. Reine Angew. Math. 335, 1–8 (1982; Zbl 0485.32019); H. Esnault and E. Viehweg, Math. Ann. 263, 75-86 (1983; Zbl 0489.14016); and Invent. Math. 78, 445–490 (1984; Zbl 0532.10020)].

MSC:
14F17 Vanishing theorems in algebraic geometry
14M10 Complete intersections
14N05 Projective techniques in algebraic geometry
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] M. Andreatta, E. L. Ballico, and A. Sommese, On the projective normality of the adjunction bundles. II, preprint. · Zbl 0937.14032
[2] Marco Andreatta and Andrew J. Sommese, On the projective normality of the adjunction bundles, Comment. Math. Helv. 66 (1991), no. 3, 362 – 367. · Zbl 0758.14035 · doi:10.1007/BF02566655 · doi.org
[3] Wolf Barth, Submanifolds of low codimension in projective space, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 409 – 413. · Zbl 0349.14024
[4] D. Bayer and D. Mumford (to appear).
[5] David Bayer and Michael Stillman, On the complexity of computing syzygies, J. Symbolic Comput. 6 (1988), no. 2-3, 135 – 147. Computational aspects of commutative algebra. · Zbl 0667.68053 · doi:10.1016/S0747-7171(88)80039-7 · doi.org
[6] David Bayer and Michael Stillman, A criterion for detecting \?-regularity, Invent. Math. 87 (1987), no. 1, 1 – 11. · Zbl 0625.13003 · doi:10.1007/BF01389151 · doi.org
[7] Aaron Bertram, Lawrence Ein, and Robert Lazarsfeld, Surjectivity of Gaussian maps for line bundles of large degree on curves, Algebraic geometry (Chicago, IL, 1989) Lecture Notes in Math., vol. 1479, Springer, Berlin, 1991, pp. 15 – 25. · Zbl 0752.14036 · doi:10.1007/BFb0086260 · doi.org
[8] D. Butler, Normal generation of vector bundles on curves (to appear).
[9] P. Deligne and A. Dimca, Filtrations de Hodge et par l’ordre du pôle pour les hypersurfaces singulaires, preprint. · Zbl 0743.14028
[10] L. Ein and R. Lazarsfeld, A theorem on the syzygies of smooth projective varieties of arbitrary dimension (to appear). · Zbl 0814.14040
[11] E. Esnault, Hodge type of subvarieties of \( {\mathbb{P}^N}\) of small degrees, preprint. · Zbl 0755.14006
[12] Hélène Esnault and Eckart Viehweg, Sur une minoration du degré d’hypersurfaces s’annulant en certains points, Math. Ann. 263 (1983), no. 1, 75 – 86 (French). · Zbl 0489.14016 · doi:10.1007/BF01457085 · doi.org
[13] Hélène Esnault and Eckart Viehweg, Dyson’s lemma for polynomials in several variables (and the theorem of Roth), Invent. Math. 78 (1984), no. 3, 445 – 490. · Zbl 0545.10021 · doi:10.1007/BF01388445 · doi.org
[14] Gerd Faltings, Ein Kriterium für vollständige Durchschnitte, Invent. Math. 62 (1981), no. 3, 393 – 401 (German). · Zbl 0456.14027 · doi:10.1007/BF01394251 · doi.org
[15] Hubert Flenner, Babylonian tower theorems on the punctured spectrum, Math. Ann. 271 (1985), no. 1, 153 – 160. · Zbl 0541.14011 · doi:10.1007/BF01455804 · doi.org
[16] -, Babylonian tower theorems for coverings (to appear). · Zbl 0697.14036
[17] Mark L. Green, Koszul cohomology and the geometry of projective varieties, J. Differential Geom. 19 (1984), no. 1, 125 – 171. · Zbl 0559.14008
[18] Phillip 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.
[19] L. Gruson, R. Lazarsfeld, and C. Peskine, On a theorem of Castelnuovo, and the equations defining space curves, Invent. Math. 72 (1983), no. 3, 491 – 506. · Zbl 0565.14014 · doi:10.1007/BF01398398 · doi.org
[20] Robin Hartshorne, Varieties of small codimension in projective space, Bull. Amer. Math. Soc. 80 (1974), 1017 – 1032. · Zbl 0304.14005
[21] Audun Holme and Michael Schneider, A computer aided approach to codimension 2 subvarieties of \?_\?,\?\?6, J. Reine Angew. Math. 357 (1985), 205 – 220. · Zbl 0581.14035 · doi:10.1515/crll.1985.357.205 · doi.org
[22] Yujiro Kawamata, A generalization of Kodaira-Ramanujam’s vanishing theorem, Math. Ann. 261 (1982), no. 1, 43 – 46. · Zbl 0476.14007 · doi:10.1007/BF01456407 · doi.org
[23] George R. Kempf, Projective coordinate rings of abelian varieties, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 225 – 235. · Zbl 0785.14025
[24] Robert Lazarsfeld, A sharp Castelnuovo bound for smooth surfaces, Duke Math. J. 55 (1987), no. 2, 423 – 429. · Zbl 0646.14005 · doi:10.1215/S0012-7094-87-05523-2 · doi.org
[25] Robert Lazarsfeld, A sampling of vector bundle techniques in the study of linear series, Lectures on Riemann surfaces (Trieste, 1987) World Sci. Publ., Teaneck, NJ, 1989, pp. 500 – 559. · Zbl 0800.14003
[26] Shigefumi Mori, Classification of higher-dimensional varieties, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 269 – 331. · Zbl 1103.14301
[27] David Mumford, Varieties defined by quadratic equations, Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969) Edizioni Cremonese, Rome, 1970, pp. 29 – 100.
[28] N. Yu. Netsvetaev, Projective varieties defined by small number of equations are complete intersections, Topology and geometry — Rohlin Seminar, Lecture Notes in Math., vol. 1346, Springer, Berlin, 1988, pp. 433 – 453. · Zbl 0664.14028 · doi:10.1007/BFb0082787 · doi.org
[29] Henry C. Pinkham, A Castelnuovo bound for smooth surfaces, Invent. Math. 83 (1986), no. 2, 321 – 332. · Zbl 0612.14029 · doi:10.1007/BF01388966 · doi.org
[30] Z. Ran, On projective varieties of codimension 2, Invent. Math. 73 (1983), no. 2, 333 – 336. · Zbl 0521.14018 · doi:10.1007/BF01394029 · doi.org
[31] Ziv Ran, Local differential geometry and generic projections of threefolds, J. Differential Geom. 32 (1990), no. 1, 131 – 137. · Zbl 0788.14037
[32] E. Sato, Babylonian tower theorem on varieties, preprint. · Zbl 0782.14040
[33] J. G. Semple and L. Roth, Introduction to Algebraic Geometry, Oxford, at the Clarendon Press, 1949. · Zbl 0041.27903
[34] F. Severi, Si alcune questioni di postualzione, Rend. Circ. Mat. Palermo 17 (1903), 73-103. · JFM 34.0700.01
[35] Eckart Viehweg, Vanishing theorems, J. Reine Angew. Math. 335 (1982), 1 – 8. · Zbl 0485.32019 · doi:10.1515/crll.1982.335.1 · doi.org
[36] Jonathan M. Wahl, The Jacobian algebra of a graded Gorenstein singularity, Duke Math. J. 55 (1987), no. 4, 843 – 871. · Zbl 0644.14001 · doi:10.1215/S0012-7094-87-05540-2 · doi.org
[37] Jonathan Wahl, Gaussian maps on algebraic curves, J. Differential Geom. 32 (1990), no. 1, 77 – 98. · Zbl 0724.14022
[38] -, Gaussian maps and tensor products of irreducible representations (to appear). · Zbl 0764.20022
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.