Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On syzygies of Segre embeddings


Author: Elena Rubei
Journal: Proc. Amer. Math. Soc. 130 (2002), 3483-3493
MSC (2000): Primary 14M25, 13D02
Published electronically: May 9, 2002
MathSciNet review: 1918824
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the syzygies of the ideals of the Segre embeddings. Let $d \in {\mathbf N}$, $ d \geq 3$; we prove that the line bundle ${\mathcal O}(1,...,1)$on the $P^1 \times ... \times P^1 $ ($d$ copies) satisfies Property $N_p$ of Green-Lazarsfeld if and only if $p \leq 3$. Besides we prove that if we have a projective variety not satisfying Property $N_p$ for some $p$, then the product of it with any other projective variety does not satisfy Property $N_p$. From this we also deduce other corollaries about syzygies of Segre embeddings.


References [Enhancements On Off] (What's this?)

  • [B-M] Ş. Bărcănescu and N. Manolache, Betti numbers of Segre-Veronese singularities, Rev. Roumaine Math. Pures Appl. 26 (1981), no. 4, 549–565. MR 627814
  • [B-S] D. Bayer, M. Stillman Macaulay: A system for computation in algebraic geometry and commutative algebra. It can be downloaded from math.columbia.edu/bayer/macaulay via anonymous ftp.
  • [C-M] A. Campillo and C. Marijuan, Higher order relations for a numerical semigroup, Sém. Théor. Nombres Bordeaux (2) 3 (1991), no. 2, 249–260. MR 1149797
  • [C-P] Antonio Campillo and Pilar Pisón, L’idéal d’un semi-groupe de type fini, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 12, 1303–1306 (French, with English and French summaries). MR 1226120
  • [G-P] F.J. Gallego, B.P. Purnaprajna Some results on rational surfaces and Fano varieties J. Reine Angew Math. 538, 25-55 (2001)
  • [Gr1-2] Mark L. Green, Koszul cohomology and the geometry of projective varieties, J. Differential Geom. 19 (1984), no. 1, 125–171. MR 739785
    Mark L. Green, Koszul cohomology and the geometry of projective varieties. II, J. Differential Geom. 20 (1984), no. 1, 279–289. MR 772134
  • [Gr3] Mark L. Green, Koszul cohomology and geometry, Lectures on Riemann surfaces (Trieste, 1987) World Sci. Publ., Teaneck, NJ, 1989, pp. 177–200. MR 1082354
  • [G-L] Mark Green and Robert Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Invent. Math. 83 (1986), no. 1, 73–90. MR 813583, 10.1007/BF01388754
  • [J-P-W] T. Józefiak, P. Pragacz, and J. Weyman, Resolutions of determinantal varieties and tensor complexes associated with symmetric and antisymmetric matrices, Young tableaux and Schur functors in algebra and geometry (Toruń, 1980), Astérisque, vol. 87, Soc. Math. France, Paris, 1981, pp. 109–189. MR 646819
  • [Las] Alain Lascoux, Syzygies des variétés déterminantales, Adv. in Math. 30 (1978), no. 3, 202–237 (French). MR 520233, 10.1016/0001-8708(78)90037-3
  • [O-P] G. Ottaviani, R. Paoletti Syzygies of Veronese embeddings Compositio Mathematica 125, 31-37 (2001) CMP 2001:09
  • [P-W] Piotr Pragacz and Jerzy Weyman, Complexes associated with trace and evaluation. Another approach to Lascoux’s resolution, Adv. in Math. 57 (1985), no. 2, 163–207. MR 803010, 10.1016/0001-8708(85)90052-0
  • [St] Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR 1363949

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 14M25, 13D02

Retrieve articles in all journals with MSC (2000): 14M25, 13D02


Additional Information

Elena Rubei
Affiliation: Dipartimento di Matematica “U. Dini”, via Morgagni 67/A, 50134 Firenze, Italia
Email: rubei@math.unifi.it

DOI: https://doi.org/10.1090/S0002-9939-02-06597-8
Received by editor(s): December 20, 2000
Received by editor(s) in revised form: July 13, 2001
Published electronically: May 9, 2002
Communicated by: Michael Stillman
Article copyright: © Copyright 2002 American Mathematical Society