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
DOI: https://doi.org/10.1090/S0002-9939-02-06597-8
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] S. Barcanescu, N. Manolache Betti numbers of Segre-Veronese singularities Rev. Roumaine Math. Pures Appl. 26 no.4, 549-565 (1981) MR 82j:13029
  • [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, C. Marijuan Higher relations for a numerical semigroup Sem. Theorie Nombres Bordeaux 3, 249-260 (1991) MR 93d:13027
  • [C-P] A. Campillo, P. Pison L'ideal d'un semigroupe de type fini Comptes Rendus Acad. Sci. Paris Serie I, 316, 1303-1306 (1993) MR 94b:20055
  • [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] M. Green Koszul cohomology and the geometry of projective varieties I,II J. Differ. Geom. 20, 125-171, 279-289 (1984) MR 85e:14022; MR 86j:14011
  • [Gr3] M. Green Koszul cohomology and geometry, in: M. Cornalba et al. (eds), Lectures on Riemann Surfaces, World Scientific Press (1989) MR 91k:14012
  • [G-L] M. Green, R. Lazarsfeld On the projective normality of complete linear series on an algebraic curve Invent. Math. 83, 73-90 (1986) MR 87g:14022
  • [J-P-W] T. Josefiak, P. Pragacz, J.Weyman Resolutions of determinantal varieties and tensor complexes associated with symmetric and antisymmetric matrices Asterisque 87-88, 109-189 (1981) MR 83j:14044
  • [Las] A. Lascoux Syzygies des variétés determinantales Adv. in Math. 30, 202-237 (1978) MR 80j:14043
  • [O-P] G. Ottaviani, R. Paoletti Syzygies of Veronese embeddings Compositio Mathematica 125, 31-37 (2001) CMP 2001:09
  • [P-W] P. Pragacz, J.Weyman Complexes associated with trace and evalutation. Another approach to Lascoux's resolution Adv. Math. 57, 163-207 (1985) MR 87f:14030
  • [St] B. Sturmfels, Gröbner bases and convex polytopes, University Lecture Series American Mathematical Society 8 (1996). MR 97b:13034

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

American Mathematical Society