Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

CLASSIFICATION OF BETTI DIAGRAMS OF VARIETIES OF ALMOST MINIMAL DEGREE

  • Lee, Wan-Seok (Department of Mathematical Sciences Korea Advanced Institute of Science and Technology) ;
  • Park, Eui-Sung (Department of Mathematics Korea University)
  • Received : 2010.05.22
  • Published : 2011.09.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this article we study the problem to determine all occurring Betti diagrams of varieties $X{\subset}\mathbb{P}^r$ of almost minimal degree, i.e. deg(X) = codim(X; $\mathbb{P}^r$)+2. We describe a realistic picture of how many different kind of Betti diagrams exist at all (Theorem 3.1). By means of the computer algebra system "SINGULAR", we obtain a complete list of all occurring Betti diagrams in the cases where codim$(X,\mathbb{P}^r){\leq}8$.

Keywords

References

  1. M. Brodmann and P. Schenzel, On varieties of almost minimal degree in small codimension, J. Algebra 305 (2006), no. 2, 789-801. https://doi.org/10.1016/j.jalgebra.2006.03.027
  2. M. Brodmann and P. Schenzel, Arithmetic properties of projective varieties of almost minimal degree, J. Algebraic Geom. 16 (2007), no. 2, 347-400. https://doi.org/10.1090/S1056-3911-06-00442-5
  3. M. L. Catalano-Johnson, The possible dimensions of the higher secant varieties, Amer. J. Math. 118 (1996), no. 2, 355-361. https://doi.org/10.1353/ajm.1996.0012
  4. D. Eisenbud and J. Harris, On varieties of minimal degree (a centennial account), Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 3-13, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987.
  5. T. Fujita, Classification Theories of Polarized Varieties, London Mathematical Society Lecture Note Series, 155. Cambridge University Press, Cambridge, 1990.
  6. J. Harris, Algebraic Geometry. A First Course, Springer-Velag New York, 1995.
  7. L. T. Hoa, On minimal free resolutions of projective varieties of degree = codimension+2, J. Pure Appl. Algebra 87 (1993), no. 3, 241-250. https://doi.org/10.1016/0022-4049(93)90112-7
  8. L. T. Hoa, J. Stuckrad, and W. Vogel, Towards a structure theory for projective varieties of degree = codimension + 2, J. Pure Appl. Algebra 71 (1991), no. 2-3, 203-231. https://doi.org/10.1016/0022-4049(91)90148-U
  9. U. Nagel, Arithmetically Buchsbaum divisors on varieties of minimal degree, Trans. Amer. Math. Soc. 351 (1999), no. 11, 4381-4409. https://doi.org/10.1090/S0002-9947-99-02357-0
  10. U. Nagel, Minimal free resolutions of projective subschemes of small degree, Syzygies and Hilbert functions, 209-232, Lect. Notes Pure Appl. Math., 254, Chapman & Hall/CRC, Boca Raton, FL, 2007.
  11. E. Park, Projective curves of degree = codimension+2, Math. Z. 256 (2007), no. 3, 685-697. https://doi.org/10.1007/s00209-007-0101-z
  12. E. Park, Smooth varieties of almost minimal degree, J. Algebra 314 (2007), no. 1, 185-208. https://doi.org/10.1016/j.jalgebra.2007.02.027
  13. E. Park, On secant loci and simple linear projections of some projective varieties, preprint.

Cited by

  1. Projective subvarieties having large Green–Lazarsfeld index vol.351, pp.1, 2012, https://doi.org/10.1016/j.jalgebra.2011.10.041
  2. On syzygies of divisors on rational normal scrolls vol.287, pp.11-12, 2014, https://doi.org/10.1002/mana.201300085