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Transactions of the American Mathematical Society

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Curvilinear base points, local complete intersection and Koszul syzygies in biprojective spaces

Authors: J. William Hoffman and Hao Hao Wang
Journal: Trans. Amer. Math. Soc. 358 (2006), 3385-3398
MSC (2000): Primary 14Q10; Secondary 13D02, 14Q05
Published electronically: February 14, 2006
MathSciNet review: 2218980
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ I = \langle f_1 , f_2 , f_3\rangle$ be a bigraded ideal in the bigraded polynomial ring $ k[s, u; t, v]$. Assume that $ I$ has codimension 2. Then $ Z = \mathbb{V}(I) \subset \mathbf{P}^{1} \times \mathbf{P}^{1}$ is a finite set of points. We prove that if $ Z$ is a local complete intersection, then any syzygy of the $ f_i$ vanishing at $ Z$, and in a certain degree range, is in the module of Koszul syzygies. This is an analog of a recent result of Cox and Schenck (2003).

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

  • 1. D. A. Cox, Equations of parametric curves and surfaces via syzygies, Contemporary Mathematics, 286, (2001), 1-20. MR 1874268 (2002i:14056)
  • 2. D. A. Cox, R. N. Goldman and M. Zhang, On the validity of implicitization by moving quadrics for rational surfaces with no base points, J. Symb. Comput., 29, (2000), 419-440. MR 1751389 (2002d:14098)
  • 3. D. A. Cox and H. Schenck, Local complete intersections in $ \mathbb{P}^2$ and Koszul syzygies, Proc. Amer. Math. Soc., 131, (2003), 2007-2014.MR 1963743 (2004b:13010)
  • 4. D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995. MR 1322960 (97a:13001)
  • 5. G.-M. Greuel, G. Pfister and H. Schönemann, SINGULAR 2.0, A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern, 2001,
  • 6. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, 1977. MR 0463157 (57:3116)
  • 7. J. Herzog, Ein Cohen-Macauly-Kriterium mit Anwedungen auf den Konormalenmodul und den Differentialmodule, Math. Z., 163, (1978), 149-162.MR 0512469 (80a:13025)
  • 8. J. W. Hoffman and H. Wang, Castelnuovo-Mumford regularity in biprojective spaces, Adv. Geom. 4 (2004), 513-536.MR 2096526
  • 9. E. Hyry, The diagonal subring and the Cohen-Macaulay property of a multigraded ring, Trans. Amer. Math. Soc., 351, 6, (1999), 2213-2232.MR 1467469 (99i:13005)
  • 10. J. H. Sampson and G. Washnitzer, A Künneth formula for coherent algebraic sheaves, Illinois J. Math., 3, (1959), 389-402. MR 0106906 (21:5636)
  • 11. J. P. Serre, Local Algebra, Springer-Verlag, 2000. MR 1771925 (2001b:13001)
  • 12. R. P. Stanley, Combinatorics and commutative algebra, Birkhäuser, 1983.MR 0725505 (85b:05002)
  • 13. H. Wang, Equations of Parametric Surfaces with Base Points via Syzygies, J. Symbolic Comput. 39 (2005), 73-101.MR 2168242

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Additional Information

J. William Hoffman
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803

Hao Hao Wang
Affiliation: Department of Mathematics, Southeast Missouri State University, Cape Girardeau, Missouri 63755

Keywords: Base points, local complete intersection, syzygy, saturation, projective space
Received by editor(s): May 19, 2003
Received by editor(s) in revised form: May 7, 2004
Published electronically: February 14, 2006
Additional Notes: The authors thank William Adkins and David Cox for numerous discussions and suggestions.
Article copyright: © Copyright 2006 American Mathematical Society

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