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

Hoffman, J.; Wang, Hao

doi:10.1090/S0002-9947-06-04119-5

Curvilinear base points, local complete intersection and Koszul syzygies in biprojective spaces
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by J. William Hoffman and Hao Hao Wang PDF

Trans. Amer. Math. Soc. 358 (2006), 3385-3398 Request permission

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

David A. Cox, Equations of parametric curves and surfaces via syzygies, Symbolic computation: solving equations in algebra, geometry, and engineering (South Hadley, MA, 2000) Contemp. Math., vol. 286, Amer. Math. Soc., Providence, RI, 2001, pp. 1–20. MR 1874268, DOI 10.1090/conm/286/04751
David Cox, Ronald Goldman, and Ming Zhang, On the validity of implicitization by moving quadrics of rational surfaces with no base points, J. Symbolic Comput. 29 (2000), no. 3, 419–440. MR 1751389, DOI 10.1006/jsco.1999.0325
David Cox and Hal Schenck, Local complete intersections in ${\Bbb P}^2$ and Koszul syzygies, Proc. Amer. Math. Soc. 131 (2003), no. 7, 2007–2014. MR 1963743, DOI 10.1090/S0002-9939-02-06804-1
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
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, http://www.singular.uni-kl.de.
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
Jürgen Herzog, Ein Cohen-Macaulay-Kriterium mit Anwendungen auf den Konormalenmodul und den Differentialmodul, Math. Z. 163 (1978), no. 2, 149–162 (German). MR 512469, DOI 10.1007/BF01214062
J. William Hoffman and Hao Hao Wang, Castelnuovo-Mumford regularity in biprojective spaces, Adv. Geom. 4 (2004), no. 4, 513–536. MR 2096526, DOI 10.1515/advg.2004.4.4.513
Eero Hyry, The diagonal subring and the Cohen-Macaulay property of a multigraded ring, Trans. Amer. Math. Soc. 351 (1999), no. 6, 2213–2232. MR 1467469, DOI 10.1090/S0002-9947-99-02143-1
J. H. Sampson and G. Washnitzer, A Künneth formula for coherent algebraic sheaves, Illinois J. Math. 3 (1959), 389–402. MR 106906, DOI 10.1215/ijm/1255455260
Jean-Pierre Serre, Local algebra, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. Translated from the French by CheeWhye Chin and revised by the author. MR 1771925, DOI 10.1007/978-3-662-04203-8
Richard P. Stanley, Combinatorics and commutative algebra, Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1983. MR 725505, DOI 10.1007/978-1-4899-6752-7
William A. Adkins, J. William Hoffman, and Hao Hao Wang, Equations of parametric surfaces with base points via syzygies, J. Symbolic Comput. 39 (2005), no. 1, 73–101. MR 2168242, DOI 10.1016/j.jsc.2004.09.007