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The Minimal Resolution Conjecture for Points on the Cubic Surface

Part of: Theory of modules and ideals Special varieties Commutative algebra: Homological methods

Published online by Cambridge University Press:  20 November 2018

M. Casanellas
M. Casanellas*
Affiliation:
DepartamentMatematica Aplicada I, ETSEIB UPC, Av. Diagonal 647, 08028-Barcelona. Spain, marta.casanellas@upc.edu
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Abstract

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In this paper we prove that a generalized version of the Minimal Resolution Conjecture given by Mustaţă holds for certain general sets of points on a smooth cubic surface $X\,\subset \,{{\mathbb{P}}^{3}}$. The main tool used is Gorenstein liaison theory and, more precisely, the relationship between the free resolutions of two linked schemes.

MSC classification

Primary: 13D02: Syzygies, resolutions, complexes
Secondary: 13C40: Linkage, complete intersections and determinantal ideals 14M05: Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) 14M07: Low codimension problems
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 1 , 01 February 2009 , pp. 29 - 49
Copyright
Copyright © Canadian Mathematical Society 2009

References

[1] Ballico, E. and Geramita, A. V., The minimal free resolution of the ideal of s general points in P3. Proceedings of the 1984 Vancuver conference on algebraic geometry, CMS Conf. Proc. 6, American Mathematical Society, Providence, RI, 1986, pp. 1–10.Google Scholar
[2] Broadmann, M. P. and Sharp, R. Y., Local cohomology: an algebraic introduction with geometric applications. Cambridge studies in advanced mathematics 60, Cambridge University Press, 1998.Google Scholar
[3] Casanellas, M., Drozd, E., and Hartshorne, R., Gorenstein liaison and ACM sheaves. J. Reine Angew. Math. 584 (2005), 149–171.Google Scholar
[4] Eisenbud, D., Homological algebra on a complete intersection, with an application to group representations. Trans. Amer. Math. Soc. 260 (1980), no. 1, 35–64.Google Scholar
[5] Eisenbud, D., Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics 150, Springer-Verlag, New-York, 1995.Google Scholar
[6] Eisenbud, D., Popescu, S., Schreyer, F.-O., and Walter, C., Exterior algebra methods for the minimal resolution conjecture. Duke Math. J. 112 (2002), no. 2, 379–395.Google Scholar
[7] Farkas, G., Mustaţă, M., and Popa, M., Divisors on Mg,g+1 and the minimal resolution conjecture for points on canonical curves. Ann. Sci. École Norm. Sup. 36 (2003), no. 4, 553–581.Google Scholar
[8] Geramita, A. V., Harima, T., Migliore, J. C., and Shin, Y. S., The Hilbert function of a level algebra. Mem. Amer. Math. Soc. 186 (2007), no. 872.Google Scholar
[9] Giuffrida, S., Maggioni, R., and Ragusa, A., Resolutions of generic points lying on a smooth quadric. Manuscripta Math. 91 (1996), no. 4, 421–444.Google Scholar
[10] Hartshorne, R., Some examples of Gorenstein liaison in codimension three. Collect. Math. 53 (2002), no. 1, 21–48.Google Scholar
[11] Hirschowitz, A. and Simpson, C., La résolution minimale de l’idéal d’un arrangement général ’un grand nombre de points dans Pn. Invent. Math. 126 (1996), no. 3, 467–503.Google Scholar
[12] Lorenzini, A., The minimal resolution conjecture. J. Algebra 156 (1993), no. 1, 5–35.Google Scholar
[13] MacLane, S., Homology. Die Grundlehren der mathematischenWissenschaften 114, Springer-Verlag, Berlin, 1963.Google Scholar
[14] Migliore, J. C., Introduction to liaison theory and deficiencymodules. Progress in Mathematics 165, Birkhäuser, Boston, MA, 1998.Google Scholar
[15] MustaţĂ, M., Graded Betti numbers of general finite subsets of points on projective varieties. Matematiche 53 (1998), suppl., 53–81.Google Scholar
[16] Walter, C., The minimal free resolution of the homogeneous ideal of s general points in P4, Math. Z. 219 (1995), no. 2, 231–234.Google Scholar
[17] Weibel, C. A., An introduction to homological algebra. Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, Cambridge, 1994.Google Scholar