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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Betti numbers for fat point ideals in the plane: A geometric approach

Authors: Alessandro Gimigliano, Brian Harbourne and Monica Idà
Journal: Trans. Amer. Math. Soc. 361 (2009), 1103-1127
MSC (2000): Primary 14C20, 13P10; Secondary 14J26, 14J60.
Published electronically: September 9, 2008
MathSciNet review: 2452836
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Abstract: We consider the open problem of determining the graded Betti numbers for fat point subschemes $ Z$ supported at general points of $ \mathbf{P}^2$. We relate this problem to the open geometric problem of determining the splitting type of the pullback of $ \Omega_{\mathbf{P}^2}$ to the normalization of certain rational plane curves. We give a conjecture for the graded Betti numbers which would determine them in all degrees but one for every fat point subscheme supported at general points of $ \mathbf{P}^2$. We also prove our Betti number conjecture in a broad range of cases. An appendix discusses many more cases in which our conjecture has been verified computationally and provides a new and more efficient computational approach for computing graded Betti numbers in certain degrees. It also demonstrates how to derive explicit conjectural values for the Betti numbers and how to compute splitting types.

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  • [AC] E. Arbarello and M. Cornalba, Footnotes to a paper of B. Segre, Math. Ann., 256 (1981), 341-362. MR 626954 (83d:14016)
  • [As] M.-G. Ascenzi, The restricted tangent bundle of a rational curve in $ \mathbf{P}^2$, Comm. Algebra 16 (1988), no. 11, 2193-2208. MR 962311 (89i:14023)
  • [BI] E. Ballico and M. Idà, On the minimal free resolution for fat point schemes of multiplicity at most $ 3$ in $ \mathbf{P}^2$, J. Pure Appl. Algebra 212 (2008), 1756-1769.
  • [BZ] A. Buckley and M. Zompatori, Linear systems of plane curves with a composite number of base points of equal multiplicity, Trans. Amer. Math. Soc. 355 (2003), no. 2, 539-549. MR 1932712 (2003j:14039)
  • [Cas] G. Castelnuovo, Ricerche generali sopra i sistemi lineari di curve piane, Mem. Accad. Sci. Torino, II 42 (1891).
  • [Cat] M. V. Catalisano, ``Fat'' points on a conic, Comm. Alg. 19(8) (1991), 2153-2168. MR 1123117 (93b:14016)
  • [CM1] C. Ciliberto and R. Miranda, Degenerations of planar linear systems, J. Reine Angew. Math. 501 (1998), 191-220. MR 1637857 (2000m:14005)
  • [CM2] C. Ciliberto and R. Miranda, Linear systems of plane curves with base points of equal multiplicity, Trans. Amer. Math. Soc. 352 (2000), no. 9, 4037-4050. MR 1637062 (2000m:14006)
  • [CM3] C. Ciliberto and R. Miranda. Nagata's conjecture for a square or nearly-square number of points, Ric. Mat. 55 (2006), no. 1, 71-78. MR 2248163 (2007d:14098)
  • [CCMO] C. Ciliberto, F. Cioffi, R. Miranda and F. Orecchia, Bivariate Hermite interpolation and linear systems of plane curves with base fat points, in: Computer mathematics, 87-102, Lecture Notes Ser. Comput., 10, World Sci. Publishing, River Edge, NJ, 2003. MR 2061827 (2005c:41002)
  • [dF] T. de Fernex, Negative curves on very general blow-ups of $ \mathbf{P}^2$, Projective varieties with unexpected properties, 199-207, Walter de Gruyter GmbH & Co. KG, Berlin, 2005. MR 2202253 (2006k:14018)
  • [E1] L. Evain, La fonction de Hilbert de la réunion de $ 4^h$ gros points génériques de $ \mathbf{P}^2$ de même multiplicité, J. Algebraic Geom. 8 (1999), no. 4, 787-796. MR 1703614 (2000e:13023)
  • [E2] L. Evain, Computing limit linear series with infinitesimal methods, Ann. Inst. Fourier 57 (2007), 1947-1974. MR 2377892
  • [F1] S. Fitchett, On Bounding the Number of Generators for Fat Point Ideals on the Projective Plane, J. Algebra, 236 (2001), 502-521. MR 1813489 (2002c:13028)
  • [F2] S. Fitchett, Corrigendum to: ``On bounding the number of generators for fat point ideals on the projective plane'' [J. Algebra 236 (2001), no. 2, 502-521], J. Algebra 276 (2004), no. 1, 417-419. MR 2054405 (2005a:13029)
  • [F3] S. Fitchett, Maps of linear systems on blow-ups of the projective plane, J. Pure Appl. Algebra 156 (2001), 1-14. MR 1807013 (2002c:14013)
  • [FHH] S. Fitchett, B. Harbourne and S. Holay, Resolutions of Fat Point Ideals Involving Eight General Points of $ \mathbf{P}^2$, J. Algebra 244 (2001), 684-705. MR 1859044 (2002g:14089)
  • [GGR] A. V. Geramita, D. Gregory and L. Roberts, Monomial ideals and points in projective space, J. Pure Appl. Alg. 40 (1986), 33-62. MR 825180 (87d:13023)
  • [G] A. Gimigliano, On linear systems of plane curves, Thesis, Queen's University, Kingston (1987).
  • [GHI1] A. Gimigliano, B. Harbourne and M. Idà, The role of the cotangent bundle in resolving ideals of fat points in the plane, in press, J. Pure Appl. Algebra (2008).
  • [GHI2] A. Gimigliano, B. Harbourne and M. Idà, Betti numbers for fat point ideals in the plane: A geometric approach, arXiv:0706.2588 [The source TEXfile includes nonprinting explicit Macaulay 2 scripts for carrying out computations and for computing splitting types.]
  • [GI] A. Gimigliano and M. Idà, The ideal resolution for generic $ 3$-fat points in $ \mathbf{P}^2$, J. Pure Appl. Algebra 187 (2004), no. 1-3, 99-128. MR 2027898 (2005e:13017)
  • [GS] D. Grayson, and M. Stillman, Macaulay 2, a software system for research in algebraic geometry, Available at
  • [Ha1] B. Harbourne, An Algorithm for Fat Points on $ \mathbf{P}^2$, Can. J. Math. 52 (2000), 123-140. MR 1745704 (2001g:13037)
  • [Ha2] B. Harbourne, The Ideal Generation Problem for Fat Points, J. Pure Appl. Alg. 145(2), 165-182 (2000). MR 1733250 (2001c:13028)
  • [Ha3] B. Harbourne, Free Resolutions of Fat Point Ideals on $ \mathbf{P}^2$, J. Pure Appl. Alg. 125, 213-234 (1998). MR 1600024 (99d:13016)
  • [Ha4] B. Harbourne, The geometry of rational surfaces and Hilbert functions of points in the plane. Can. Math. Soc. Conf. Proc., vol. 6 (1986), 95-111. MR 846019 (87k:14041)
  • [Ha5] B. Harbourne, Very ample divisors on rational surfaces, Math. Ann. 272, 139-153 (1985). MR 794097 (86k:14026)
  • [HHF] B. Harbourne, S. Holay and S. Fitchett, Resolutions of ideals of quasiuniform fat point subschemes of $ {\bf P}^2$, Trans. Amer. Math. Soc. 355 (2003), no. 2, 593-608. MR 1932715 (2004c:13019)
  • [HR] B. Harbourne and J. Roé. Linear systems with multiple base points in $ {\bf P}^2$, Adv. Geom. 4 (2004), 41-59. MR 2155364 (2006b:13038)
  • [Hi1] A. Hirschowitz, Une conjecture pour la cohomologie des diviseurs sur les surfaces rationelles génériques, Journ. Reine Angew. Math. 397 (1989), 208-213. MR 993223 (90g:14021)
  • [Hi2] A. Hirschowitz. La méthode d'Horace pour l'interpolation à plusieurs variables, Manus. Math. 50 (1985), 337-388. MR 784148 (86j:14013)
  • [Ia] A. Iarrobino, Inverse system of a symbolic power III: Thin algebras and fat points, Compositio Math. 108 (1997), no. 3, 319-356. MR 1473851 (98k:13017)
  • [I] M. Idà, The minimal free resolution for the first infinitesimal neighborhoods of $ n$ general points in the plane, J. Alg. 216 (1999), 741-753. MR 1692957 (2000d:13022)
  • [Mi] T. Mignon, Systèmes de courbes planes à singularités imposées: Le cas des multiplicités inférieures ou égales à quatre, J. Pure Appl. Algebra 151 (2000), no. 2, 173-195. MR 1775572 (2001g:14048)
  • [Mu1] D. Mumford, Varieties defined by quadratic equations, in: Questions on algebraic varieties, Corso C.I.M.E. 1969 Rome: Cremonese, 1970, 30-100. MR 0282975 (44:209)
  • [Mu2] D. Mumford, Lectures on curves on an algebraic surface, Princeton 1966. MR 0209285 (35:187)
  • [N1] M. Nagata, On the $ 14$-th problem of Hilbert, Amer. J. Math. 81 (1959), 766-772. MR 0105409 (21:4151)
  • [N2] M. Nagata, On rational surfaces, II, Mem. Coll. Sci. Univ. Kyoto, Ser. A Math. 33 (1960), 271-293. MR 0126444 (23:A3740)
  • [P] F. Palatini, Sulla rappresentazione delle forme ternerie mediante la somma di potenze di forme lineare, Atti Accad. Naz. Lincei Rend. (Serie V) vol. 12, n. 3 (1903).
  • [R] J. Roé, Limit linear systems and applications, preprint (math/0602213), 2006.
  • [S] B. Segre, Alcune questioni su insiemi finiti di punti in Geometria Algebrica, Atti del Convegno Internaz. di Geom. Alg., Torino (1961). MR 0169119 (29:6372)
  • [Y] S. Yang, Linear series in $ \mathbf{P}^2$ with base points of bounded multiplicity, J. Algebraic Geom. 16 (2007), no. 1, 19-38. MR 2257318 (2007i:14011) Additional Citations for the Appendix:
  • [A.Ar] M. Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962), 485-497. MR 0146182 (26:3704)
  • [A.H1] B. Harbourne, Blowings-up of $ \mathbf{P}^2$ and their blowings-down, Duke Math. J. 52, 129-148 (1985). MR 791295 (86m:14026)
  • [A.H2] B. Harbourne, Complete linear systems on rational surfaces, Trans. Amer. Math. Soc. 289, 213-226 (1985). MR 779061 (86h:14030)

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Additional Information

Alessandro Gimigliano
Affiliation: Dipartimento di Matematica e CIRAM, Università di Bologna, 40126 Bologna, Italy

Brian Harbourne
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588-0130

Monica Idà
Affiliation: Dipartimento di Matematica, Università di Bologna, 40126 Bologna, Italy

Keywords: Graded Betti numbers, fat points, splitting types.
Received by editor(s): December 29, 2006
Received by editor(s) in revised form: June 15, 2007
Published electronically: September 9, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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