Resolutions of ideals of quasiuniform fat point subschemes of
Authors:
Brian Harbourne, Sandeep Holay and Stephanie Fitchett
Journal:
Trans. Amer. Math. Soc. 355 (2003), 593608
MSC (2000):
Primary 13P10, 14C99; Secondary 13D02, 13H15
Published electronically:
October 4, 2002
MathSciNet review:
1932715
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The notion of a quasiuniform fat point subscheme is introduced and conjectures for the Hilbert function and minimal free resolution of the ideal defining are put forward. In a large range of cases, it is shown that the Hilbert function conjecture implies the resolution conjecture. In addition, the main result gives the first determination of the resolution of the th symbolic power of an ideal defining general points of when both and are large (in particular, for infinitely many for each of infinitely many , and for infinitely many for every ). Resolutions in other cases, such as ``fat points with tails'', are also given. Except where an explicit exception is made, all results hold for an arbitrary algebraically closed field . As an incidental result, a bound for the regularity of is given which is often a significant improvement on previously known bounds.
 [AH]
J. Alexander and A. Hirschowitz. An asymptotic vanishing theorem for generic unions of multiple points, Invent. Math. 140 (2000), no. 2, 303325. MR 2001i:14024
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G. Campanella. Standard bases of perfect homogeneous polynomial ideals of height , J. Algebra 101 (1986), 4760. MR 87f:13005
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M. V. Catalisano. ``Fat'' points on a conic, Comm. Algebra 19 (1991), no. 8, 21532168. MR 93b:14016
 [CM1]
C. Ciliberto and R. Miranda. Degenerations of planar linear systems, J. Reine Angew. Math. 501 (1998), 191220. MR 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), 40374050. MR 2000m:14006
 [DGM]
E. D. Davis, A. V. Geramita, and P. Maroscia. Perfect homogeneous ideals: Dubreil's theorems revisited, Bull. Sci. Math. (2) 108 (1984), 143185. MR 86m:13024
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P. Dubreil. Sur quelques propriétés des systèmes de points dans le plan et des courbes gauches algébriques, Bull. Soc. Math. France, 61 (1933), 258283.
 [E]
L. Evain. La fonction de Hilbert de la réunion de gros points génériques de 2 de même multiplicité, J. Algebra Geom. 8 (1999), 787796. MR 2000e:13023
 [F1]
S. Fitchett. On bounding the number of generators for fat point ideals on the projective plane, J. Algebra 236 (2001), 502521. MR 2002c:13028
 [F2]
S. Fitchett. Maps of linear systems on blow ups of the projective plane, J. Pure and Applied Algebra 156 (2001), 114. MR 2002c:14013
 [FHH]
S. Fitchett, B. Harbourne and S. Holay. Resolutions of fat point ideals involving eight general points of 2, J. Algebra 244 (2001), 684705. MR 2002g:14089
 [GGR]
A. V. Geramita, D. Gregory and L. Roberts. Monomial ideals and points in projective space, J. Pure and Appl. Algebra 40 (1986), 3362. MR 87d:13023
 [GO]
A. V. Geramita, and F. Orrechia. Minimally generating ideals defining certain tangent cones, J. Algebra 78 (1982), 3657. MR 84e:13028
 [Gi1]
A. Gimigliano. Our thin knowledge of fat points, Queen's papers in Pure and Applied Mathematics, no. 83, The Curves Seminar at Queen's, vol. VI (1989). MR 91a:14007
 [Gi2]
A. Gimigliano. Regularity of linear systems of plane curves, J. Algebra 124 (1989), 447460. MR 90i:14005
 [Ha1]
B. Harbourne. The geometry of rational surfaces and Hilbert functions of points in the plane, Canad. Math. Soc. Conf. Proc. 6 (1986), 95111. MR 87k:14041
 [Ha2]
B. Harbourne. Iterated blowups and moduli for rational surfaces, in: Algebraic Geometry, Sundance 1986, Lecture Notes in Math., 1311 (1988), 101117. MR 90b:14009
 [Ha3]
B. Harbourne. Points in Good Position in 2, in: Zerodimensional schemes, Proceedings of the International Conference held in Ravello, Italy, June 813, 1992, De Gruyter, 1994. MR 95m:14005
 [Ha4]
B. Harbourne. Anticanonical rational surfaces, Trans. Amer. Math. Soc. 349 (1997), 11911208. MR 97f:14007
 [Ha5]
B. Harbourne. Free Resolutions of Fat Point Ideals on 2, J. Pure and Applied Algebra 125 (1998), 213234. MR 99d:13016
 [Ha6]
B. Harbourne. The ideal generation problem for fat points, J. Pure and Applied Algebra 145 (2000), 165182. MR 2001c:13028
 [Ha7]
B. Harbourne. An Algorithm for Fat Points on 2, Canad. J. Math. 52 (2000), 123140. MR 2001g:13037
 [Ha8]
B. Harbourne. Problems and Progress: A survey on fat points in 2, Queen's papers in Pure and Applied Mathematics, The Curves Seminar at Queen's, vol. 123, 2002. CMP 2002:12
 [HR]
B. Harbourne and J. Roé. Linear systems with multiple base points in 2, preprint (2001).
 [Hi]
A. Hirschowitz. Une conjecture pour la cohomologie des diviseurs sur les surfaces rationelles génériques, Journ. Reine Angew. Math. 397 (1989), 208213. MR 90g:14021
 [Id]
M. Idà. The minimal free resolution for the first infinitesimal neighborhoods of general points in the plane, J. Algebra 216 (1999), 741753. MR 2000d:13022
 [Mi]
R. Miranda. Linear Systems of Plane Curves, Notices Amer. Math. Soc. 46(2) (1999), 192202. MR 99m:14012
 [Mu]
D. Mumford. Varieties defined by quadratic equations, in: Questions on Algebraic Varieties, Corso C.I.M.E. 1969 Rome: Cremonese, 1970, 29100. MR 44:209
 [N1]
M. Nagata. On the 14th problem of Hilbert, Amer. J. Math. 81 (1959), 766772. MR 21:4151
 [N2]
M. Nagata. On rational surfaces, II, Mem. Coll. Sci. Univ. Kyoto, Ser. A Math. 33 (1960), 271293. MR 23:A3740
 [R]
J. Roé. Linear systems of plane curves with imposed multiple points, Illinois J. Math. 45 (2001), 895906.
 [X]
G. Xu. Ample line bundles on smooth surfaces, Jour. Reine Angew. Math. 469 (1995), 199209. MR 96k:14003
 [AH]
 J. Alexander and A. Hirschowitz. An asymptotic vanishing theorem for generic unions of multiple points, Invent. Math. 140 (2000), no. 2, 303325. MR 2001i:14024
 [Cam]
 G. Campanella. Standard bases of perfect homogeneous polynomial ideals of height , J. Algebra 101 (1986), 4760. MR 87f:13005
 [Cat]
 M. V. Catalisano. ``Fat'' points on a conic, Comm. Algebra 19 (1991), no. 8, 21532168. MR 93b:14016
 [CM1]
 C. Ciliberto and R. Miranda. Degenerations of planar linear systems, J. Reine Angew. Math. 501 (1998), 191220. MR 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), 40374050. MR 2000m:14006
 [DGM]
 E. D. Davis, A. V. Geramita, and P. Maroscia. Perfect homogeneous ideals: Dubreil's theorems revisited, Bull. Sci. Math. (2) 108 (1984), 143185. MR 86m:13024
 [Du]
 P. Dubreil. Sur quelques propriétés des systèmes de points dans le plan et des courbes gauches algébriques, Bull. Soc. Math. France, 61 (1933), 258283.
 [E]
 L. Evain. La fonction de Hilbert de la réunion de gros points génériques de 2 de même multiplicité, J. Algebra Geom. 8 (1999), 787796. MR 2000e:13023
 [F1]
 S. Fitchett. On bounding the number of generators for fat point ideals on the projective plane, J. Algebra 236 (2001), 502521. MR 2002c:13028
 [F2]
 S. Fitchett. Maps of linear systems on blow ups of the projective plane, J. Pure and Applied Algebra 156 (2001), 114. MR 2002c:14013
 [FHH]
 S. Fitchett, B. Harbourne and S. Holay. Resolutions of fat point ideals involving eight general points of 2, J. Algebra 244 (2001), 684705. MR 2002g:14089
 [GGR]
 A. V. Geramita, D. Gregory and L. Roberts. Monomial ideals and points in projective space, J. Pure and Appl. Algebra 40 (1986), 3362. MR 87d:13023
 [GO]
 A. V. Geramita, and F. Orrechia. Minimally generating ideals defining certain tangent cones, J. Algebra 78 (1982), 3657. MR 84e:13028
 [Gi1]
 A. Gimigliano. Our thin knowledge of fat points, Queen's papers in Pure and Applied Mathematics, no. 83, The Curves Seminar at Queen's, vol. VI (1989). MR 91a:14007
 [Gi2]
 A. Gimigliano. Regularity of linear systems of plane curves, J. Algebra 124 (1989), 447460. MR 90i:14005
 [Ha1]
 B. Harbourne. The geometry of rational surfaces and Hilbert functions of points in the plane, Canad. Math. Soc. Conf. Proc. 6 (1986), 95111. MR 87k:14041
 [Ha2]
 B. Harbourne. Iterated blowups and moduli for rational surfaces, in: Algebraic Geometry, Sundance 1986, Lecture Notes in Math., 1311 (1988), 101117. MR 90b:14009
 [Ha3]
 B. Harbourne. Points in Good Position in 2, in: Zerodimensional schemes, Proceedings of the International Conference held in Ravello, Italy, June 813, 1992, De Gruyter, 1994. MR 95m:14005
 [Ha4]
 B. Harbourne. Anticanonical rational surfaces, Trans. Amer. Math. Soc. 349 (1997), 11911208. MR 97f:14007
 [Ha5]
 B. Harbourne. Free Resolutions of Fat Point Ideals on 2, J. Pure and Applied Algebra 125 (1998), 213234. MR 99d:13016
 [Ha6]
 B. Harbourne. The ideal generation problem for fat points, J. Pure and Applied Algebra 145 (2000), 165182. MR 2001c:13028
 [Ha7]
 B. Harbourne. An Algorithm for Fat Points on 2, Canad. J. Math. 52 (2000), 123140. MR 2001g:13037
 [Ha8]
 B. Harbourne. Problems and Progress: A survey on fat points in 2, Queen's papers in Pure and Applied Mathematics, The Curves Seminar at Queen's, vol. 123, 2002. CMP 2002:12
 [HR]
 B. Harbourne and J. Roé. Linear systems with multiple base points in 2, preprint (2001).
 [Hi]
 A. Hirschowitz. Une conjecture pour la cohomologie des diviseurs sur les surfaces rationelles génériques, Journ. Reine Angew. Math. 397 (1989), 208213. MR 90g:14021
 [Id]
 M. Idà. The minimal free resolution for the first infinitesimal neighborhoods of general points in the plane, J. Algebra 216 (1999), 741753. MR 2000d:13022
 [Mi]
 R. Miranda. Linear Systems of Plane Curves, Notices Amer. Math. Soc. 46(2) (1999), 192202. MR 99m:14012
 [Mu]
 D. Mumford. Varieties defined by quadratic equations, in: Questions on Algebraic Varieties, Corso C.I.M.E. 1969 Rome: Cremonese, 1970, 29100. MR 44:209
 [N1]
 M. Nagata. On the 14th problem of Hilbert, Amer. J. Math. 81 (1959), 766772. MR 21:4151
 [N2]
 M. Nagata. On rational surfaces, II, Mem. Coll. Sci. Univ. Kyoto, Ser. A Math. 33 (1960), 271293. MR 23:A3740
 [R]
 J. Roé. Linear systems of plane curves with imposed multiple points, Illinois J. Math. 45 (2001), 895906.
 [X]
 G. Xu. Ample line bundles on smooth surfaces, Jour. Reine Angew. Math. 469 (1995), 199209. MR 96k:14003
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Additional Information
Brian Harbourne
Affiliation:
Department of Mathematics and Statistics, University of NebraskaLincoln, Lincoln, Nebraska 685880323
Email:
bharbour@math.unl.edu
Sandeep Holay
Affiliation:
Department of Mathematics, Southeast Community College, Lincoln, Nebraska 68508
Email:
sholay@southeast.edu
Stephanie Fitchett
Affiliation:
Florida Atlantic University, Honors College, Jupiter, Florida 33458
Email:
sfitchet@fau.edu
DOI:
http://dx.doi.org/10.1090/S0002994702031240
PII:
S 00029947(02)031240
Keywords:
Ideal generation conjecture,
symbolic powers,
resolution,
fat points,
maximal rank.
Received by editor(s):
December 31, 2000
Received by editor(s) in revised form:
May 2, 2002
Published electronically:
October 4, 2002
Additional Notes:
The first author benefitted from a National Science Foundation grant.
Article copyright:
© Copyright 2002
American Mathematical Society
