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37
Linear Systems of Plane Curves
 Notices of the AMS
, 1999
"... Interpolation with polynomials is a subject that has occupied mathematicians ’ minds for millenia. The general problem can be informally phrased as: Given a set of points {(xi,yi)} in the plane, find a ..."
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Cited by 21 (1 self)
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Interpolation with polynomials is a subject that has occupied mathematicians ’ minds for millenia. The general problem can be informally phrased as: Given a set of points {(xi,yi)} in the plane, find a
Linear systems with multiple base points
 in P 2 , Adv. Geom
"... Abstract: Given positive integers m1, m2,..., mn, and t, and n points of P 2 in general position, consider the linear system of curves of degree t which have multiplicity at least mi at the ith point. In this paper bounds are given for the minimal degree such that the linear system is nonempty and ..."
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Cited by 13 (8 self)
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Abstract: Given positive integers m1, m2,..., mn, and t, and n points of P 2 in general position, consider the linear system of curves of degree t which have multiplicity at least mi at the ith point. In this paper bounds are given for the minimal degree such that the linear system is nonempty and for the minimal degree such that it is regular, often improving substantially what was previously known. As an application, for certain m and n (both of which can be large) the Hilbert function and minimal free resolution are determined for symbolic powers I (m) for the ideal I defining n general points of P 2; very few cases were known previously with both m and n large. We work over an algebraically closed field of characteristic zero. I.
Problems and Progress: A survey on fat points in P²
, 2001
"... This paper, which expands on a talk given at the International Workshop on Fat ..."
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Cited by 13 (2 self)
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This paper, which expands on a talk given at the International Workshop on Fat
Free Resolutions of Fat Point Ideals on P 2
 Journal of Pure and Applied Algebra
, 1998
"... Minimal free resolutions for homogeneous ideals corresponding to certain 0dimensional subschemes of P 2 defined by sheaves of complete ideals are determined implicitly. All work is over an algebraically closed field of arbitrary characteristic. I. ..."
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Cited by 9 (1 self)
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Minimal free resolutions for homogeneous ideals corresponding to certain 0dimensional subschemes of P 2 defined by sheaves of complete ideals are determined implicitly. All work is over an algebraically closed field of arbitrary characteristic. I.
On Nagata’s Conjecture
 J. Alg
"... Abstract: This paper gives an improved lower bound on the degrees d such that for general points p1,..., pn ∈ P 2 and m> 0 there is a plane curve of degree d vanishing at each pi with multiplicity at least m. In this paper we work over an arbitrary algebraically closed field. For positive integers m ..."
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Cited by 6 (3 self)
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Abstract: This paper gives an improved lower bound on the degrees d such that for general points p1,..., pn ∈ P 2 and m> 0 there is a plane curve of degree d vanishing at each pi with multiplicity at least m. In this paper we work over an arbitrary algebraically closed field. For positive integers m and n, define d(m, n) to be the least integer d such that for general points p1,..., pn ∈ P2 there is a curve of degree d vanishing at each point pi with multiplicity at least m. For n ≥ 10, Nagata [N1] conjectures that d(m, n)> m √ n, and proves this when n> 9 is a square. (For n ≤ 9, applying methods of [N2] it can be
systems in P 2 with base points of bounded multiplicity
 Department of Mathematics, Harvard University
"... Abstract. We present a proof of the HarbourneHirschowitz conjecture for linear systems with multiple points of order 7 or less. This uses a wellknown degeneration of the plane developed by Ciliberto and Miranda as well as a combinatorial game that arises from specializing points onto lines. Conten ..."
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Cited by 5 (0 self)
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Abstract. We present a proof of the HarbourneHirschowitz conjecture for linear systems with multiple points of order 7 or less. This uses a wellknown degeneration of the plane developed by Ciliberto and Miranda as well as a combinatorial game that arises from specializing points onto lines. Contents
A counterexample to a conjecture on linear systems
"... Abstract. In his paper [1] Ciliberto proposes a conjecture in order to characterize special linear systems of P n through multiple base points. In this note we give a counterexample to this conjecture by showing that there is a substantial difference between the speciality of linear systems on P 2 a ..."
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Cited by 4 (2 self)
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Abstract. In his paper [1] Ciliberto proposes a conjecture in order to characterize special linear systems of P n through multiple base points. In this note we give a counterexample to this conjecture by showing that there is a substantial difference between the speciality of linear systems on P 2 and those of P 3.
Curves having one place at infinity and linear systems on rational surfaces
 J. Pure Appl. Algebra
"... Abstract. Denoting by Ld(m0, m1,..., mr) the linear system of plane curves passing through r + 1 generic points p0, p1,..., pr of the projective plane with multiplicity mi (or larger) at each pi, we prove the HarbourneHirschowitz Conjecture for linear systems Ld(m0, m1,..., mr) determined by a wide ..."
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Cited by 4 (4 self)
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Abstract. Denoting by Ld(m0, m1,..., mr) the linear system of plane curves passing through r + 1 generic points p0, p1,..., pr of the projective plane with multiplicity mi (or larger) at each pi, we prove the HarbourneHirschowitz Conjecture for linear systems Ld(m0, m1,..., mr) determined by a wide family of systems of multiplicities m = (mi) r i=0 and arbitrary degree d. Moreover, we provide an algorithm for computing a bound of the regularity of an arbitrary system m and we give its exact value when m is in the above family. To do that, we prove an H 1vanishing theorem for line bundles on surfaces associated with some pencils “at infinity”. 1.