Migliore-Mir\'o-Roig-Nagel [Trans. A.M.S. 2011, arXiv: 0811.1023] show that the weak Lefschet... more Migliore-Mir\'o-Roig-Nagel [Trans. A.M.S. 2011, arXiv: 0811.1023] show that the weak Lefschetz property (WLP) can fail for an ideal I in K[x_1,x_2,x_3,x_4] generated by powers of linear forms. This is in contrast to the analogous situation in K[x_1,x_2,x_3], where WLP always holds [H.Schenck, A.Seceleanu, Proc. A.M.S. 2010, arXiv:0911.0876]. We use the inverse system dictionary to connect I to an ideal of
The notion of a quasiuniform fat point subscheme Z P2 is introduced and conjectures for the Hilbe... more The notion of a quasiuniform fat point subscheme Z P2 is introduced and conjectures for the Hilbert function and minimal free resolution of the ideal I dening Z 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 rst determination of the
Conjectures for the Hilbert function of the m-th symbolic power of the ideal of n general points ... more Conjectures for the Hilbert function of the m-th symbolic power of the ideal of n general points of P2 are verified for infinitely many m for each square n > 9, using an approach developed by the authors in a previous paper. In those cases that n is even, conjectures for the resolution are also verified. Previously, by work of
Working over the complex field and formalizing and sharpening approaches introduced by several au... more Working over the complex field and formalizing and sharpening approaches introduced by several authors, we give a method for verifying when a divisor on a blow up of P^2 at general points is nef. The method is useful both theoretically and when doing computer computations. The main application is to obtain significantly improved explicit lower bounds for multipoint Seshadri constants
The paper (10) raised the question of what the possible Hilbert functions are for fat point subsc... more The paper (10) raised the question of what the possible Hilbert functions are for fat point subschemes of the form 2p1+···+2pr, for all possible choices of r distinct points in P2. We study this problem for r points in P2 over an al- gebraically closed field k of arbitrary characteristic in case either r 8 or the points lie on
Given an immersion $\phi: P^1 \to \P^2$, we give new approaches to determining the splitting of t... more Given an immersion $\phi: P^1 \to \P^2$, we give new approaches to determining the splitting of the pullback of the cotangent bundle. We also give new bounds on the splitting type for immersions which factor as $\phi: P^1 \cong D \subset X \to P^2$, where $X \to P^2$ is obtained by blowing up $r$ distinct points $p_i \in P^2$. As
Our results concern minimal graded free resolutions of fat point ideals for points in a hyperplan... more Our results concern minimal graded free resolutions of fat point ideals for points in a hyperplane. Suppose, for example, that I(m,d) is the ideal defining r given points of multiplicity m in the projective space P^d. Assume that the given points lie in a hyperplane P^{d-1} in P^d, and that the ground field k is algebraically closed of characteristic 0. We give an explicit minimal graded free resolution of I(m,d) in k[P^d] in terms of the minimal graded free resolutions of the ideals I(j,d-1) in k[P^{d-1}] with j < m+1. As a corollary, we give the following formula for the Poincare polynomial P_{m,d} of I(m,d) in terms of the Poincare polynomials P_{j,d-1} of I(j,d-1): P_{m,d} = (1 + XT)(\Sigma_{0<j\le m} T^{m-j}(P_{j,d-1} - 1)) + 1 + XT^m.
Bulletin of The Belgian Mathematical Society-simon Stevin, 2009
We describe an approach for computing arbitrarily accurate estimates for multi-point Seshadri con... more We describe an approach for computing arbitrarily accurate estimates for multi-point Seshadri constants for $n$ generic points of ${\bf P}^{2}$. We apply the approach to obtain improved estimates. We work over an algebraically closed field of characteristic 0.
The graded Betti numbers of the minimal free resolution (and also therefore the Hilbert function)... more The graded Betti numbers of the minimal free resolution (and also therefore the Hilbert function) of the ideal of a fat point subscheme Z of P2 are determined whenever Z is supported at any 6 or fewer distinct points. A broad range of cases in which the points can be infinitely near, related to the classification of normal cubic surfaces,
Bulletin of The Belgian Mathematical Society-simon Stevin - BULL BELG MATH SOC-SIMON STEV, 2009
It is an open problem to determine the Hilbert function and graded Betti numbers for the ideal of... more It is an open problem to determine the Hilbert function and graded Betti numbers for the ideal of a fat point subscheme supported at general points of the projective plane. In fact, there is not yet even a general explicit conjecture for the graded Betti numbers. Here we formulate explicit asymptotic conjectures for both problems. We work over an algebraically closed field $K$ of arbitrary characteristic.
Migliore-Mir\'o-Roig-Nagel [Trans. A.M.S. 2011, arXiv: 0811.1023] show that the weak Lefschet... more Migliore-Mir\'o-Roig-Nagel [Trans. A.M.S. 2011, arXiv: 0811.1023] show that the weak Lefschetz property (WLP) can fail for an ideal I in K[x_1,x_2,x_3,x_4] generated by powers of linear forms. This is in contrast to the analogous situation in K[x_1,x_2,x_3], where WLP always holds [H.Schenck, A.Seceleanu, Proc. A.M.S. 2010, arXiv:0911.0876]. We use the inverse system dictionary to connect I to an ideal of
The notion of a quasiuniform fat point subscheme Z P2 is introduced and conjectures for the Hilbe... more The notion of a quasiuniform fat point subscheme Z P2 is introduced and conjectures for the Hilbert function and minimal free resolution of the ideal I dening Z 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 rst determination of the
Conjectures for the Hilbert function of the m-th symbolic power of the ideal of n general points ... more Conjectures for the Hilbert function of the m-th symbolic power of the ideal of n general points of P2 are verified for infinitely many m for each square n > 9, using an approach developed by the authors in a previous paper. In those cases that n is even, conjectures for the resolution are also verified. Previously, by work of
Working over the complex field and formalizing and sharpening approaches introduced by several au... more Working over the complex field and formalizing and sharpening approaches introduced by several authors, we give a method for verifying when a divisor on a blow up of P^2 at general points is nef. The method is useful both theoretically and when doing computer computations. The main application is to obtain significantly improved explicit lower bounds for multipoint Seshadri constants
The paper (10) raised the question of what the possible Hilbert functions are for fat point subsc... more The paper (10) raised the question of what the possible Hilbert functions are for fat point subschemes of the form 2p1+···+2pr, for all possible choices of r distinct points in P2. We study this problem for r points in P2 over an al- gebraically closed field k of arbitrary characteristic in case either r 8 or the points lie on
Given an immersion $\phi: P^1 \to \P^2$, we give new approaches to determining the splitting of t... more Given an immersion $\phi: P^1 \to \P^2$, we give new approaches to determining the splitting of the pullback of the cotangent bundle. We also give new bounds on the splitting type for immersions which factor as $\phi: P^1 \cong D \subset X \to P^2$, where $X \to P^2$ is obtained by blowing up $r$ distinct points $p_i \in P^2$. As
Our results concern minimal graded free resolutions of fat point ideals for points in a hyperplan... more Our results concern minimal graded free resolutions of fat point ideals for points in a hyperplane. Suppose, for example, that I(m,d) is the ideal defining r given points of multiplicity m in the projective space P^d. Assume that the given points lie in a hyperplane P^{d-1} in P^d, and that the ground field k is algebraically closed of characteristic 0. We give an explicit minimal graded free resolution of I(m,d) in k[P^d] in terms of the minimal graded free resolutions of the ideals I(j,d-1) in k[P^{d-1}] with j < m+1. As a corollary, we give the following formula for the Poincare polynomial P_{m,d} of I(m,d) in terms of the Poincare polynomials P_{j,d-1} of I(j,d-1): P_{m,d} = (1 + XT)(\Sigma_{0<j\le m} T^{m-j}(P_{j,d-1} - 1)) + 1 + XT^m.
Bulletin of The Belgian Mathematical Society-simon Stevin, 2009
We describe an approach for computing arbitrarily accurate estimates for multi-point Seshadri con... more We describe an approach for computing arbitrarily accurate estimates for multi-point Seshadri constants for $n$ generic points of ${\bf P}^{2}$. We apply the approach to obtain improved estimates. We work over an algebraically closed field of characteristic 0.
The graded Betti numbers of the minimal free resolution (and also therefore the Hilbert function)... more The graded Betti numbers of the minimal free resolution (and also therefore the Hilbert function) of the ideal of a fat point subscheme Z of P2 are determined whenever Z is supported at any 6 or fewer distinct points. A broad range of cases in which the points can be infinitely near, related to the classification of normal cubic surfaces,
Bulletin of The Belgian Mathematical Society-simon Stevin - BULL BELG MATH SOC-SIMON STEV, 2009
It is an open problem to determine the Hilbert function and graded Betti numbers for the ideal of... more It is an open problem to determine the Hilbert function and graded Betti numbers for the ideal of a fat point subscheme supported at general points of the projective plane. In fact, there is not yet even a general explicit conjecture for the graded Betti numbers. Here we formulate explicit asymptotic conjectures for both problems. We work over an algebraically closed field $K$ of arbitrary characteristic.
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