Resolutions of ideals of six fat points in P^2

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 P^2 are determined whenever Z is supported at any 6 or fewer distinct points. All results hold over an algebraically closed field k of arbitrary characteristic.Comment: 21 pp., final versio

Resolutions of ideals of fat points with support in a hyperplane

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.Comment: 10 pages; to appear in Proc. Amer. Math. Soc.; some expositional changes; added a reference to paper of Geramita, Migliore and Sabourin (math.AC/0411445

Configuration types and cubic surfaces

This paper is a sequel to the paper \cite{refGH}. We relate the matroid notion of a combinatorial geometry to a generalization which we call a configuration type. Configuration types arise when one classifies the Hilbert functions and graded Betti numbers for fat point subschemes supported at $n\le8$ essentially distinct points of the projective plane. Each type gives rise to a surface $X$ obtained by blowing up the points. We classify those types such that $n=6$ and $-K_X$ is nef. The surfaces obtained are precisely the desingularizations of the normal cubic surfaces. By classifying configuration types we recover in all characteristics the classification of normal cubic surfaces, which is well-known in characteristic 0 \cite{refBW}. As an application of our classification of configuration types, we obtain a numerical procedure for determining the Hilbert function and graded Betti numbers for the ideal of any fat point subscheme $Z=m_1p_1+...+m_6p_6$ such that the points $p_i$ are essentially distinct and $-K_X$ is nef, given only the configuration type of the points $p_1,...,p_6$ and the coefficients $m_i$.Comment: 14 pages, final versio

Combinatorial bounds on Hilbert functions of fat points in projective space

We study Hilbert functions of certain non-reduced schemes A supported at finite sets of points in projective space, in particular, fat point schemes. We give combinatorially defined upper and lower bounds for the Hilbert function of A using nothing more than the multiplicities of the points and information about which subsets of the points are linearly dependent. When N=2, we give these bounds explicitly and we give a sufficient criterion for the upper and lower bounds to be equal. When this criterion is satisfied, we give both a simple formula for the Hilbert function and combinatorially defined upper and lower bounds on the graded Betti numbers for the ideal defining A, generalizing results of Geramita-Migliore-Sabourin (2006). We obtain the exact Hilbert functions and graded Betti numbers for many families of examples, interesting combinatorially, geometrically, and algebraically. Our method works in any characteristic. AWK scripts implementing our results can be obtained at http://www.math.unl.edu/~bharbourne1/CHT/Example.html .Comment: 23 pages; changes have been made following suggestions of the referee; explicit statements are now included for dimensions greater than 2, hence the title no longer mentions the plane; however the content is largely the same as in the previous version; this version is to appear in the Journal of Pure and Applied Algebr

On the containment problem

The purpose of this note is to provide an overview of the containment problem for symbolic and ordinary powers of homogeneous ideals, related conjectures and examples. We focus here on ideals with zero dimensional support. This is an area of ongoing active research. We conclude the note with a list of potential promising paths of further research.Comment: 13 pages, 1 figur

Free Resolutions of Fat Point Ideals on P2P^2

By defining a fat point subscheme of $P^2$ to be a 0-dimensional subscheme defined by a sheaf of integrally closed ideals one extends the notion of fat point subschemes to allow infinitely near points. With this notion of fat points, this preprint determines the number of generators in each degree in a minimal homogeneous set of generators for the homogeneous ideal defining any fat point subscheme supported at points of any plane conic, smooth or not. Special cases for points on a cubic are also studied. All work is over an algebraically closed field of arbitrary characteristic.Comment: plain tex, 15 pp. The preprint itself is not contained in the Duke archive; plainTeX textfile and dvi versions of this preprint can instead be obtained via the author's www site, http://www.math.unl.edu/~bharbour/ . Comments and requests can be directed to [email protected]

Are symbolic powers highly evolved?

Searching for structural reasons behind old results and conjectures of Chudnovksy regarding the least degree of a nonzero form in an ideal of fat points in projective N-space, we make conjectures which explain them, and we prove the conjectures in certain cases, including the case of general points in the projective plane. Our conjectures were also partly motivated by the Eisenbud-Mazur Conjecture on evolutions, which concerns symbolic squares of prime ideals in local rings, but in contrast we consider higher symbolic powers of homogeneous ideals in polynomial rings.Comment: 13 pages; for version 3 a minor change was made to the acknowledgments but no change was made to mathematical content; for version 2 a reference to a paper of Esnault and Viehweg has been added; related to this a new section, 4.2, has been included with additional questions. Otherwise, version 2 is the same as version