arXiv subjects

Brian Harbourne

Publications and source records attributed to Brian Harbourne.

At least 55 records · Page 3Linked to original sources

arXiv2011

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.

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arXiv2011

Inverse systems, Gelfand-Tsetlin patterns and the weak Lefschetz property

Migliore-Miró-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 fat points and show that failure of WLP for powers of linear forms is connected to the geometry of the associated fat point scheme. Recent results of Sturmfels-Xu in [J. Eur. Math. Soc. 2010, arXiv:0803.0892] allow us to relate WLP to Gelfand-Tsetlin patterns. See the paper "On the weak Lefschetz property for powers of linear forms" by Migliore-Miró-Roig-Nagel [arXiv:1008.2149] for related results.

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arXiv2011

Recent developments and open problems in linear series

In the week 3--9, October 2010, the Mathematisches Forschungsinstitut at Oberwolfach hosted a mini workshop Linear Series on Algebraic Varieties. These notes contain a variety of interesting problems which motivated the participants prior to the event, and examples, results and further problems which grew out of discussions during and shortly after the workshop. A lot of arguments presented here are scattered in the literature or constitute folklore. It was one of our aims to have a usable and easily accessible collection of examples and results.

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arXiv2010

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 .

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arXiv2010

A primer on Seshadri constants

Seshadri constants express the so called local positivity of a line bundle on a projective variety. They were introduced by Demailly. The original idea of using them towards a proof of the Fujita conjecture failed but they quickly became a subject of intensive study quite in their own right. Lazarsfeld's book "Positivity in Algebraic Geometry" contains a whole chapter devoted to local positivity and serves as a very enjoyable introduction to Seshadri constants. Since this book has appeared, the subject witnessed quite a bit of development. It is the aim of these notes to give an account of recent progress as well as to discuss many open questions and provide some examples.

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arXiv2009

Global aspects of the geometry of surfaces

These notes (prepared for the author's lectures at the Cracow Summer School on Linear Systems organized by S. Mueller-Stach and T. Szemberg, held March 23-27, 2009 at the Pedagogical University of Cracow under the sponsorship of the Deutsche Forschungsgemeinschaft) present a number of open problems on the theory of smooth projective algebraic surfaces, and put into historical context recent work on a range of topics, including Mori dream spaces and the finite generation of the Cox ring, Seshadri constants, and the resurgence of homogeneous ideals and the problem of which ordinary powers of homogeneous ideals contain given symbolic powers of those ideals. These notes include many exercises, with solutions.

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arXiv2009

Comparing powers and symbolic powers of ideals

We develop tools to study the problem of containment of symbolic powers $I^{(m)}$ in powers $I^r$ for a homogeneous ideal $I$ in a polynomial ring $k[{\bf P}^N]$ in $N+1$ variables over an algebraically closed field $k$. We obtain results on the structure of the set of pairs $(r,m)$ such that $I^{(m)}\subseteq I^r$. As corollaries, we show that $I^2$ contains $I^{(3)}$ whenever $S$ is a finite generic set of points in ${\bf P}^2$ (thereby giving a partial answer to a question of Huneke), and we show that the containment theorems of Ein-Lazarsfeld-Smith and Hochster-Huneke are optimal for every fixed dimension and codimension.

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arXiv2009

The resurgence of ideals of points and the containment problem

Given a symbolic power of a homogeneous ideal in a polynomial ring, we study the problem of determining which powers of the ideal contain it. For ideals defining 0-dimensional subschemes of projective space, as an immediate corollary of our previous paper (arXiv:0706.3707) we give a complete solution in terms of the least degrees of nonzero elements of the symbolic powers, under the condition that the ideal is generated in a single degree and such that that degree is the regularity of the ideal (see Corollary 1.2). Using geometric methods, we also give complete solutions for ideals of points in the projective plane, when those points lie on an irreducible conic or when the points are general and there are at most 9 points. Finally, we give new examples of ideals all of whose symbolic powers are ordinary powers of the ideal.

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arXiv2007

Discrete behavior of Seshadri constants on surfaces

Working over C, we show that, apart possibly from a unique limit point, the possible values of multi-point Seshadri constant for general points on smooth projective surfaces form a discrete set. In addition to its theoretical interest, this result is of practical value, which we demonstrate by giving significantly improved explicit lower bounds for Seshadri constants on P^2 and new results about ample divisors on blow ups of P^2 at general points.

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arXiv2007

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

We consider the open problem of determining the graded Betti numbers for fat point subschemes supported at general points of the projective plane. We relate this problem to the open geometric problem of determining the splitting type of the pullback of the cotangent bundle on the plane 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 the plane. 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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arXiv2007

The role of the cotangent bundle in resolving ideals of fat points in the plane

We study the connection between the generation of a fat point scheme supported at general points in the plane and the behaviour of the cotangent bundle with respect to some rational curves particularly relevant for the scheme. We put forward two conjectures, giving examples and partial results in support of them.

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arXiv2005

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)(Σ_{0<j\le m} T^{m-j}(P_{j,d-1} - 1)) + 1 + XT^m.

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arXiv2005

The (unexpected) importance of knowing $α$

In order to determine the Hilbert function of the ideal of a fat point subscheme of projective space, we show that it is enough to determine, both for the subscheme itself and the subschemes obtained from it by successively adjoining to it additional general points, the least degrees in which the ideals of the subschemes are nonzero. This point of view gives rise to the following conjecture: if C is any reduced irreducible curve on a surface X obtained by blowing up P^2 at any finite number of generic points, then C^2 > g-2, where g is the arithmetic genus of C. This conjecture is shown to be equivalent to the usual well-known conjectures for computing dimensions of complete linear systems on X.

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arXiv2003

Linear systems with multiple base points in P2

Given positive integers $m_1, m_2, ..., m_n$, and $n$ general points $p_i$ of ${\bf CP}^2$, bounds are given for the least degree $t$ among plane curves passing through each point $p_i$ with multiplicity at least $m_i$, and for the least $t$ such that the $n$ multiple points impose independent conditions on curves of degree $t$, often improving substantially what was previously known. As an application, the Hilbert function (resp., minimal free resolution) is determined for symbolic powers $I^{(m)}$ for the ideal $I$ defining $n$ general points of ${\bf CP}^2$ for infinitely many m for each square n (resp., for infinitely many m for each even square n). Four graphs are included showing other values of m and n for which results are given.

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arXiv2001

Hilbert functions and resolutions for ideals of n = s^2 fat points in P2

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 Evain (for the Hilbert function) and of Harbourne, Holay and Fitchett (for the resolution), these conjectures were known for infinitely many m for a given n > 9 only for n being a power of 4.

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arXiv2001

Seshadri constants and very ample divisors on algebraic surfaces

A broadly applicable geometric approach for constructing nef divisors on blow ups of algebraic surfaces at n general points is given; it works for all surfaces in all characteristics for any n. This construction is used to obtain substantial improvements for currently known lower bounds for n point Seshadri constants. Remarks are included about a range of applications to classical problems involving linear systems on P2.

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arXiv2001

Resolutions of fat point ideals involving 8 general points of P2

The main result provides an algorithm for determining the minimal free resolution of ideals of fat point subschemes of ${\bf P}^2$ involving up to 8 general points with arbitrary multiplicities; the results hold over algebraically closed fields of any characteristic. The algorithm, which works by giving a formula in certain cases and a reduction to these cases otherwise, does not involve Gröbner bases, and so is very fast, even for very large multiplicities. Partial information is also obtained in certain cases with $n>8$.

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arXiv2001

Problems and progress: survey on fat points in P2

This paper surveys certain problems involving numerical characters for ideals I(Z) defining fat points subschemes $Z=m_1p_1+...+m_np_n$ for general points $p_i\in {\bf P}^2$. It also presents some new results, and includes a suite of MACAULAY 2 scripts for computing actual or conjectured values of (or bounds on) these characters. One such script, for example, is findres, which implements an algorithm due to Fitchett, Harbourne and Holay for computing the syzygy modules in a minimal free resolution of the ideal I(Z) for any such Z with n<=8; since findres does not rely on a Gröbner basis calculation, it is very fast and can handle very large multiplicities.

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