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Brian Harbourne

Publications and source records attributed to Brian Harbourne.

At least 37 records · Page 2Linked to original sources
arXiv2017

Negative curves on symmetric blowups of the projective plane, resurgences and Waldschmidt constants

The Klein and Wiman configurations are highly symmetric configurations of lines in the projective plane arising from complex reflection groups. One noteworthy property of these configurations is that all the singularities of the configuration have multiplicity at least three. In this paper we study the surface X obtained by blowing up the projective plane in the singular points of one of these line configurations. We study invariant curves on X in detail, with a particular emphasis on curves of negative self-intersection. We use the representation theory of the stabilizers of the singular points to discover several invariant curves of negative self-intersection on X, and use these curves to study Nagata-type questions for linear series on X. The homogeneous ideal I of the collection of points in the configuration is an example of an ideal where the symbolic cube of the ideal is not contained in the square of the ideal; ideals with this property are seemingly quite rare. The resurgence and asymptotic resurgence are invariants which were introduced to measure such failures of containment. We use our knowledge of negative curves on X to compute the resurgence of I exactly. We also compute the asymptotic resurgence and Waldschmidt constant exactly in the case of the Wiman configuration of lines, and provide estimates on both for the Klein configuration.

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arXiv2017

Asymptotics of linear systems, with connections to line arrangements

The main focus of these notes is recent work on linear systems in which line arrangements play a role, including problems such as semi-effectivity, containment problems of symbolic powers of homogeneous ideals in their powers, bounded negativity, and a new perspective on the SHGH Conjecture. Along the way we will be concerned with asymptotic invariants such as Waldschmidt constants, resurgences and H-constants.

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arXiv2017

Line arrangements and configurations of points with an unusual geometric property

The SHGH conjecture proposes a solution to the question of how many conditions a general union of fat points imposes on the complete linear system of curves in $\mathbb P^2$ of fixed degree $d$, and it is known to be true in many cases. We propose a new problem, namely to understand the number of conditions imposed by a general union of fat points on the incomplete linear system defined by the condition of passing through a given finite set of points $Z$ (not general). Motivated by work of Di Gennaro-Ilardi-Vallès and Faenzi-Vallès, we give a careful analysis for the case where there is a single general fat point, which has multiplicity $d-1$. There is an expected number of conditions imposed by this fat point, and we study those $Z$ for which this expected value is not achieved. We show, for instance, that if $Z$ is in linear general position then such unexpected curves do not exist. We give criteria for the occurrence of such unexpected curves and describe the range of values of $d$ for which they occur. Unexpected curves have a very particular structure, which we describe, and they are often unique for a given set of points. In particular, we give a criterion for when they are irreducible, and we exhibit examples both where they are reducible and where they are irreducible. Furthermore, we relate properties of $Z$ to properties of the arrangement of lines dual to the points of $Z$. In particular, we obtain a new interpretation of the splitting type of a line arrangement. Finally, we use our results to establish a Lefschetz-like criterion for Terao's conjecture on the freeness of line arrangements.

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arXiv2016

The Waldschmidt constant for squarefree monomial ideals

Given a squarefree monomial ideal $I \subseteq R =k[x_1,\ldots,x_n]$, we show that $\widehatα(I)$, the Waldschmidt constant of $I$, can be expressed as the optimal solution to a linear program constructed from the primary decomposition of $I$. By applying results from fractional graph theory, we can then express $\widehatα(I)$ in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of $I$. Moreover, expressing $\widehatα(I)$ as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on $\widehatα(I)$, thus verifying a conjecture of Cooper-Embree-Hà-Hoefel for monomial ideals in the squarefree case. As an application, we compute the Waldschmidt constant and the resurgence for some families of squarefree monomial ideals. For example, we determine both constants for unions of general linear subspaces of $\mathbb{P}^n$ with few components compared to $n$, and we find the Waldschmidt constant for the Stanley-Reisner ideal of a uniform matroid.

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arXiv2016

The Halphen cubics of order two

For each $m\ge 1$, Roulleau and Urzúa give an implicit construction of a configuration of $4(3m^2-1)$ complex plane cubic curves. This construction was crucial for their work on surfaces of general type. We make this construction explicit by proving that the Roulleau-Urzúa configuration consists precisely of the Halphen cubics of order $m$, and we determine specific equations of the cubics for $m=1$ (which were known) and for $m=2$ (which are new).

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arXiv2016

Very general monomial valuations of $\mathbb{P}^2$ and a Nagata type conjecture

It is well known that multi-point Seshadri constants for a small number $s$ of points in the projective plane are submaximal. It is predicted by the Nagata conjecture that their values are maximal for $s\geq 9$ points. Tackling the problem in the language of valuations one can make sense of $s$ points for any positive real $s\geq 1$. We show somewhat surprisingly that a Nagata-type conjecture should be valid for $s\geq 8+1/36$ points and we compute explicitly all Seshadri constants (expressed here as the asymptotic maximal vanishing element) for $s\leq 7+1/9$.

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arXiv2014

Bounded Negativity and Arrangements of Lines

The Bounded Negativity Conjecture predicts that for any smooth complex surface $X$ there exists a lower bound for the selfintersection of reduced divisors on $X$. This conjecture is open. It is also not known if the existence of such a lower bound is invariant in the birational equivalence class of $X$. In the present note we introduce certain constants $H(X)$ which measure in effect the variance of the lower bounds in the birational equivalence class of $X$. We focus on rational surfaces and relate the value of $H({\mathbb P}^2)$ to certain line arrangements. Our main result is Theorem 3.3 and the main open challenge is Problem 3.10.

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arXiv2014

Resurgences for ideals of special point configurations in ${\bf P}^N$ coming from hyperplane arrangements

Symbolic powers of ideals have attracted interest in commutative algebra and algebraic geometry for many years, with a notable recent focus on containment relations between symbolic powers and ordinary powers. Several invariants have been introduced and studied in the latter context, including the resurgence and asymptotic resurgence. There have been exciting new developments in this area recently. It had been expected for several years that $I^{Nr-N+1}\subseteq I^r$ should hold for the ideal $I$ of any finite set of points in ${\bf P}^N$ for all $r>0$, but in the last year various counterexamples have now been constructed, all involving point sets coming from hyperplane arrangements. In the present work, we compute their resurgences and obtain in particular the first examples where the resurgence and the asymptotic resurgence are not equal.

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arXiv2013

Regina Lectures on Fat Points

These notes are a record of lectures given in the Workshop on Connections Between Algebra and Geometry at the University of Regina, May 29--June 1, 2012. The lectures were meant as an introduction to current research problems related to fat points for an audience that was not expected to have much background in commutative algebra or algebraic geometry (although sections 8 and 9 of these notes demand somewhat more background than earlier sections).

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arXiv2013

On plane rational curves and the splitting of the tangent bundle

Given an immersion $ϕ: P^1 \to ¶^2$, 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 $ϕ: P^1 \cong D \subset X \to P^2$, where $X \to P^2$ is obtained by blowing up $r$ distinct points $p_i \in P^2$. As applications in the case that the points $p_i$ are generic, we give a complete determination of the splitting types for such immersions when $r \leq 7$. The case that $D^2=-1$ is of particular interest. For $r \leq8$ generic points, it is known that there are only finitely many inequivalent $ϕ$ with $D^2=-1$, and all of them have balanced splitting. However, for $r=9$ generic points we show that there are infinitely many inequivalent $ϕ$ with $D^2=-1$ having unbalanced splitting (only two such examples were known previously). We show that these new examples are related to a semi-adjoint formula which we conjecture accounts for all occurrences of unbalanced splitting when $D^2=-1$ in the case of $r=9$ generic points $p_i$. In the last section we apply such results to the study of the resolution of fat point schemes.

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arXiv2013

Containment results for ideals of various configurations of points in P^N

Guided by evidence coming from a few key examples and attempting to unify previous work of Chudnovsky, Esnault-Viehweg, Eisenbud-Mazur, Ein-Lazarsfeld-Smith, Hochster-Huneke and Bocci-Harbourne, Harbourne and Huneke recently formulated a series of conjectures that relate symbolic and regular powers of ideals of fat points in ${\bf P}^N$. In this paper we propose another conjecture along the same lines (Conjecture 3.9), and we verify it and the conjectures of Harbourne and Huneke for a variety of configurations of points.

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arXiv2013

Inclics, galaxies, star configurations and Waldschmidt constants

This paper introduces complexes of linear varieties, called inclics (for INductively Constructible LInear ComplexeS). As examples, we study galaxies (these are constructed starting with a star configuration to which we add general points in a larger projective space). By assigning an order of vanishing (i.e., a multiplicity) to each member of the complex, we obtain fat linear varieties (fat points if all of the linear varieties are points). The scheme theoretic union of these fat linear varieties gives an inclic scheme $X$. For such a scheme, we show there is an inductive procedure for computing the Hilbert function of its defining ideal $I_X$, regardless of the choice of multiplicities. As an application, we show how our results allow the computation of the Hilbert functions of, for example, symbolic powers $(I_X)^{(m)}$ for arbitrary $m$ of many new examples of radical ideals $(I_X)$, and we explicitly compute the Waldschmidt constants $γ(I_X)$ for galactic inclics $X$.

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arXiv2013

Containment Counterexamples for ideals of various configurations of points in ${\bf P}^N$

When $I$ is the radical homogeneous ideal of a finite set of points in projective $N$-space, ${\bf P}^N$, over a field $K$, it has been conjectured that $I^{(rN-N+1)}$ should be contained in $I^r$ for all $r\geq 1$. Recent counterexamples show that this can fail when N=r=2. We study properties of the resulting ideals. We also show that failures occur for infinitely many $r$ in every characteristic $p>2$ when N=2, and we find additional positive characteristic failures when $N>2$.

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arXiv2012

Linear subspaces, symbolic powers and Nagata type conjectures

Prompted by results of Guardo, Van Tuyl and the second author for lines in projective 3 space, we develop asymptotic upper bounds for the least degree of a homogeneous form vanishing to order at least m on a union of disjoint r dimensional planes in projective n space for n at least 2r+1. These considerations lead to new conjectures that suggest that the well known conjecture of Nagata for points in the projective plane is not sporadic, but rather a special case of a more general phenomenon.

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arXiv2012

Fat lines in P^3: powers versus symbolic powers

We study the symbolic and regular powers of ideals I for a family of special configurations of lines in P^3. For this family, we show that I^(m) = I^m for all integers m if and only if I^(3) = I^3. We use these configurations to answer a question of Huneke that asks whether I^(m) = I^m for all m if equality holds when m equals the big height of the ideal I.

math.AC↗
arXiv2012

Asymptotic resurgences for ideals of positive dimensional subschemes of projective space

Recent work of Ein-Lazarsfeld-Smith and Hochster-Huneke raised the problem of determining which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci-Harbourne defined a quantity called the resurgence to address this problem for homogeneous ideals in polynomial rings, with a focus on zero dimensional subschemes of projective space; the methods and results obtained there have much less to say about higher dimensional subschemes. Here we take the first steps toward extending this work to higher dimensional subschemes. We introduce new asymptotic versions of the resurgence and obtain upper and lower bounds on them for ideals of smooth subschemes, generalizing what is done by Bocci-Harbourne. We apply these bounds to ideals of unions of general lines in ${\bf P}^N$. We also pose a Nagata type conjecture for symbolic powers of ideals of lines in ${\bf P}^3$.

math.AG↗
arXiv2012

Symbolic powers versus regular powers of ideals of general points in P^1 x P^1

Recent work of Ein-Lazarsfeld-Smith and Hochster-Huneke raised the problem of which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci-Harbourne developed methods to address this problem, which involve asymptotic numerical characters of symbolic powers of the ideals. Most of the work done up to now has been done for ideals defining 0-dimensional subschemes of projective space. Here we focus on certain subschemes given by a union of lines in ${\bf P}^3$ which can also be viewed as points in ${\bf P}^1\times {\bf P}^1$. We also obtain results on the closely related problem, studied by Hochster and by Li-Swanson, of determining situations for which each symbolic power of an ideal is an ordinary power.

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