136 research outputs found
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).Comment: 32 pages, 3 figure
Negative curves on algebraic surfaces
We study curves of negative self-intersection on algebraic surfaces. We
obtain results for smooth complex projective surfaces X on the number of
reduced, irreducible curves C of negative self-intersection C^2. The only known
examples of surfaces for which C^2 is not bounded below are in positive
characteristic, and the general expectation is that no examples can arise over
the complex numbers. Indeed, we show that the idea underlying the examples in
positive characteristic cannot produce examples over the complex number field.
The previous version of this paper claimed to give a counterexample to the
Bounded Negativity Conjecture. The idea of the counterexample was to use Hecke
translates of a smooth Shimura curve in order to create an infinite sequence of
curves violating the Bounded Negativity Conjecture. To this end we applied
Hirzebruch Proportionality to all Hecke translates, simultaneously
desingularized by a version of Jaffee's Lemma which exists in the literature
but which turns out to be false. Indeed, in the new version of the paper, we
show that only finitely many Hecke translates of a special subvariety of a
Hilbert modular surface remain smooth. This new result is based on work done
jointly with Xavier Roulleau, who has been added as an author. The other
results in the original posting of this paper remain unchanged.Comment: 14 pages, X. Roulleau added as author, counterexample to Bounded
Negativity Conjecture withdrawn and replaced by a proof that there are only
finitely many smooth Shimura curves on a compact Hilbert modular surface; the
other results in the original posting of this paper remain unchange
The Waldschmidt constant for squarefree monomial ideals
Given a squarefree monomial ideal , we show
that , the Waldschmidt constant of , can be expressed as
the optimal solution to a linear program constructed from the primary
decomposition of . By applying results from fractional graph theory, we can
then express in terms of the fractional chromatic number of
a hypergraph also constructed from the primary decomposition of . Moreover,
expressing as the solution to a linear program enables us
to prove a Chudnovsky-like lower bound on , thus verifying a
conjecture of Cooper-Embree-H\`a-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 with few
components compared to , and we find the Waldschmidt constant for the
Stanley-Reisner ideal of a uniform matroid.Comment: 26 pages. This project was started at the Mathematisches
Forschungsinstitut Oberwolfach (MFO) as part of the mini-workshop "Ideals of
Linear Subspaces, Their Symbolic Powers and Waring Problems" held in February
2015. Comments are welcome. Revised version corrects some typos, updates the
references, and clarifies some hypotheses. To appear in the Journal of
Algebraic Combinatoric
Matroid Configurations and Symbolic Powers of Their Ideals
Star configurations are certain unions of linear subspaces of projective space that have been studied extensively. We develop a framework for studying a substantial generalization, which we call matroid configurations, whose ideals generalize Stanley-Reisner ideals of matroids. Such a matroid configuration is a union of complete intersections of a fixed codimension. Relating these to the Stanley-Reisner ideals of matroids and using methods of liaison theory allows us, in particular, to describe the Hilbert function and minimal generators of the ideal of, what we call, a hypersurface configuration. We also establish that the symbolic powers of the ideal of any matroid configuration are Cohen-Macaulay. As applications, we study ideals coming from certain complete hypergraphs and ideals derived from tetrahedral curves. We also consider Waldschmidt constants and resurgences. In particular, we determine the resurgence of any star configuration and many hypersurface configurations. Previously, the only non-trivial cases for which the resurgence was known were certain monomial ideals and ideals of finite sets of points. Finally, we point out a connection to secant varieties of varieties of reducible forms
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