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Line arrangements and configurations of points with an unexpected geometric property

Part of: Projective and enumerative geometry Commutative algebra: Homological methods Cycles and subschemes (Co)homology theory Algebraic combinatorics

Published online by Cambridge University Press:  10 September 2018

D. Cook II ,
B. Harbourne ,
J. Migliore  and
U. Nagel
D. Cook II
Affiliation:
Google LLC, 111 8th Avenue, New York, NY 10011, USA email dcook.math@gmail.com
B. Harbourne
Affiliation:
Department of Mathematics, University of Nebraska, Lincoln, NE 68588-0130, USA email bharbourne1@unl.edu
J. Migliore
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA email migliore.1@nd.edu
U. Nagel
Affiliation:
Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, KY 40506-0027, USA email uwe.nagel@uky.edu
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Abstract

We propose here a generalization of the problem addressed by the SHGH conjecture. The SHGH conjecture posits a solution to the question of how many conditions a general union $X$ of fat points imposes on the complete linear system of curves in $\mathbb{P}^{2}$ of fixed degree $d$, in terms of the occurrence of certain rational curves in the base locus of the linear subsystem defined by $X$. As a first step towards a new theory, we show that rational curves play a similar role in a special case of a generalized problem, which asks how many conditions are imposed by a general union of fat points on linear subsystems defined by imposed base points. Moreover, motivated by work of Di Gennaro, Ilardi and Vallès and of Faenzi and Vallès, we relate our results to the failure of a strong Lefschetz property, and we give a Lefschetz-like criterion for Terao’s conjecture on the freeness of line arrangements.

Keywords

unexpected curvesfat pointsline arrangementsstrong Lefschetz propertylinear systemsstable vector bundlesplitting type

MSC classification

Primary: 14N20: Configurations and arrangements of linear subspaces
Secondary: 13D02: Syzygies, resolutions, complexes 14C20: Divisors, linear systems, invertible sheaves 14N05: Projective techniques 05E40: Combinatorial aspects of commutative algebra 14F05: Sheaves, derived categories of sheaves and related constructions
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 10 , October 2018 , pp. 2150 - 2194
Copyright
© The Authors 2018 

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