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Unexpected curves, line arrangements, and Lefschetz properties Nagel, Uwe
Description
We discuss connections between Lefschetz properties and the study of Hilbert functions of (fat) points as well as the theory of line arrangements. To this end, we begin by considering a finite set Z of points in the plane with the property that, for some integer j, the dimension of the linear system of plane curves of degree j + 1 through the points of Z and having multiplicity j at a general point is unexpectedly large. We give criteria for the occurrence of such unexpected curves and describe the range of their degrees. Inspired by work of Di Gennaro, Ilardi, and Vall`es, we relate properties of Z to properties of the arrangement of lines dual to the points of Z. In particular, we get a new interpretation of the splitting type of a line arrangement, and we show that the existence of an unexpected curve is equivalent to the failure of a certain Lefschetz property. This implies a Lefschetz-like criterion for Terao’s conjecture on the freeness of line arrangements. This is based on joint work with D. Cook II, B. Harbourne, and J. Migliore
Item Metadata
Title |
Unexpected curves, line arrangements, and Lefschetz properties
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2016-03-18T09:46
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Description |
We discuss connections between Lefschetz properties and the study of Hilbert functions of (fat) points as well
as the theory of line arrangements. To this end, we begin by considering a finite set Z of points in the plane with the
property that, for some integer j, the dimension of the linear system of plane curves of degree j + 1 through the points
of Z and having multiplicity j at a general point is unexpectedly large. We give criteria for the occurrence of such
unexpected curves and describe the range of their degrees. Inspired by work of Di Gennaro, Ilardi, and Vall`es, we relate
properties of Z to properties of the arrangement of lines dual to the points of Z. In particular, we get a new interpretation
of the splitting type of a line arrangement, and we show that the existence of an unexpected curve is equivalent to the
failure of a certain Lefschetz property. This implies a Lefschetz-like criterion for Terao’s conjecture on the freeness of
line arrangements. This is based on joint work with D. Cook II, B. Harbourne, and J. Migliore
|
Extent |
36 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: University of Kentucky
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Series | |
Date Available |
2016-09-20
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0314362
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International