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Logarithmic vector fields and Severi Strata in the discriminant
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Cadman, Paul, Mond, David and van straten, Duco (2017) Logarithmic vector fields and Severi Strata in the discriminant. In: Singularities in Geometry, Topology, Foliations and Dynamics. Published in: Singularities in Geometry, Topology, Foliations and Dynamics pp. 55-76. ISBN 9783319393391. doi:10.1007/978-3-319-39339-1
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Official URL: https://doi.org/10.1007/978-3-319-39339-1
Abstract
The discriminant, D, in the base of a miniversal deformation of an irreducible plane curve singularity, is partitioned according to the genus of the (singular) fibre, or, equivalently, by the sum of the delta invariants of the singular points of the fibre. The members of the partition are known as the Severi strata. The smallest is the δ-constant stratum, D(δ), where the genus of the fibre is 0. It is well known, by work of Givental’ and Varchenko, to be Lagrangian with respect to the symplectic form Ω obtained by pulling back the intersection form on the cohomology of the fibre via the period mapping. We show that the remaining Severi strata are also co-isotropic with respect to Ω, and moreover that the coefficients of the expression of Ω with respect to a basis of Ω 2(log D) are equations for D(δ). Similarly, the coeδcients of Ω ^k with respect to a basis for Ω 2k (log D) are equations for D(δ − k + 1). These equations allow us to show that for E 6 and E 8, D(δ) is Cohen-Macaulay (this was already shown by Givental’ for A 2k ), and that, as far as we can calculate, for A 2k all of the Severi strata are Cohen-Macaulay.
Item Type: | Conference Item (Paper) | ||||||
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Subjects: | Q Science > QA Mathematics | ||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
Series Name: | Trends in Mathematics | ||||||
Journal or Publication Title: | Singularities in Geometry, Topology, Foliations and Dynamics | ||||||
Publisher: | Birkhauser | ||||||
ISBN: | 9783319393391 | ||||||
Book Title: | Singularities in Geometry, Topology, Foliations and Dynamics : A Celebration of the 60th Birthday of José Seade, Merida, Mexico, December 2014 | ||||||
Editor: | Cisneros-Molina, J. L. and Lê, D.T. and Oka, M. and Snoussi, J. | ||||||
Official Date: | 14 February 2017 | ||||||
Dates: |
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Number of Pages: | 231 | ||||||
Page Range: | pp. 55-76 | ||||||
DOI: | 10.1007/978-3-319-39339-1 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||
Date of first compliant deposit: | 8 December 2017 | ||||||
Conference Paper Type: | Paper | ||||||
Title of Event: | Singularities in Geometry, Topology, Foliations and Dynamics | ||||||
Type of Event: | Conference |
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