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On a class of spaces of skew-symmetric forms related to Hamiltonian systems of conservation laws

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posted on 2022-12-07, 14:35 authored by L Manivel, Evgeny FerapontovEvgeny Ferapontov

It was shown in Ferapontov et al. (Lett Math Phys 108(6):1525–1550, 2018) that the classification of n-component systems of conservation laws possessing a third-order Hamiltonian structure reduces to the following algebraic problem: classify n-planes H in ∧ 2(Vn+2) such that the induced map Sym2H⟶ ∧ 4Vn+2 has 1-dimensional kernel generated by a non-degenerate quadratic form on H. This problem is trivial for n= 2 , 3 and apparently wild for n≥ 5. In this paper we address the most interesting borderline case n= 4. We prove that the variety V parametrizing those 4-planes H is an irreducible 38-dimensional PGL(V6) -invariant subvariety of the Grassmannian G(4 , ∧ 2V6). With every H∈ V we associate a characteristic cubic surface SH⊂ PH, the locus of rank 4 two-forms in H. We demonstrate that the induced characteristic map σ: V/ PGL(V6) ⤏ Mc, where Mc denotes the moduli space of cubic surfaces in P3, is dominant, hence generically finite. Based on Manivel and Mezzetti (Manuscr Math 117:319–331, 2005), a complete classification of 4-planes H∈ V with the reducible characteristic surface SH is given.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

manuscripta mathematica

Volume

172

Issue

1-2

Pages

599-620

Publisher

Springer

Version

  • AM (Accepted Manuscript)

Rights holder

© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature

Publisher statement

This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1007/s00229-022-01425-8

Acceptance date

2022-08-15

Publication date

2022-09-06

Copyright date

2022

ISSN

0025-2611

eISSN

1432-1785

Language

  • en

Depositor

Prof Evgeny Ferapontov. Deposit date: 1 December 2022

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