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Foliations with isolated singularities on Hirzebruch surfaces

Carlos Galindo ORCID logo, Francisco Monserrat ORCID logo and Jorge Olivares ORCID logo
From the journal Forum Mathematicum

Abstract

We study foliations on Hirzebruch surfaces Sδ and prove that, similarly to those on the projective plane, any can be represented by a bi-homogeneous polynomial affine 1-form. In case has isolated singularities, we show that, for δ=1, the singular scheme of does determine the foliation, with some exceptions that we describe, as is the case of foliations in the projective plane. For δ1, we prove that the singular scheme of does not determine the foliation. However, we prove that, in most cases, two foliations and given by sections s and s have the same singular scheme if and only if s=Φ(s), for some global endomorphism Φ of the tangent bundle of Sδ.

MSC 2010: 32S65; 32L10

Communicated by Jan Bruinier


Funding source: Ministerio de Ciencia e Innovación

Award Identifier / Grant number: PGC2018-096446-B-C22

Award Identifier / Grant number: RED2018-102583-T

Funding source: Generalitat Valenciana

Award Identifier / Grant number: AICO-2019-223

Funding source: Universitat Jaume I

Award Identifier / Grant number: UJI-2018-10

Funding source: Consejo Nacional de Ciencia y Tecnología

Award Identifier / Grant number: CVU 10069

Funding statement: The first two authors are partially supported by the Spanish Government MICINN/FEDER/AEI/UE, grants PGC2018-096446-B-C22 and RED2018-102583-T, as well as by Generalitat Valenciana, grant AICO-2019-223 and Universitat Jaume I, grant UJI-2018-10. The third author was partially supported by CONACYT: Estancias Sabáticas Vinculadas a la Consolidación de Grupos de Investigación, CVU 10069.

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Received: 2021-05-31
Published Online: 2021-10-10
Published in Print: 2021-11-01

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