Deformations and moduli of irregular canonical covers with K2 = 4pg − 8
In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying K 2 = 4pg − 8, for any even integer pg ≥ 4. These surfaces also have unbounded irregularity q. We carry out our study by investigating the deformations of the canonical morphis...
| Autores: | , , , |
|---|---|
| Tipo de recurso: | artículo |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Universidad Complutense de Madrid (UCM) |
| Repositorio: | Docta Complutense |
| Idioma: | inglés |
| OAI Identifier: | oai:docta.ucm.es:20.500.14352/87709 |
| Acceso en línea: | https://hdl.handle.net/20.500.14352/87709 |
| Access Level: | acceso abierto |
| Palabra clave: | 512.774 Deformation of morphisms Moduli of surfaces of general type Canonical covers or surfaces of minimal degree Geometria algebraica 1201.01 Geometría Algebraica |
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Deformations and moduli of irregular canonical covers with K2 = 4pg − 8Bangere, PurnaprajnaGallego Rodrigo, Francisco JavierMukherjee, JayanRaychaudhury, Debaditya512.774Deformation of morphismsModuli of surfaces of general typeCanonical covers or surfaces of minimal degreeGeometria algebraica1201.01 Geometría AlgebraicaIn this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying K 2 = 4pg − 8, for any even integer pg ≥ 4. These surfaces also have unbounded irregularity q. We carry out our study by investigating the deformations of the canonical morphism ϕ : X → PN, where ϕ is a quadruple Galois cover of a smooth surface of minimal degree. These canonical covers are classified in Gallego and Purnaprajna (Trans Am Math Soc 360(10):5489-5507, 2008) into four distinct families, one of which is the easy case of a product of curves. The main objective of this article is to study the deformations of the other three, non trivial, unbounded families. We show that any deformation of ϕ factors through a double cover of a ruled surface and, hence, is never birational. More interestingly, we prove that, with two exceptions, a general deformation of ϕ is two-to-one onto its image, whose normalization is a ruled surface of appropriate irregularity. We also show that, with the exception of one family, the deformations of X are unobstructed even though H2(TX) does not vanish. Consequently, X belongs to a unique irreducible component of the Gieseker moduli space. These irreducible components are uniruled. As a result of all this, we show the existence of infinitely many moduli spaces, satisfying the strict Beauville inequality pg > 2q − 4, with an irreducible component that has a proper "quadruple" sublocus where the degree of the canonical morphism jumps up. These components are above the Castelnuovo line, but nonetheless parametrize surfaces with non birational canonical morphisms. The existence of jumping subloci is a contrast with the moduli of surfaces with K 2 = 2pg − 4, studied by Horikawa. Irreducible moduli components with a jumping sublocus also present a similarity and a difference to the moduli of curves of genus g ≥ 3, for, like in the case of curves, the degree of the canonical morphism goes down outside a closed sublocus but, unlike in the case of curves, it is never birational. Finally, our study shows that there are infinitely many moduli spaces with an irreducible component whose general elements have non birational canonical morphism and another irreducible component whose general elements have birational canonical map.SpringerUniversidad Complutense de Madrid20232023-03-2120232023-03-21journal articlehttp://purl.org/coar/resource_type/c_6501info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/20.500.14352/87709reponame:Docta Complutenseinstname:Universidad Complutense de Madrid (UCM)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:docta.ucm.es:20.500.14352/877092026-06-02T12:44:21Z |
| dc.title.none.fl_str_mv |
Deformations and moduli of irregular canonical covers with K2 = 4pg − 8 |
| title |
Deformations and moduli of irregular canonical covers with K2 = 4pg − 8 |
| spellingShingle |
Deformations and moduli of irregular canonical covers with K2 = 4pg − 8 Bangere, Purnaprajna 512.774 Deformation of morphisms Moduli of surfaces of general type Canonical covers or surfaces of minimal degree Geometria algebraica 1201.01 Geometría Algebraica |
| title_short |
Deformations and moduli of irregular canonical covers with K2 = 4pg − 8 |
| title_full |
Deformations and moduli of irregular canonical covers with K2 = 4pg − 8 |
| title_fullStr |
Deformations and moduli of irregular canonical covers with K2 = 4pg − 8 |
| title_full_unstemmed |
Deformations and moduli of irregular canonical covers with K2 = 4pg − 8 |
| title_sort |
Deformations and moduli of irregular canonical covers with K2 = 4pg − 8 |
| dc.creator.none.fl_str_mv |
Bangere, Purnaprajna Gallego Rodrigo, Francisco Javier Mukherjee, Jayan Raychaudhury, Debaditya |
| author |
Bangere, Purnaprajna |
| author_facet |
Bangere, Purnaprajna Gallego Rodrigo, Francisco Javier Mukherjee, Jayan Raychaudhury, Debaditya |
| author_role |
author |
| author2 |
Gallego Rodrigo, Francisco Javier Mukherjee, Jayan Raychaudhury, Debaditya |
| author2_role |
author author author |
| dc.contributor.none.fl_str_mv |
Universidad Complutense de Madrid |
| dc.subject.none.fl_str_mv |
512.774 Deformation of morphisms Moduli of surfaces of general type Canonical covers or surfaces of minimal degree Geometria algebraica 1201.01 Geometría Algebraica |
| topic |
512.774 Deformation of morphisms Moduli of surfaces of general type Canonical covers or surfaces of minimal degree Geometria algebraica 1201.01 Geometría Algebraica |
| description |
In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying K 2 = 4pg − 8, for any even integer pg ≥ 4. These surfaces also have unbounded irregularity q. We carry out our study by investigating the deformations of the canonical morphism ϕ : X → PN, where ϕ is a quadruple Galois cover of a smooth surface of minimal degree. These canonical covers are classified in Gallego and Purnaprajna (Trans Am Math Soc 360(10):5489-5507, 2008) into four distinct families, one of which is the easy case of a product of curves. The main objective of this article is to study the deformations of the other three, non trivial, unbounded families. We show that any deformation of ϕ factors through a double cover of a ruled surface and, hence, is never birational. More interestingly, we prove that, with two exceptions, a general deformation of ϕ is two-to-one onto its image, whose normalization is a ruled surface of appropriate irregularity. We also show that, with the exception of one family, the deformations of X are unobstructed even though H2(TX) does not vanish. Consequently, X belongs to a unique irreducible component of the Gieseker moduli space. These irreducible components are uniruled. As a result of all this, we show the existence of infinitely many moduli spaces, satisfying the strict Beauville inequality pg > 2q − 4, with an irreducible component that has a proper "quadruple" sublocus where the degree of the canonical morphism jumps up. These components are above the Castelnuovo line, but nonetheless parametrize surfaces with non birational canonical morphisms. The existence of jumping subloci is a contrast with the moduli of surfaces with K 2 = 2pg − 4, studied by Horikawa. Irreducible moduli components with a jumping sublocus also present a similarity and a difference to the moduli of curves of genus g ≥ 3, for, like in the case of curves, the degree of the canonical morphism goes down outside a closed sublocus but, unlike in the case of curves, it is never birational. Finally, our study shows that there are infinitely many moduli spaces with an irreducible component whose general elements have non birational canonical morphism and another irreducible component whose general elements have birational canonical map. |
| publishDate |
2023 |
| dc.date.none.fl_str_mv |
2023 2023-03-21 2023 2023-03-21 |
| dc.type.none.fl_str_mv |
journal article http://purl.org/coar/resource_type/c_6501 |
| dc.type.openaire.fl_str_mv |
info:eu-repo/semantics/article |
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article |
| dc.identifier.none.fl_str_mv |
https://hdl.handle.net/20.500.14352/87709 |
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https://hdl.handle.net/20.500.14352/87709 |
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Inglés eng |
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Inglés |
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eng |
| dc.rights.none.fl_str_mv |
open access http://purl.org/coar/access_right/c_abf2 Attribution 4.0 International http://creativecommons.org/licenses/by/4.0/ |
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info:eu-repo/semantics/openAccess |
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open access http://purl.org/coar/access_right/c_abf2 Attribution 4.0 International http://creativecommons.org/licenses/by/4.0/ |
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openAccess |
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application/pdf |
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Springer |
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Springer |
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reponame:Docta Complutense instname:Universidad Complutense de Madrid (UCM) |
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Universidad Complutense de Madrid (UCM) |
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Docta Complutense |
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