Explicity birational geometry of Fano 3-folds of high index

We complete the analysis on the birational rigidity of quasismooth Fano 3-fold deformation families appearing in the Graded Ring Database as a complete intersection. When such a deformation family X has Fano index at least 2 and is minimally embedded in a weighted projective space in codimension 2,...

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Main Author: Tiago Duarte Guerreiro
Format: Default Thesis
Published: 2022
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Online Access:https://dx.doi.org/10.26174/thesis.lboro.19078169.v1
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spelling rr-article-190781692022-02-03T11:47:27Z Explicity birational geometry of Fano 3-folds of high index Tiago Duarte Guerreiro (11510986) Other mathematical sciences not elsewhere classified Birational Geometry Fano 3-folds Sarkisov Links Minimal Model Program Mathematical Sciences not elsewhere classified We complete the analysis on the birational rigidity of quasismooth Fano 3-fold deformation families appearing in the Graded Ring Database as a complete intersection. When such a deformation family X has Fano index at least 2 and is minimally embedded in a weighted projective space in codimension 2, we determine which cyclic quotient singularity is a maximal centre. If a cyclic quotient singularity is a maximal centre, we construct a Sarkisov link to a non-isomorphic Mori fibre space or a birational involution. We define linear cyclic quotient singularities on X and prove that these are maximal centres by explicitly computing Sarkisov links centred at them. It turns out that each X has a linear cyclic quotient singularity leading to a new birational model. As a consequence, we show that if X is birationally rigid then its Fano index is 1. If the new birational model is a strict Mori fibre space, we determine its fibration type explicitly. In this case, a general member of X is birational to a del Pezzo fibration of degrees 1, 2 or 3 or to a conic bundle Y/S where S is a weighted projective plane with at most A<sub>2</sub> singularities. 2022-02-03T11:47:27Z Text Thesis 10.26174/thesis.lboro.19078169.v1 https://figshare.com/articles/thesis/Explicity_birational_geometry_of_Fano_3-folds_of_high_index/19078169 CC BY-NC-ND 4.0
institution Loughborough University
collection Figshare
topic Other mathematical sciences not elsewhere classified
Birational Geometry
Fano 3-folds
Sarkisov Links
Minimal Model Program
Mathematical Sciences not elsewhere classified
spellingShingle Other mathematical sciences not elsewhere classified
Birational Geometry
Fano 3-folds
Sarkisov Links
Minimal Model Program
Mathematical Sciences not elsewhere classified
Tiago Duarte Guerreiro
Explicity birational geometry of Fano 3-folds of high index
description We complete the analysis on the birational rigidity of quasismooth Fano 3-fold deformation families appearing in the Graded Ring Database as a complete intersection. When such a deformation family X has Fano index at least 2 and is minimally embedded in a weighted projective space in codimension 2, we determine which cyclic quotient singularity is a maximal centre. If a cyclic quotient singularity is a maximal centre, we construct a Sarkisov link to a non-isomorphic Mori fibre space or a birational involution. We define linear cyclic quotient singularities on X and prove that these are maximal centres by explicitly computing Sarkisov links centred at them. It turns out that each X has a linear cyclic quotient singularity leading to a new birational model. As a consequence, we show that if X is birationally rigid then its Fano index is 1. If the new birational model is a strict Mori fibre space, we determine its fibration type explicitly. In this case, a general member of X is birational to a del Pezzo fibration of degrees 1, 2 or 3 or to a conic bundle Y/S where S is a weighted projective plane with at most A2 singularities.
format Default
Thesis
author Tiago Duarte Guerreiro
author_facet Tiago Duarte Guerreiro
author_sort Tiago Duarte Guerreiro (11510986)
title Explicity birational geometry of Fano 3-folds of high index
title_short Explicity birational geometry of Fano 3-folds of high index
title_full Explicity birational geometry of Fano 3-folds of high index
title_fullStr Explicity birational geometry of Fano 3-folds of high index
title_full_unstemmed Explicity birational geometry of Fano 3-folds of high index
title_sort explicity birational geometry of fano 3-folds of high index
publishDate 2022
url https://dx.doi.org/10.26174/thesis.lboro.19078169.v1
_version_ 1796098668965658624