Hilbert schemes and toric degenerations for low degree Fano threefolds
type: Item
instance of: scholarly article
title: Hilbert schemes and toric degenerations for low degree Fano threefolds
zbMATH Open document ID: 1445.14006
author: Jan Arthur Christophersen
author: Nathan Owen Ilten
published in: Journal für die Reine und Angewandte Mathematik
publication date: 2016-08-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1202.0510
reviewed by: Klaus Altmann
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14M25
zbMATH DE Number: 6620064
zbMATH Keywords: toric degenerations
zbMATH Keywords: Fano threefolds
zbMATH Keywords: reflexive polytopes
zbMATH Keywords: Hilbert schemes
describes a project that uses: 4ti2
describes a project that uses: TOPCOM
describes a project that uses: Macaulay2
describes a project that uses: Graded Ring Database
describes a project that uses: VersalDeformations
MaRDI profile type: Publication
OpenAlex ID: W3103663961
arXiv ID: 1202.0510
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DOI: 10.1515/CRELLE-2014-0011
review text: Smooth Fano threefolds \(V\) with very ample anticanonical divisor \(-K_V\) lead to points \([V]\in\mathcal H_d\) in the Hilbert scheme of degree \(d:=-K_V^3\) subvarieties of \(\mathbb{P}(|-K_V|)\). These point lie on unique irreducible components.NEWLINEMaximal families of those Fanos, i.e.\ the set of those components of \(\mathcal H_d\) are completely classified.NEWLINEFor small degree \(d\), these families can be distinguished by \((d,b_2,b_3)\) where \(b_i\) denote the \(i\)-th Betti number of \(V\). NEWLINEThe dimensions of the corresponding components, i.e.\ \(h^0(\mathcal N_{V})\), are known, too.NEWLINENEWLINEOn the other hand, toric Fano threefolds \(X\) of degree \(d\) with Gorenstein singularities correspond to three-dimensional reflexive polytopes. They are classified, too.NEWLINEThey do also provide points \([X]\in\mathcal H_d\) which might, however, sit in several components. Presenting this incidence for \(d\leq 12\) is the main point of the paper. Thus, it provides a complete overview about toric degenerations of the above \(V\) which is important from the view point of mirror symmetry.NEWLINENEWLINEOne of the tools is the identification of certain Stanley-Reisner schemes \(S\) wich provide points \([S]\in\mathcal H_d\), too.NEWLINEThese are the most degenerate versions. Degenerations from \(X\) to \(S\) are understood by unimodular triangulations of the reflexive polytopes.NEWLINEMoreover, deformation theory of \(X\) is used because the components of \(\mathcal H_d\) containing \([X]\) are identified by the components of the tangent cone of \(\mathcal H_d\) in \([X]\). NEWLINEThe equations of the latter are obtained via explicit computeralgebra computations.
description: scientific article; zbMATH DE number 6620064