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Fano 3-folds in codimension 4, Tom and Jerry. Part I

Part of: Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  14 May 2012

Gavin Brown ,
Michael Kerber  and
Miles Reid
Gavin Brown
Affiliation:
School of Mathematics, Loughborough University, LE11 3TU, UK (email: [email protected])
Michael Kerber
Affiliation:
Institute of Science and Technology (IST) Austria, 3400 Klosterneuburg, Austria (email: [email protected])
Miles Reid
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (email: [email protected])
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Abstract

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We introduce a strategy based on Kustin–Miller unprojection that allows us to construct many hundreds of Gorenstein codimension 4 ideals with 9×16 resolutions (that is, nine equations and sixteen first syzygies). Our two basic games are called Tom and Jerry; the main application is the biregular construction of most of the anticanonically polarised Mori Fano 3-folds of Altınok’s thesis. There are 115 cases whose numerical data (in effect, the Hilbert series) allow a Type I projection. In every case, at least one Tom and one Jerry construction works, providing at least two deformation families of quasismooth Fano 3-folds having the same numerics but different topology.

Keywords

Mori theoryFano 3-foldunprojectionSarkisov program

MSC classification

Secondary: 14J45: Fano varieties
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 4 , July 2012 , pp. 1171 - 1194
Copyright
Copyright © Foundation Compositio Mathematica 2012

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