Flipping surfaces

Hacking, Paul; Tevelev, Jenia; Urzúa, Giancarlo

doi:10.1090/jag/682

Flipping surfaces

Authors: Paul Hacking, Jenia Tevelev and Giancarlo Urzúa
Journal: J. Algebraic Geom. 26 (2017), 279-345
DOI: https://doi.org/10.1090/jag/682
Published electronically: August 26, 2016
MathSciNet review: 3606997
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Abstract | References | Additional Information

Abstract: We study semistable extremal $3$-fold neighborhoods, which are fundamental building blocks of birational geometry, following earlier work of Mori, Kollár, and Prokhorov. We classify possible flips and extend Mori’s algorithm for computing flips of extremal neighborhoods of type $k2A$ to more general $k1A$ neighborhoods. The novelty of our approach is to show that $k1A$ belongs to the same deformation family as $k2A$; in fact we explicitly construct the universal family of extremal neighborhoods. This construction follows very closely Mori’s division algorithm, which can be interpreted as a sequence of mutations in the cluster algebra. We identify, in the versal deformation space of a cyclic quotient singularity, the locus of deformations such that the total space admits a (terminal) antiflip. We show that these deformations come from at most two irreducible components of the versal deformation space. As an application, we give an algorithm for computing stable one-parameter degenerations of smooth projective surfaces (under some conditions) and describe several components of the Kollár–Shepherd-Barron–Alexeev boundary of the moduli space of smooth canonically polarized surfaces of geometric genus zero.

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Additional Information

Paul Hacking
Affiliation: Department of Mathematics and Statistics, University of Massachusetts Amherst, 710 N. Pleasant Street, Amherst, Massachusetts 01003-9305
MR Author ID: 737867
Email: hacking@math.umass.edu

Jenia Tevelev
Affiliation: Department of Mathematics and Statistics, University of Massachusetts Amherst, 710 N. Pleasant Street, Amherst, Massachusetts 01003-9305
MR Author ID: 607263
Email: tevelev@math.umass.edu

Giancarlo Urzúa
Affiliation: Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Campus San Joaquín, Avenida Vicuña Mackenna 4860, Santiago, Chile
MR Author ID: 797224
Email: urzua@mat.uc.cl

Received by editor(s): March 7, 2014
Received by editor(s) in revised form: March 6, 2015, and September 19, 2015
Published electronically: August 26, 2016
Article copyright: © Copyright 2016 University Press, Inc.