Simple embeddings of rational homology balls and antiflips

Park, Heesang; Shin, Dongsoo; Urzúa, Giancarlo

doi:10.2140/agt.2021.21.1857

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Simple embeddings of rational homology balls and antiflips

Heesang Park, Dongsoo Shin and Giancarlo Urzúa

Algebraic & Geometric Topology 21 (2021) 1857–1880

DOI: 10.2140/agt.2021.21.1857

Abstract

Let $V$ be a regular neighborhood of a negative chain of $2$ –spheres (ie an exceptional divisor of a cyclic quotient singularity), and let $B_{p, q}$ be a rational homology ball which is smoothly embedded in $V$ . Assume that the embedding is simple, ie the corresponding rational blowup can be obtained by just a sequence of ordinary blowups from $V$ . Then we show that this simple embedding comes from the semistable minimal model program (MMP) for $3$ –dimensional complex algebraic varieties under certain mild conditions. That is, one can find all simply embedded $B_{p, q}$ ’s in $V$ via a finite sequence of antiflips applied to a trivial family over a disk. As applications, simple embeddings are impossible for chains of $2$ –spheres with self-intersections equal to $- 2$ . We also show that there are (infinitely many) pairs of disjoint $B_{p, q}$ ’s smoothly embedded in regular neighborhoods of (almost all) negative chains of $2$ –spheres. Along the way, we describe how MMP gives (infinitely many) pairs of disjoint rational homology balls $B_{p, q}$ embedded in blown-up rational homology balls $B_{n, a} ♯ \bar{{ℂ ℙ}^{2}}$ (via certain divisorial contractions), and in the Milnor fibers of certain cyclic quotient surface singularities. This generalizes results of Khodorovskiy (2012, 2014), H Park, J Park and D Shin (2016) and Owens (2018) by means of a uniform point of view.

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Keywords

antiflip, Mori sequence, rational homology ball

Mathematical Subject Classification 2010

Primary: 57R40, 57R55

Secondary: 14B07, 14E30

References

Publication

Received: 11 December 2019

Revised: 22 April 2020

Accepted: 24 July 2020

Published: 18 August 2021

Authors

Heesang Park
Department of Mathematics Konkuk University Seoul South Korea	School of Mathematics Korea Institute for Advanced Study Seoul South Korea

Dongsoo Shin
Department of Mathematics Chungnam National University Daejeon South Korea	School of Mathematics Korea Institute for Advanced Study Seoul South Korea

Giancarlo Urzúa
Facultad de Matemáticas Pontificia Universidad Católica de Chile Santiago Chile

Volume 21, issue 4 (2021)