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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

Let V be a regular neighborhood of a negative chain of 2–spheres (ie an exceptional divisor of a cyclic quotient singularity), and let Bp,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 Bp,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 Bp,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 Bp,q embedded in blown-up rational homology balls Bn,a2¯ (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.

antiflip, Mori sequence, rational homology ball
Mathematical Subject Classification 2010
Primary: 57R40, 57R55
Secondary: 14B07, 14E30
Received: 11 December 2019
Revised: 22 April 2020
Accepted: 24 July 2020
Published: 18 August 2021
Heesang Park
Department of Mathematics
Konkuk University
South Korea
School of Mathematics
Korea Institute for Advanced Study
South Korea
Dongsoo Shin
Department of Mathematics
Chungnam National University
South Korea
School of Mathematics
Korea Institute for Advanced Study
South Korea
Giancarlo Urzúa
Facultad de Matemáticas
Pontificia Universidad Católica de Chile