#### Volume 21, issue 4 (2021)

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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
##### 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 $V\phantom{\rule{-0.17em}{0ex}}$. Assume that the embedding is simple, ie the corresponding rational blowup can be obtained by just a sequence of ordinary blowups from  $V\phantom{\rule{-0.17em}{0ex}}$. 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 ${B}_{n,a}♯\overline{{ℂℙ}^{2}}$ (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.

##### Keywords
antiflip, Mori sequence, rational homology ball
##### Mathematical Subject Classification 2010
Primary: 57R40, 57R55
Secondary: 14B07, 14E30