Simple embeddings of rational homology balls and antiflips

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,B-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,B-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. 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 B-n,B-a (sic) CP2 (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.