Simple embeddings of rational homology balls and antiflips

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 $B_{n,a}♯\overline{{ℂ\mathbb{P}}^2}$ (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.