Multiplicity-One Cox Rings of Toric Point Blow-Ups

Abstract

Let $X$ be a complete toric variety and let $e$ be the identity of its open torus. The Cox ring of ${\rm Bl}_eX$ is described by saturated powers of the toric-point lattice ideal $I_X$. We give a finite criterion for generation in Rees multiplicity one in terms of analytic spreads of projected lattice ideals and zero divisors in the associated graded ring. We apply this criterion first to fake weighted projective spaces, proving that multiplicity-one generation is equivalent to $I_X$ being a complete intersection. For projective toric surfaces, we prove that multiplicity-one generation holds if and only if it holds for every Picard-number-one toric surface dominated by $X$.