2014/07/01 by Lindsay Crowl Erickson, Erickson, Lindsay
Computer Science · Mathematics · #14B05 #14B07 #14E15 #14J17 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1407.0110
openalex publication_date 2014/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Rational pairs, recently introduced by Kollár and Kovács, generalize rational singularities to pairs (X,D). Here X is a normal variety and D is a reduced divisor on X. Integral to the definition of a rational pair is the notion of a thrifty resolution, also defined by Kollár and Kovács, and in order to work with rational pairs it is often necessary to know whether a given resolution is thrifty. In this paper we present several foundational results that are helpful for identifying thrifty resolutions and analyzing their behavior. We also show that general hyperplane sections of rational pairs are again rational. In 1978, Elkik proved that rational singularities are deformation invariant. Our main result is an analogue of this theorem for rational pairs: given a flat family X→ S and a Cartier divisor D on X, if the fibers over a smooth point s∈ S form a rational pair, then (X,D) is also rational near the fiber Xs.