Gorenstein resolutions of $3$-dimensional terminal singularities

Abstract

Let $X$ be a $3$-dimensional terminal singularity of index $\geq 2$. We shall construct projective birational morphisms $f : Y \to X$ such that $Y$ has only Gorenstein terminal singularities and that $f$ factors the minimal resolution of a general member of $\lvert-K_{X}\rvert$. We also study prime divisors of $f$, especially the discrepancies of these prime divisors.