A remark on partial resolutions of $3$-dimensional terminal singularities

Abstract

Let $X$ be a $3$-dimensional terminal singularity of index $\geq 2$. We study projective birational morphisms $\varphi : Y \to X$ such that the exceptional divisor of $\varphi$ consists of all prime divisors with discrepancies $< 1$ (resp.\ $\leq 1$) over $X$.