Strong rational connectedness of toric varieties

Abstract

In this paper, we prove that: For any given finitely many distinct points $P_1,...,P_r$ and a closed subvariety $S$ of codimension $\geq 2$ in a complete toric variety over a uncountable (characteristic 0) algebraically closed field, there exists a rational curve $f:\mathbb{P}^1\to X$ passing through $P_1,...,P_r$, disjoint from $S\setminus \{P_1,...,P_r\}$ (see Main Theorem). As a corollary, we prove that the smooth loci of complete toric varieties are strongly rationally connected.