Strong rational connectedness of Toric Varieties

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

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Published 20 February 2012