Strong rational connectedness of toric varieties

TL;DR

This paper proves that complete toric varieties over uncountable algebraically closed fields are strongly rationally connected, meaning rational curves can pass through multiple points and avoid certain subvarieties, extending understanding of their geometric properties.

Contribution

It establishes the strong rational connectedness of smooth loci in complete toric varieties, generalizing previous results and providing new insights into their geometric structure.

Findings

Existence of rational curves passing through multiple points

Rational curves avoiding specified subvarieties

Smooth loci of complete toric varieties are strongly rationally connected

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.