Rationally connected non-Fano type varieties

TL;DR

This paper explores the relationship between rationally connected varieties and Fano type varieties, providing examples of rationally connected varieties not birationally equivalent to Fano type, thus addressing a key open question.

Contribution

It demonstrates that not all rationally connected varieties are birationally equivalent to Fano type varieties, offering new insights into their classification.

Findings

Existence of rationally connected varieties of dimension ≥3 not birationally equivalent to Fano type

Counterexamples to the assumption that all rationally connected varieties are of Fano type

Clarification of the relationship between rational connectedness and Fano type properties

Abstract

Varieties of Fano type are very well behaved with respect to MMP, and they are known to be rationally connected. We study the relation between classes of rationally connected varieties and varieties of Fano type. We give examples of rationally connected varieties of dimension $\geqslant 3$ which are not birationally equivalent to Fano type varieties, thereby answering the question of Cascini and Gongyo.