
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 which are not birationally equivalent to Fano type varieties, thereby answering the question of Cascini and Gongyo.
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