Rationally connected non-Fano type varieties

Abstract

Varieties of Fano type are very well behaved with respect to the MMP, and they are known to be rationally connected. We study a relation between the classes of rationally connected varieties and varieties of Fano type. It is known that these classes are birationally equivalent in dimension 2. We give examples of rationally connected varieties of dimension 3 which are not birational to varieties of Fano type, thereby answering the question of Cascini and Gongyo [2, Question 5.2].

The rationality problem for conic bundles

“…There are analogs of these results for surfaces over non-closed fields. Similar question for del Pezzo fibrations was studied by I. Krylov [Kry18]. 14.6.…”

Birational geometry of algebraic varieties fibred into Fano double spaces

“…Theorem 0.1. Assume that dim F 4 and every fibre F of the projection π is a variety with at most quadratic singularities of rank at least 4, and moreover codim(Sing F ⊂ F ) 4. Assume further that every fibre F satisfies the conditions (h), (hd) and (v), whereas the fibre space V /S satisfies the K-condition K 2condition.…”

“…Finally, let us point out the recent work [4], where by means of the results of [6,7] (see also [8,Chapter 7]) the problem of existence of rationally connected varieties that are non-Fano type varieties, stated in [3], was solved.…”

A Remark on Rationally Connected Varieties and Mori Dream Spaces

“…However, the converse is not true (the blow-up of P 2 in 10 very general points provides an obvious counterexample). In the recent paper [Kry15] it is shown that there exist smooth rationally connected varieties of dimension n ≥ 4 that are not birationally equivalent to a variety of Fano type.…”

“…Krylov then asked the following question. Question 1.1 (see [11,Remark 5.7]). Let X be a rationally connected variety.…”