Fano Hypersurfaces With Arbitrarily Large Degrees of Irrationality

Abstract

We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index $e$ , then the degree of irrationality of a very general complex Fano hypersurface of index $e$ and dimension n is bounded from below by a constant times $\sqrt{n}$ . To our knowledge, this gives the first examples of rationally connected varieties with degrees of irrationality greater than 3. The proof follows a degeneration to characteristic $p$ argument, which Kollár used to prove nonrationality of Fano hypersurfaces. Along the way, we show that in a family of varieties, the invariant ‘the minimal degree of a dominant rational map to a ruled variety’ can only drop on special fibers. As a consequence, we show that for certain low-dimensional families of varieties, the degree of irrationality also behaves well under specialization.

The polarized degree of irrationality of $K3$ surfaces

“…The case when X is a curve leads to the classical notion of gonality, whereas for dim(X) ≥ 2 very little is known in general and the problem has recently received considerable attention. The main results in this direction are for hypersurfaces [5,6,9,21,25] or abelian varieties [2,8,11,15,16,20,22,23,26] or hyper-Kähler varieties [24] or more specific examples [1,13]. Giving a sharp lower bound is in general a very difficult problem.…”

Higher index Fano varieties with finitely many birational automorphisms

“…The arguments of Totaro and Schreieder both involve the specialization property of decomposition of the diagonal, which was developed by Voisin [Voi13] and expanded upon in work of Colliot-Thélène and Pirutka [CP16]. In other degree ranges, by studying the positivity properties of these forms in more detail, the authors demonstrated that the degrees of irrationality of complex Fano hypersurfaces can be arbitrarily large and, in a different range, the degrees of possible rational endomorphisms on complex Fano hypersurfaces must satisfy certain congruence conditions (see [CS20, CS21]).…”