The degree of irrationality of hypersurfaces in various Fano varieties

Abstract

IThe degree of irrationality of an n-dimensional algebraic variety X , denoted irr(X ), is the minimal degree of a dominant rational map φ : X P n .The aim of this paper is to compute the degree of irrationality of hypersurfaces in various Fano varieties: quadrics, cubic threefolds, cubic fourfolds, complete intersection threefolds of type (2,2), products of projective spaces, and Grassmannians. Throughout we work with varieties over C.Recently there has been a great deal of interest in understanding different measures of irrationality of higher dimensional varieties. Bastianelli, Cortini, and De Poi conjectured ([BCDP14, Conj. 1.5]) that if X is a very general d hypersurfacewith d ≥ 2n + 1, then irr(X ) = d − 1. They proved their conjecture in the case X is a surface or threefold. This conjecture was proved in full by Bastianelli, De Poi, Ein, Lazarsfeld, and the second author in ([BDPE + 15]). Gounelas and Kouvidakis ([GK17]) computed the covering gonality and the degree of irrationality of the Fano surface of a generic cubic threefold. Bastianelli, Ciliberto, Flamini, and Supino ([BCFS17]) computed the covering gonality of a very general hypersurface in P n+1 . Recently, Voisin ([Voi18]) proved that the covering gonality of a very general n-dimensional abelian variety goes to infinity with n.In this paper we show that the ideas in the proof of [BDPE + 15, Thm. C] can be extended to compute the degree of irrationality of hypersurfaces in many Fano varieties. For example, let Q ⊂ P n+2 be a smooth quadric in projective space.be a very general hypersurface in Q with X ∈ |O Q (d )|. If d ≥ 2n then irr(X ) = d .We have other results for hypersurfaces in cubic threefolds and cubic fourfolds.Theorem B. Let X = X d ⊂ Z ⊂ P n+2 be a smooth complete intersection of type (3, d ) in a smooth cubic hypersurface.

Measures of irrationality for hypersurfaces of large degree

“…In higher dimensions one might therefore want to consider first the case that is a very general complete intersection of the stated multidegrees. In the codimension two case, for instance, Stapleton [Sta17] shows that if is a very general complete intersection of type such that , then …”

“…One might have imagined that the irrationality degree grows linearly in , but Stapleton [Sta17] observes that in fact so at best the growth is sublinear. Recalling that for every (Example 1.7), the conjecture would yield a natural family of examples showing that the covering gonality and the degree of irrationality capture very different phenomena.…”

Geometry of Points Satisfying Cayley-Bacharach Conditions and Applications

“…Let Γ be a set of points CB(k). In [SU,Theorem 1.9] the authors prove that Γ lies on a curve of degree at most 2 provided that |Γ| ≤ 5 2 k + 1. We note that this bound is smaller than 3k − 1 (i.e.…”

A Cayley-Bacharach theorem and plane configurations

“…Several recent papers have exploited the Bastianelli result to compute the gonality, or the degree of irrationality, a higher dimensional analog of gonality, of several classes of smooth complete intersection varieties in projective space (see e.g. [BCD14], [BDE + 17], [SU20], [HLU20]). The theme is that if the canonical bundle of X is sufficiently positive (as, for example, in the case of high degree complete intersections), the fibers are forced to lie in special positions.…”