We study various measures of irrationality for hypersurfaces of large degree in projective space and other varieties. These include the least degree of a rational covering of projective space, and the minimal gonality of a covering family of curves. The theme is that positivity properties of canonical bundles lead to lower bounds on these invariants. In particular, we prove that if $X\subset \mathbb P^{n+1}$ is a very general smooth hypersurface of dimension $n$ and degree $d\ge 2n+1$, then any dominant rational mapping $f\colon X\dashrightarrow \mathbb P^n$ must have degree at least $d-1$. We also propose a number of open problems, and we show how our methods lead to simple new proofs of results of Ran and Beheshti–Eisenbud concerning varieties of multi-secant lines

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In this paper, we examine linear conditions on finite sets of points in projective space implied by the Cayley–Bacharach condition. In particular, by bounding the number of points satisfying the Cayley–Bacharach condition, we force them to lie on unions of low-dimensional linear spaces. These results are motivated by investigations into degrees of irrationality of complete intersections, which are controlled by minimum-degree rational maps to projective space. As an application of our main theorem, we describe the fibers of such maps for certain complete intersections of codimension two.

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.

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