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

It is well known since M. Noether that the gonality of a smooth plane curve of degree d at least 4 is d-1. Given a k-dimensional complex projective variety X, the most natural extension of gonality is probably the degree of irrationality, that is the minimum degree of a dominant rational to a k-dimensional projective space. In this paper we are aimed at extending the assertion on plane curves to smooth hypersurfaces in the n-dimensional projective space in terms of degree of irrationality. We prove that both surfaces in P^3 and threefolds in P^4 of sufficiently large degree d have degree of irrationality d-1, except for finitely many cases we classify, whose degree of irrationality is d-2. To this aim we use Mumford's technique of induced differentials and we shift the problem to study first order congruences of lines of P^n. In particular, we also slightly improve the description of such congruences in P^4 and we provide a bound on degree of irrationality of hypersurfaces of arbitrary dimension

We study congruences of lines Xω defined by a sufficiently general choice of an alternating 3-form ω in n+1 dimensions, as Fano manifolds of index 3 and dimension n−1. These congruences include the G2-variety for n=6 and the variety of reductions of projected ℙ2×ℙ2 for n=7.\ud We compute the degree of Xω as the n-th Fine number and study the Hilbert scheme of these congruences proving that the choice of ω bijectively corresponds to Xω except when n=5. The fundamental locus of the congruence is also studied together with its singular locus: these varieties include the Coble cubic for n=8 and the Peskine variety for n=9.\ud The residual congruence Y of Xω with respect to a general linear congruence containing Xω is analysed in terms of the quadrics containing the linear span of Xω. We prove that Y is Cohen-Macaulay but non-Gorenstein in codimension 4. We also examine the fundamental locus G of Y of which we determine the singularities and the irreducible components

The existence is proved of two new families of sextic threefolds in P 5 , which are not quadratically normal. These threefolds arise naturally in the realm of first order congruences of lines as focal loci and in the study of the completely exceptional Monge-Ampère equations. One of these families comes from a smooth congruence of multidegree (1, 3, 3) which is a smooth Fano fourfold of index two and genus 9.

The goal of this article is to compare the coefficients in the expansion of the permanent with those in the expansion of the determinant of a three-lines circulant matrix. As an application we solve a conjecture stated in [17] concerning the minimality of GT-systems.2010 Mathematics Subject Classification. 15B05, 15A05, 13E10, 14M25.