Title: Irrationality maps of divisors come from fibrations in curves

Abstract: The degree of irrationality of an algebraic variety is an interesting measure of the birational complexity of a variety. E.g. a variety is rational exactly when it has degree of irrationality one, and in dimension one the degree of irrationality coincides with the gonality. An influential result of Bastianelli et al. says that for high degree hypersurfaces in projective space, any degree of irrationality map extends to a "congruence of lines" (i.e. a rational fibration of projective space in lines). Said differently, any degree of irrationality map of a hypersurface factors through a rational fibration of projective space in lines. Somewhat surprisingly, Levinson, Ullery, and I prove this principle generalizes to high degree divisors in arbitrary varieties, which is the main subject of this talk. As a consequence, Lebovitz and I show that the degree of irrationality of a smooth and sufficiently positive divisor in a toric variety is asymptotic to the lattice width of the associated polytope.