Gonality of curves on fundamental loci of first order congruences (Appendix to article of Ein, Lazarsfeld and Ullery)

TL;DR

This paper proves the existence of low-gonality curves on fundamental loci of first order congruences and refines a theorem relating to dominant rational maps from hypersurfaces to projective space.

Contribution

It introduces new existence results for low-gonality curves on specific geometric loci and refines bounds on degrees of rational maps from hypersurfaces.

Findings

Existence of families of low-gonality curves on fundamental loci.

Refinement of bounds on degrees of rational maps from hypersurfaces.

Characterization of when the degree bound is attained.

Abstract

This note is an appendix to 'Measures of irrationality for hypersurfaces of large degree' by L. Ein, R. Lazarsfeld and B. Ullery. We prove an existence result for families of curves having low gonality, and lying on fundamental loci of first order congruences of lines in $\mathbb{P}^{n+1}$. As an application, we follow the ideas of the main paper, and we present a slight refinement of a theorem included in it. In particular, we show that given a very general hypersurface $X\subset \mathbb{P}^{n+1}$ of degree $d \geq 3n-2 \geq 7$, and a dominant rational map $f \colon X \dashrightarrow \mathbb{P}^{n}$, then $deg(f) \geq d-1$, and equality holds if and only if $f$ is the projection from a point of $X$.