Curves on complete intersections and measures of irrationality

TL;DR

This paper investigates the minimal degrees of curves on complete intersections, proving a classical conjecture about their lower bounds and confirming a conjecture on measures of irrationality for these varieties.

Contribution

It proves a classical conjecture on the degree bounds of curves on general complete intersections and verifies a conjecture on measures of irrationality.

Findings

Degree of any curve on a general complete intersection is bounded below by the degree of the variety.

Confirmed a conjecture relating to measures of irrationality for complete intersections.

Established bounds that relate geometric properties of curves to the defining polynomials.

Abstract

We study the minimal degrees and gonalities of curves on complete intersections. We prove a classical conjecture which asserts that the degree of any curve on a general complete intersection $X \subseteq \mathbb{P}^N$ cut out by polynomials of large degrees is bounded from below by the degree of $X$. As an application, we verify a conjecture of Bastianelli--De Poi--Ein--Lazarsfeld--Ullery on measures of irrationality for complete intersections.