Tilt stability and the degree of irrationality of surfaces on threefolds

TL;DR

This paper investigates the relationship between tilt stability conditions on surfaces within threefolds and the irrationality degree of those surfaces, providing bounds and specific examples such as intersections of quadrics.

Contribution

It establishes a link between tilt stability conjectures and the degree of irrationality for surfaces on threefolds, including explicit calculations for certain complete intersections.

Findings

If the Bogomolov-Gieseker inequality holds, the irrationality degree is either large or tied to low-degree curve nets.

The degree of irrationality for the intersection of three quadrics in P^5 is exactly 4.

Provides conditions under which the minimal degree of rational maps is constrained.

Abstract

Let $S$ be a smooth projective surface on a smooth threefold $X$ such that $X$ has Picard rank 1 and NS$(S)$ is generated by the restriction of divisors from X. We show that if $X$ satisfies the Bogomolov-Gieseker type inequality for tilt semistable objects conjectured by Bayer-Macr\`i-Stellari, then the minimum degree of a dominant rational map $S\dashrightarrow\mathbb{P}^2$ is either relatively large or determined by a net of curves of low degree on $S$. As one application, we prove that the complete intersection of three very general quadrics in $\mathbb{P}^5$ has degree of irrationality 4.