# 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.

## Key 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.

---
Source: https://tomesphere.com/paper/1907.13084