Tilt stability and the degree of irrationality of surfaces on threefolds

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-Macrì-Stellari, then the minimum degree of a dominant rational map $S⇢{\mathbb{P}}^2$ 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.

The characterization of the length of the base locus as ab^2H^3-e on page 3 line 3 is inaccurate, as can be seen if the base locus of phi is e.g. a fat point. This invalidates the rest of the paper