Smoothing surfaces on fourfolds

TL;DR

This paper extends classical results on the smoothness of surfaces arising from rank drops of morphisms between vector bundles on fourfolds, including cases where the sheaves are not vector bundles or globally generated.

Contribution

It proves new variants of a Bertini-type theorem for morphisms between sheaves on fourfolds, relaxing previous assumptions and providing applications to linkage classes of surfaces.

Findings

General maps drop rank along smooth surfaces.

Extended results to non-vector bundle sheaves.

Demonstrated smoothability of surfaces in linkage classes, including Horrocks-Mumford.

Abstract

If $\mathcal E, \mathcal F$ are vector bundles of ranks $r-1,r$ on a smooth fourfold $X$ and $\mathcal{Hom}(\mathcal E,\mathcal F)$ is globally generated, it is well known that the general map $\phi: \mathcal E \to \mathcal F$ is injective and drops rank along a smooth surface. Chang improved on this with a filtered Bertini theorem. We strengthen these results by proving variants in which (a) $\mathcal F$ is not a vector bundle and (b) $\mathcal{Hom}(\mathcal E,\mathcal F)$ is not globally generated. As an application, we give examples of even linkage classes of surfaces on $\mathbb P^4$ in which all integral surfaces are smoothable, including the linkage classes associated with the Horrocks-Mumford surface.