Smoothing surfaces on fourfolds

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.