Smoothing surfaces on fourfolds

If $\mathcal{E},\mathcal{F}$ are vector bundles of ranks $r-1,r$ on a smooth fourfold $X$ and $\mathcal{H}\mathcal{o}\mathcal{m}(\mathcal{E},\mathcal{F})$ 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) $\mathcal{H}\mathcal{o}\mathcal{m}(\mathcal{E},\mathcal{F})$ 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.