Let $f:E\to F$ be a morphism of vector bundles on an irreducible algebraic variety $X$. Does anybody know any results about the irreducibility or smoothness of the degeneracy locus of $f$? I know only the Connectedness Theorem due to Fulton.
The degeneracy locus can be reducible, and even nonreduced. For instance, take $X= \mathbb{P}^2$ and consider a morphism $\mathcal{O}(1)^2 \stackrel{f} \to \mathcal{O}^2$. $f$ is given by a $2 \times 2$ matrix of linear forms, so its degeneracy locus is a conic. For a general choice of $f$ this conic will be smooth, but for special choice of the matrix it can become singular or even a double line. The best result I am aware of can be found in Ottaviani's book "Varieta' proiettive di codimensione piccola" [projective varieties of small codimension, unfortunately I do not think an english translation is available]. Set $D_k(f):=\{x \in X \;  \; \textrm{rank}(f_x) \leq k \}$ Then we have the following Theorem (of Bertini's type) Set $\textrm{rank}(E)=m$, $\textrm{rank}(F)=n$. Assume that $E^{*} \otimes F$ is globally generated. Then for the generic morphism $f \colon E \to F$, the locus $D_k(f)$ is either empty or it has the expected codimension $(mk)(nk)$, and the singular locus of $D_k(f)$ is contained in $D_{k1}(f)$. In particular, if $\dim X < (mk+1)(nk+1)$ then $D_k(f)$ is smooth for a general choice of $f$. 


At Barbara's request, I am posting this as an answer (with a correction). Assume that $X$ is CohenMacaulay. If $D_{k}(f)$ has the expected dimension, then it is CohenMacaulay. This is, of course, far from being nonsingular, but limits how bad the singularities can be (e.g., no embedded points and all irreducible components have the same dimension). I learned about this result from Chapter 4 of Geometry of algebraic curves. I do not have the book handy, but I think this result was originally proven in [HochsterEagon, "A class of perfect determinantal ideals"].


