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.



Irreducibility\smoothness of the degeneracy locus
The degeneracy locus can be reducible, and even non-reduced. 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 $(m-k)(n-k)$, and the singular locus of $D\_k(f)$ is contained in $D\_{k-1}(f)$.


In particular, if


$\dim X < (m-k+1)(n-k+1)$


then $D\_k(f)$ is smooth for a general choice of $f$. 


