Question

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.

Answer 1

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 $f$.

Answer 2

At Barbara's request, I am posting this as an answer (with a correction).

Assume that $X$ is Cohen-Macaulay. If $D_{k}(f)$ has the expected dimension, then it is Cohen-Macaulay. This is, of course, far from being non-singular, 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 [Hochster-Eagon, "A class of perfect determinantal ideals"].

Chapter 4 has a general discussion of determinantal varieties that might be helpful. I seem to remember that they prove, if $D_{k}(f)$ has expected dimension, then the singular locus of $D_{k}(f)$ is contained in $D_{k-1}(f)$ (part of the theorem cited by Francesco Polizzi, but without genericity), but I could be misremembering.
I was misremembering.