Connectedness of degeneracy loci

Question

Let $E, F$ be vector bundles of ranks $e, f$ on a smooth variety $X$ of dimension $n$.

Let $\varphi : E \to F$ be a morphism and let $k$ be such that $n > (e-k)(f-k)$.

Fulton-Lazarsfeld's theorem gives that if $E^* \otimes F$ is ample, then the degeneracy locus $$D_k(\varphi)=\{x \in X : {\rm rank} \varphi(x) \le k\}$$ is connected.

Is $D_k(\varphi)$ still connected if we only assume that $E^* \otimes F$ is big and globally generated?

I am especially interested in the case when $E$ is trivial and $\varphi$ is a general morphism.

Answer 1

On page 2 of this paper there is an example. Pick two integers $a \ge 2, n \ge 3$. Take $X=\mathbb P^1 \times \mathbb P^{n-1}$, $E={\mathcal O}_X$, $F=p_1^*{\mathcal O}_{\mathbb P^1}(a) \oplus p_2^*{\mathcal O}_{\mathbb P^{n-1}}(1)$ and $\varphi : {\mathcal O}_X \to F$ given by a general section $s \in H^0(F)$. The authors prove that the zero locus of $s$ (that is $D_0(\varphi)$) is not connected and that $F$ is big (and globally generated) by computing the $n$-th Segre class $s_n(F^*)=a>0$.

Answer 2

This answer explains some positive results. Let $B$ be a smooth complex variety, and let $\pi:Y\to B$ be a smooth, projective morphism with connected fibers of dimension $n$. Let $E_B$ and $F_B$ be locally free $\mathcal{O}_Y$-modules of ranks $e$ and $f$. Let $\varphi_B:E_B\to F_B$ be a morphism of $\mathcal{O}_Y$-modules. For every nonnegative integer $k\leq \text{min}(e,f)$, let $D_{\leq k}(\varphi_B)$ denote the maximal closed subscheme of $Y$ on which the rank of $\varphi_B$ is $\leq k$. Denote by $B^o\subseteq B$ the maximal open subscheme such that for every nonnegative integer $\ell\leq k$, the scheme $D_{\leq \ell}(\varphi_B)\setminus D_{\leq (\ell-1)}(\varphi_B)$ is smooth over $B^o$ and every (nonempty) connected component of every geometric fiber has dimension $n-(e-\ell)(f-\ell)$. In other words, $B^o$ is the maximal open subscheme of $B$ over which each $D_{\leq \ell}(\varphi_B)$ is transversal. Denote by $Z\subset B$ the closed complement of $B^o$. Denote by $E\subset Z$ the union of all irreducible components of $Z$ that are divisors in $B$.

Now let $X$ be a smooth, projective complex variety. Let $E$ and $F$ be locally free $\mathcal{O}_X$-modules of ranks $e$ and $f$ such that the locally free $\mathcal{O}_X$-module $\textit{Hom}_{\mathcal{O}_X}(E,F)$ is globally generated. Denote by $B$ the complex affine space of the (finite-dimensional) complex vector space $\text{Hom}_{\mathcal{O}_X}(E,F)$, i.e., there is a universal morphism of coherent sheaves $\varphi_B:\text{pr}_2^*E\to \text{pr}_2^*F$ on the product variety $B\times X$. Denote the product $B\times X$ by $Y$ with its first projection $\pi$ from $Y$ to $B$. Denote $\text{pr}_2^*E$ and $\text{pr}_2^*F$ by $E_B$ and $F_B$. Denote the universal $\mathcal{O}_Y$-module homomorphism from $E_B$ to $F_B$ by $\varphi_B$.

Proposition. (1) The open subscheme $B^o$ of $B$ is dense, i.e., it is not empty.
(2) Assume that $n-(e-k)(f-k)$ is positive and that $D_{\leq k}(\varphi_B)$ surjects to $B$. Then every (nonempty) connected component of the fiber of $D_{\leq k}(\varphi_B)$ over each generic point of $E$ has pure dimension $n-(e-k)(f-k)$, but it may be nonreduced.
(3) Assume also that the greatest common divisor equals $1$ for the multiplicities of the irreducible components of each connected component of the fiber of $D_{\leq k}(\varphi_B)$ for each generic point of $E$. Then every fiber of $D_{\leq k}$ over every point of $B$ is connected.

Proof. The statements (1) and (2) follow from the method of incidence correspondences. Form the geometric vector bundle $\rho:X\to X$ associated to the locally free sheaf $\textit{Hom}_{\mathcal{O}_X}(E,F)$ with its universal morphism $\phi:\rho^*E\to \rho^*F$. By the usual theory of generic determinantal varieties, each locally closed subschemes $D_{\leq \ell}(\phi)\setminus D_{\leq (\ell-1)}(\phi)$ of $H$ is smooth over $X$ and every nonempty fiber has pure dimension $ef - (e-\ell)(f-\ell)$. Moreover, this fiber is irreducible if the dimension $ef-(e-\ell)(f-\ell)$ is positive. Since $\textit{Hom}_{\mathcal{O}_X}(E,F)$ is globally generated, the induced morphism from $B\times X$ to $H$ is smooth and geometrically surjective with irreducible geometric fibers. Thus, the inverse image of each locally closed subscheme $D_{\leq \ell}(\varphi)\setminus D_{\leq (\ell-1)}(\varphi)$ is smooth and every irreducible component has pure codimension $(e-\ell)(f-\ell)$ in $B\times X$. Combined with generic smoothness we deduce (1).

If $B$-flatness of $D_{\leq k}(\varphi)$ fails over a divisor $E_i$ in $B$, then the dimension of the inverse image of $E_i$ equals the full dimension of $D_{\leq k}(\varphi)$. Thus, the irreducible components of the inverse image of $E_i$ give irreducible components of $D_{\leq k}(\varphi)$. Since $D_{\leq k}(\varphi)$ is irreducible of pure codimension $(e-k)(f-k)$, there is only one such irreducible component. By hypothesis, this component surjects onto $B$, contradicting non-flatness. This implies (2).

Finally, (3) follows from Minoccheri's connectedness theorem. Since $B$ is simply connected, the finite part of the Stein factorization of $D_{\leq k}(\varphi_B)$ over $B$ either maps isomorphically to $B$, i.e., all fibers of $D_{\leq k}(\varphi_B)$ are connected, or it is branched over a nonempty, irreducible divisor $E_i$ in $B$. The inverse image of $E_i$ in the Stein factorization has an irreducible component of multiplicity $m>1$. This then gives connected components of the fibers of $D_{\leq k}(\varphi_B)$ over $E_i$ such that every irreducible component has multiplicity divisible by $m$, contrary to hypothesis. QED

Note, this does explain the examples in the comments. In each example, it is straightforward to check that the degeneracy loci do become every nonreduced over a divisor in $B$.