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Let $X$ be a smooth complex projective variety of dimension $n \ge 4$ and let $F$ and $E$ be two (holomorphic) vector bundles of rank $f$ and $e$ over $X$. Given a morphism $\varphi: F \to E$ of vector bundles, we denote $D_k(\varphi)= \{ x \in X \mid rank (\varphi(x)) \le k\}$ the $k$-th degeneracy locus of $\varphi$.

Suppose we have $dim(D_2(\varphi))=2$ and $D_1(\varphi)=\emptyset$. Is it known whether $D_2(\varphi)$ is smooth (I am actually interested in the case where $n=4, f=4, e=3$)?

Thanks in advance!

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