Let $X$ be a Cohen-Macaulay projective variety (say over an algebraically closed field of characteristic 0), and $E,F$ two vector bundles such that $E^*\otimes F$ is globally generated. Consider the subscheme $V$ of $X\times H^0(X, E^*\otimes F)$ parametrizing the degeneracy loci $D_k(\phi)=\{x \in X \; |\; \mathrm{rk}(\phi_x)\leq k \}$, i.e. $V=\{(x,\phi)\;|\; x\in D_k(\phi)\}$. Is $V$ Cohen-Macaulay?
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$\begingroup$ Do you mean $V = \{ (x,\phi) \mid x \in D_k(\phi) \}$? $\endgroup$– SashaCommented Apr 6 at 8:33
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$\begingroup$ Yes, I mean that! $\endgroup$– klerkCommented Apr 6 at 8:37
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First, let $$ M_k \subset \mathrm{Mat}(e,f) $$ be the variety of matrices of size $e \times f$ and rank at most $k$. This variety is well-known to be Cohen--Macaulay.
Now consider the projection $\pi \colon V \to X$. Since the evaluation morphism $$ H^0(X,E^* \otimes F) \to (E^* \otimes F)_x \cong \mathrm{Mat}(e,f) $$ is surjective, the fiber of $\pi$ over a point $x \in X$ is a cone over $M_k$, hence Cohen--Macaulay. This shows that $\pi$ is a fibration over a Cohen--Macaulay base with Cohen--Macaulay fibers, hence Cohen--Macaulay.
