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klerk
klerk
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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 $V$ inof $X\times H^0(X, E^*\otimes F)$ parametrizing the degeneracy loci $D_k(\phi)=\{x \in X \; |\; \mathrm{rk}(\phi_x)\leq k \}$$D_k(\phi)=\{x \in X \; |\; \mathrm{rk}(\phi_x)\leq k \}$, fori.e. $\phi \in H^0(X, E^*\otimes F)$$V=\{(x,\phi)\;|\; x\in D_k(\phi)\}$. Is $V$ Cohen-Macaulay?

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 $V$ in $X\times H^0(X, E^*\otimes F)$ parametrizing the degeneracy loci $D_k(\phi)=\{x \in X \; |\; \mathrm{rk}(\phi_x)\leq k \}$, for $\phi \in H^0(X, E^*\otimes F)$. Is $V$ Cohen-Macaulay?

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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klerk
klerk
  • 115
  • 8

Is the subscheme parametrizing the k-th degeneracy loci Cohen-Macaulay?

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 $V$ in $X\times H^0(X, E^*\otimes F)$ parametrizing the degeneracy loci $D_k(\phi)=\{x \in X \; |\; \mathrm{rk}(\phi_x)\leq k \}$, for $\phi \in H^0(X, E^*\otimes F)$. Is $V$ Cohen-Macaulay?