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Hi,I have a reference request regarding associated points of sheaves. I'll be more specific, assume that we are given the following exact sequence on $\mathbb{P}^d_{\mathbb{C}}$: $0 \to \mathcal{O}(-1)^n \to \mathcal{O}^m \to \mathcal{C} \to 0$. I'm studying the degeneracy locus of $\mathcal{C}$. The case of interest to me is not the generic case, i.e. when the degeneracy locus does not have the expected codimension. I was wondering whether there are some known results regarding embedded points, specifically under what conditions do they appear.

Thank you!

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  • $\begingroup$ So you first choose a map $\mathcal O(-1)^n \to \mathcal O^m$, which I guess is a $n \times m$ matrix of sections of $\mathcal O(1)$, and look at the degeneracy locus. Is this just the vanishing set of the determinant ideal, i.e. the intersection of $\left(\begin{array}{c} m \\ n \end{array}\right)$ sections of $\mathcal O(m)$? You are looking for embedded points of this determinant ideal? $\endgroup$
    – Will Sawin
    Commented Mar 25, 2013 at 23:09
  • $\begingroup$ Precisely. That is what I am looking for. $\endgroup$
    – shamovic
    Commented Mar 26, 2013 at 6:06

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