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Let $C$ be complex projective curve (Cohen-Macaulay at least). Let $\mathcal{F}$ and $\mathcal{G}$ be coherent sheaves on $C$.

Is there any way to express the degree of $\underline{Hom}_{\mathcal{O}_C}(\mathcal{F},\mathcal{G})$ in terms of the degrees of $\mathcal{F}$ and $\mathcal{G}$?

For locally free sheaves this is obvious but I couldn't prove a generalization to the coherent case.

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No, the degree may depend not only on the degrees (and ranks) of $F$ and $G$. For instance, let $C$ be a smooth curve, $F = O_x$ and $G = O_y$. Then $$ \mathcal{H}om(F,G) = \begin{cases} O_x, & \text{if $x = y$},\\ 0, & \text{otherwise} \end{cases} $$ and its degree is 1 in the first case and 0 in the other, while the degree of $F$ and $G$ is equal to 1 (and rank equals 0) in both cases.

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  • $\begingroup$ Thanks for the answer. Do you know what happens in the specific case where $C\subset \mathbb{P}^n$, $F=\det \mathcal{N}$ the determinant of the normal sheaf and $G = \omega_{\mathbb{P}^n}|_C$ is the restriction of the canonical sheaf? $\endgroup$ – Alan Muniz Feb 24 at 16:00
  • $\begingroup$ How bad is your curve and how do you define the normal sheaf? $\endgroup$ – Sasha Feb 24 at 16:51
  • $\begingroup$ It can be Gorenstein. I know that in local complete intersection the normal sheaf is well behaved, however I'm a bit lost in a more general setting. $\endgroup$ – Alan Muniz Feb 24 at 16:55
  • $\begingroup$ So, how do you define the normal sheaf and its determinant? $\endgroup$ – Sasha Feb 24 at 17:04
  • $\begingroup$ The normal sheaf is defined $(I/I^2)^\ast$ where $I$ is the sheaf of ideals. And I just realized that I do not know how to define the determinant as I do not know is a locally free resolution of the normal sheaf exists $\endgroup$ – Alan Muniz Feb 24 at 17:19

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