# Computing degree of hom sheaf of coherent sheaves.

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.

## 1 Answer

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.

• 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? – Alan Muniz Feb 24 at 16:00
• How bad is your curve and how do you define the normal sheaf? – Sasha Feb 24 at 16:51
• 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. – Alan Muniz Feb 24 at 16:55
• So, how do you define the normal sheaf and its determinant? – Sasha Feb 24 at 17:04
• 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 – Alan Muniz Feb 24 at 17:19