**Motivation:**

For a given rank 2 vector bundle we want to know how many theta-characteristic valued twisted endomorphisms it has.

**Setting:**

Let $C$ be a smooth algebraic curve over a field of characteristic $\neq 2$ and let $\vartheta$ be a theta characteristic on $C$ ($\vartheta^2 \cong K_C$), which does not have any nontrivial global sections. Furthermore we are given line bundles $L_1,L_2$ on $C$ and we consider an extension $[E] \in \operatorname{Ext}^1(L_2,L_1)$. We take this exact sequence, dualize it and tensor it with $\vartheta \otimes L_2$. We find the exact sequence:

$$0 \to \vartheta \to \vartheta \otimes E^\vee \to \vartheta \otimes L_1^{-1}\otimes L_2 \to 0$$

Looking at the long exact sequence in cohomology and using $H^0(C,\vartheta) = H^1(C,\vartheta) = 0$ we find: $H^0(C,\vartheta \otimes L_1^{-1}\otimes L_2) \cong H^0(C,\vartheta \otimes E^\vee \otimes L_2)$ **(1)**. If we tensor our original short exact sequence by $\vartheta\otimes L_1^{-1}$ we get

$$0 \to \vartheta \to \vartheta \otimes L_1^{-1}\otimes E \to \vartheta \otimes L_1^{-1} \otimes L_2 \to 0.$$

From the long exact sequence we derive $H^0(C,\vartheta\otimes L_1^{-1}\otimes E) \cong H^0(C,\vartheta\otimes L_1^{-1}\otimes L_2)$ **(2)**. Now we tensor the original short exact sequence by $\vartheta \otimes E^\vee$ and then look at the long exact sequence:

$$0 \to H^0(\vartheta\otimes E^\vee \otimes L_1) \to H^0(\vartheta \otimes E^\vee \otimes E) \to H^0(\vartheta \otimes E^\vee\otimes L_2)\xrightarrow{\delta}$$ $$ H^1(\vartheta\otimes E^\vee \otimes L_1) \to H^1(\vartheta \otimes E^\vee \otimes E) \to H^1(\vartheta \otimes E^\vee\otimes L_2) \to 0$$

Now to learn something about $H^0(\vartheta \otimes E^\vee \otimes E)$ one has to understand $\delta$. If we apply (1), (2) and Serre duality we get the following new map

$$\Delta: H^0(\vartheta \otimes L_1^{-1}\otimes L_2) \cong H^0(\vartheta \otimes E^\vee \otimes L_2) \xrightarrow{\delta} H^1(\vartheta \otimes E^\vee \otimes L_1) \cong H^0(\vartheta \otimes E \otimes L_1^\vee)^\vee \cong H^0(\vartheta \otimes L_1^{-1} \otimes L_2)^\vee$$.

**The Questions:**

Can one understand (or describe in a different way) this map $\Delta$ in terms of the extension. In other words what is the map $\operatorname{Ext}^1(L_2,L_1) \to \left( H^0(L_1^{-1}\otimes L_2)^\vee \right)^{\otimes 2}$? What is the rank of $\Delta$? Furthermore i suspect that $\Delta$ viewed as a bilinear form is antisymmetric, because the parity of $h^0(\operatorname{End}(E)\otimes \vartheta)$ is constant in families by a result of Mumford.