Sheaf cohomology version of cup product.

Question

There is a sheaf cohomology version of the cup product? I just want to understand the following product. Let $(C,p_1,\ldots,p_n)$ be a smooth projective curve with marked points, $T_C$ its tangent sheaf and $K_C$ the cotangent. Then for a fist order deformation $\phi\in H^1(C,T_C(-p_1-\ldots-p_n))$ and an element $s\in H^0(C,K_C(p_1+\ldots+p_n))$ the cup product $\phi\cdot s$ should land in $H^1(C,{\mathcal{O}}_C)$. Any comment would be appreciate...