Let $(C, mathcal{O}_C)$ be a reduced irreducible curve and $(tilde{C},mathcal{O}_{tilde{C}})$ its normalisation with $v : tilde{C} rightarrow C$. Then we have an imoprtant skyscraper sheaf $v_* tilde{C} / C$.

According to **II.11** in *Compact Complex Surfaces* by Barth&Peters&Van de Ven, the arithmetic gena satisfies $g(C) = g(tilde{C}) + delta(C)$ where $delta(C) = sum_{xin C} mathrm{dim}_{mathbb{C}}(v_* mathcal{O}_{tilde{C}} / mathcal{O}_C)$. From this we can see that the sheaf $v_* tilde{C} / C$ plays a role in describing the geometry of normalisation.

My question is, what can you say about this sheaf. For example, can you tell the way to compute the dimension? (In fact, this is a skyscrape so dimension is almost the all information.)