The problem in the following : given an algebraic curve $C$, it's well-known that a smooth projective model of $C$ can be construct as the set of discrete valuations $v$ on it's function field $\mathbb{C}(C)$. At a smooth point will correspond an unique valuation (namely the vanishing order), and at a singular point one will get (eventually) several ones (one for each branch intersecting?).

Consider the sheaf of differentials $\Omega_{\mathbb{C}[C]/\mathbb{C}}$ ; given a valuation $v$ corresponding to $x$, what would be the significance in terms of the singularity at $x$ of the torsion part of the $\mathbb{O}_v$-module $\Omega_{\mathbb{C}[C]/\mathbb{C}} \otimes \mathbb{O}_v$?

More precisely, the latter will be isomorphic to $\mathbb{O}_v \oplus_i \frac{\mathbb{O}_v}{m_v^{n_i}}$, where $m_v$ is the maximal ideal of $\mathbb{O}_v$. The question is : what mean the $n_i$'s?

I'm not a specialist, so any help, comment or correction will be appreciated. Thank you :)


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