# Normalisations of singular plane algebraic curves

In what follows assume that the base field is $\mathbb{C}$

Background: Let $C$ be an irreducible plane algebraic curve, $S$ the set of singular points. There exists a Riemann surface $\tilde{C}$ and a holomorphic mapping $\sigma : \tilde{C} \to C$ such that $\sigma^{-1}(S)$ is finite and $\sigma : \tilde{C} \backslash \sigma^{-1}(S) \to C \backslash S$ is a biholomorphism.

We say $(\tilde{C}, \sigma)$ is the normalisation of $C$.

Question: I know the normalisations of some smooth plane algebraic curves (e.g the normalisation of a non singular elliptic curve is a torus)

Can anyone give me some examples of the normalisations of singular plane algebraic curves? What I am ideally looking for is an example of a singular plane algebraic curve and the corresponding Riemann surface and map.

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You get lots of examples by considering rational curves in the plane. –  Mariano Suárez-Alvarez Dec 23 '10 at 23:59
This question is not suitable for the site. You should try asking on math.stackexchange.com –  Andrea Ferretti Dec 24 '10 at 0:01
@Mariano: I fixed the title. –  Daniel Barter Dec 24 '10 at 0:10
@Daniel: take any smooth projective curve (which is automatically also a Riemann surface), say, in $\mathbb P^3$ and project it to $\mathbb P^2$. This map will be the normalization of the image curve (although it may be an isomorphism). –  Sándor Kovács Dec 24 '10 at 1:52
This answer to a previous question mathoverflow.net/questions/1504/… (in particular the last two paragraphs) may be relevant. (Although I was talking about it from the point of view of affine algebraic curves, not Riemann surfaces.) –  Alison Miller Dec 24 '10 at 8:21
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Here are some easy examples. The curve $xy=0$ is two lines glued at a point, its normalization is the disjoint union of two spheres. The cuspidal cubic $y^2 = x^3$ has a sphere as its normalization. The nodal cubic $y^2 = x^3 + x^2$ also has a sphere as its normalization.

In general, for a curve of degree d in the plane, its arithmetic genus can be calculated by the formula $(d-1)(d-2)/2$. This is larger than the genus of the normalization by the delta invariant of the singularity, which is $\dim \mathcal{O}_{\tilde{C}} / \mathcal{O}_C$. This torsion sheaf is supported at the singularities, and may be calculated locally if desired.

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