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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 – 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… (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

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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Expanding on Mariano's answer, my favorite examples are the ones I learned in Walker's plane algebraic curves: if the degree of C is d and there are (1/2)(d-1)(d-2) singular points, then the normalization is isomorphic to the projective line. Moreover Walker gives a rule for finding the normalization map, if you know where the singular points are.

E.g. any irreducible plane quartic with three singular points has P^1 as normalization. Moreover the rational inverse of the normalization map is defined by the P^2 of conics passing through all 3 singular points.

Stronger: if the sum of the numbers (1/2)n(n-1) over all singular points of (variable) multiplicity n, equals (1/2)(d-1)(d-2), then the same conclusion holds. E.g. an irreducible plane quartic with one triple point also has normalization isomorphic to P^1.

It is an interesting comment on education in geometry that some of us literally learned derived functor sheaf cohomology, schemes, Kodaira vanishing, Hirzebruch (or Grothendieck) Riemann Roch, and perhaps general resolution of singularities, before encountering these basic phenomena in the plane.

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