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.
Best Answer
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.