Geometry of an algebraic curve

Arithmetic of an algebraic curve (e.g., rational points) is outside the scope of this article, but see arithmetic surface. In fact, we will work over the complex numbers, which allows us to use the Hodge theory. Especially in the context of degeneration/deformation of curves, it is imperative to talk about "reducible" curves with singularities.

Genus of a stable curve

Proposition — Let X be a connected curve with δ nodes and ν irreducible components, each having genus g_i. Then X admits a dualizing sheaf and the arithmetic genus g of X is given by

g = \sum_{1}^{ν} g_{i} - ν + 1 - δ .

Proof: Let ${\tilde{X}}_{i}$ be the normalization of an irreducible component $X_{i}$ of X. Then, by the Riemann–Roch formula,

χ (𝒪_{X}) = \sum_{1}^{ν} χ (𝒪_{{\tilde{X}}_{i}}) + δ = \sum_{i = 1}^{ν} (1 - g_{i}) + δ .

Line bundles and dual graph

Let X be a possibly singular curve. Then

0 \to ℂ^{*} \to (ℂ^{*})^{r} \to Γ (X, ℱ) \to Pic (X) \to Pic (\tilde{X}) \to 0 .

where r is the number of irreducible components of X, $π : \tilde{X} \to X$ is the normalization and $ℱ = π_{*} 𝒪_{\tilde{X}} / 𝒪_{X}$ . (To get this use the fact $Pic (X) = H^{1} (X, 𝒪_{X}^{*})$ and $Pic (\tilde{X}) = H^{1} (\tilde{X}, 𝒪_{\tilde{X}}^{*}) = H^{1} (X, π_{*} 𝒪_{\tilde{X}}^{*}) .$ )

Taking the long exact sequence of the exponential sheaf sequence gives the degree map:

\deg : Pic (X) \to H^{2} (X; ℤ) ≃ ℤ^{r} .

By definition, the Jacobian variety J(X) of X is the identity component of the kernel of this map. Then the previous exact sequence gives:

0 \to ℂ^{*} \to (ℂ^{*})^{r} \to Γ (\tilde{X}, ℱ) \to J (X) \to J (\tilde{X}) \to 0 .

We next define the dual graph of X; a one-dimensional CW complex defined as follows. (related to whether a curve is of compact type or not)

Flat families of curves

By which we mean

π : 𝒳 \to S

such that

$π$ is flat (but need not be proper)
$π^{- 1} (t)$ is a smooth curve for all t ≠ 0.

By the degeneration or specialization as t → 0 we mean

Automorphism group

The automorphism group of a smooth curve of genus g has order at most 84(g − 1).

Vanishing sequence

Given a linear series V on a curve X, the image of it under ${ord}_{p}$ is a finite set and following the tradition we write it as

a_{0} (V, p) < a_{1} (V, p) < \dots < a_{r} (V, p) .

This sequence is called the vanishing sequence. For example, $a_{0} (V, p)$ is the multiplicity of a base point p. We think of higher $a_{i} (V, p)$ as encoding information about inflection of the Kodaira map $φ_{V}$ . The ramification sequence is then

b_{i} (V, p) = a_{i} (V, p) - i .

Their sum is called the ramification index of p. The global ramification is given by the following formula:

Plücker formula —

\sum_{p \in X} \sum_{0}^{r} b_{i} (V, p) = (r + 1) (d + r (g - 1)) .

Hurwitz scheme

By definition, Hurwitz scheme $ℋ_{d, g}$ is the scheme parametrizing pairs ( $C, π : C \to 𝐏^{1}$ ) where C is a smooth curve of genus g and π has degree d.

Severi variety

A variety in a Hilbert scheme that parametrizes curves in projective space with given degree and geometric genus and at most node singularities. Its dimension is 3d + g − 1.

It is a difficult^{[citation needed]} theorem that Severi variety is a variety; i.e., irreducible.

References

Sources

Joe Harris and Ian Morrison. Moduli of curves.
Kollár, János, "Chapter 1", Book on Moduli of Surfaces

External links

https://www2.bc.edu/maksym-fedorchuk/papers/thesis.pdf – possible source?
http://arxiv.org/pdf/1503.04465v2.pdf

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