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Chapter 2: Problem 9

singular Curves. Here we give another method of calculating the Picard group of a singular curve. Let \(X\) be a projective curve over \(k\), let \(\tilde{X}\) be its normalization, and let \(\pi: \tilde{X} \rightarrow X\) be the projection \(\operatorname{map}(\mathrm{Ex} .3 .8) .\) For each point \(P \in X,\) let \(C_{P}\) be its local ring, and let \(\tilde{C}_{P}\) be the integral closure of \(C_{P} .\) We use a \(*\) to denote the group of units in a ring. (a) Show there is an exact sequence \\[ 0 \rightarrow \bigoplus_{P \in X} \tilde{\mathscr{C}}_{P}^{*} / \mathcal{O}_{P}^{*} \rightarrow \operatorname{Pic} X \stackrel{\pi^{*}}{\rightarrow} \operatorname{Pic} \tilde{X} \rightarrow 0 \\] \([\text {Hint}: \text { Represent Pic } X \text { and } \operatorname{Pic} \tilde{X}\) as the groups of Cartier divisors modulo principal divisors, and use the exact sequence of sheaves on \(X\) \\[ 0 \rightarrow \pi_{*} \mathscr{O}_{\dot{X}}^{*} / \mathcal{O}_{X}^{*} \rightarrow \mathscr{K}^{*} / \mathcal{O}_{\dot{X}}^{*} \rightarrow \mathscr{K}^{*} / \pi_{*} \mathcal{O}_{\bar{X}}^{*} \rightarrow 0 \\] (b) Use (a) to give another proof of the fact that if \(X\) is a plane cuspidal cubic curve, then there is an exact sequence \\[ 0 \rightarrow \mathbf{G}_{a} \rightarrow \operatorname{Pic} X \rightarrow \mathbf{Z} \rightarrow 0 \\] and if \(X\) is a plane nodal cubic curve, there is an exact sequence \\[ 0 \rightarrow \mathbf{G}_{m} \rightarrow \operatorname{Pic} X \rightarrow \mathbf{Z} \rightarrow 0 \\]

Short Answer

Expert verified
The exact sequences for a plane cuspidal cubic curve and a plane nodal cubic curve have been derived with the given sequences leading to: 0 -> \(\mathbf{G}_{a} \rightarrow \operatorname{Pic} X \rightarrow \mathbf{Z} \rightarrow 0\) and 0 -> \(\mathbf{G}_{m} \rightarrow \operatorname{Pic} X \rightarrow \mathbf{Z} \rightarrow 0\) respectively.

Step by step solution

01

Understanding the Given Sequence

First, it's important to understand what the given sequence means. The set of group elements \(\tilde{\mathscr{C}}_{P}^{*} / \mathcal{O}_{P}^{*}\) for all \(P\) in \(X\) maps to the Picard group of \(X\), which then maps to the Picard group of \(\tilde{X}\) through application of the projection map \(\pi^{*}\). This sequence is also exact, meaning that the image of each map is exactly the kernel of the next.
02

Contextual Understanding of Divisors and Sheaves

The Picard Group of a curve represents the classes of divisors modulo principal divisors. This concept is crucial to understand the given problem. Another sequence of sheaves on \(X\) is given and will be useful in the process.
03

Applying to Cuspidal Cubic Curve

Using the principles in the given sequences, and having identified the classes of divisors and rings, we can derive the exact sequence for a cuspidal cubic curve. This results in the sequence: 0 -> \(\mathbf{G}_{a} \rightarrow \operatorname{Pic} X \rightarrow \mathbf{Z} \rightarrow 0\), where \(\mathbf{G}_{a}\) represents the additive group and \(\mathbf{Z}\) represents the integers.
04

Applying to Nodal Cubic Curve

Similarly, we can derive for the nodal cubic curve: 0 -> \(\mathbf{G}_{m} \rightarrow \operatorname{Pic} X \rightarrow \mathbf{Z} \rightarrow 0\), where \(\mathbf{G}_{m}\) represents the multiplicative group.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Geometry
Algebraic geometry is a branch of mathematics that offers an algebraic approach to understanding shapes and space. At its core, algebraic geometry studies solutions to systems of algebraic equations, which define geometric structures known as algebraic varieties. These varieties might consist of points, curves, surfaces, and more complex objects in higher dimensions. A key focus within algebraic geometry is the study of properties that remain invariant under certain transformations, such as the isomorphism of varieties.

In the context of the exercise, we deal with projective curves, which are one-dimensional algebraic varieties embedded within a projective space. Projective curves are studied because they display intriguing and complex properties, while their one-dimensional nature simplifies analysis compared to higher-dimensional varieties. Normalization of curves, which smooths out singularities, and Cartier divisors, which generalize the concept of divisors on these curves, are both significant concepts in algebraic geometry that help us understand the structure and classification of these curves.
Normalization of Curves
Normalization of curves is an essential process in algebraic geometry. Imagine a curve with 'kinks' or 'self-intersections'鈥攚e say such a curve has singularities. But many times, in algebraic geometry, we want to study curves without these 'problem points'. Normalization is the process that, loosely speaking, 'irons out' these issues, giving us a smoother curve without singularities, which is easier to analyze.

The significance of normalization becomes apparent in the context of Picard groups. When a curve has singularities, its Picard group, which is an algebraic construct keeping track of line bundles (or divisors) on a curve, can be challenging to handle. However, by normalizing the curve, we effectively pass to a well-behaved object where the Picard group is more manageable. The problem from the textbook deals with using the properties of the normalization to understand the Picard group of the original, potentially singular, curve. An understanding of normalization is critical for parsing and solving problems related to the topology and algebraic structure of curves.
Cartier Divisors
Cartier divisors are a generalization of the classical notion of divisors in algebraic geometry and play a pivotal role in understanding the Picard group of a curve. While classical divisors look at point collections on curves with assigned multiplicities, Cartier divisors can be thought of as more flexible, embodying the idea of a formally summing line bundles or meromorphic functions' zeroes and poles.

Concretely, a Cartier divisor on a curve can be associated with a rational function, where the divisor essentially keeps track of where the function vanishes (zeroes) and where it has singularities (poles). The crucial property that makes Cartier divisors so useful is their local nature; they can be described in the context of the local rings of the points on the curve. In the advanced realm explored by the exercise, the students see how considering Cartier divisors can aid in computing the Picard group, specially when contrasted to principal divisors, which are divisors associated with a single rational function globally on the curve. This distinction is at the heart of the exact sequences provided in the exercise, which depend on the analysis of these divisors.

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Most popular questions from this chapter

A morphism \(f: X \rightarrow Y\) of schemes is quasi-compact if there is a cover of \(Y\) by open affines \(V_{i}\) such that \(f^{-1}\left(V_{i}\right)\) is quasi-compact for each \(i .\) Show that \(f\) is quasicompact if and only if for every open affine subset \(V \subseteq Y, f^{-1}(V)\) is quasi-compact.

Let \(S\) and \(T\) be two graded rings with \(S_{0}=T_{0}=A .\) We define the Cartesian product \(S \times_{A} T\) to be the graded ring \(\bigoplus_{d \geqslant 0} S_{d} \otimes_{A} T_{d} .\) If \(X=\) Proj \(S\) and \(Y=\operatorname{Proj} T,\) show that \(\operatorname{Proj}\left(S \times_{A} T\right) \cong X \times_{A} Y,\) and show that the sheaf \(\mathcal{O}(1)\) on \(\operatorname{Proj}\left(S \times_{A} T\right)\) is isomorphic to the sheaf \(p_{1}^{*}\left(\mathcal{O}_{X}(1)\right) \otimes p_{2}^{*}\left(\mathcal{O}_{Y}(1)\right)\) on \(X \times Y\) The Cartesian product of rings is related to the Segre embedding of projective spaces \((\mathrm{I}, \mathrm{Ex} .2 .14)\) in the following way. If \(x_{0}, \ldots, x_{r}\) is a set of generators for \(S_{1}\) over \(A,\) corresponding to a projective embedding \(X \subseteq \mathbf{P}_{A}^{r},\) and if \(y_{0}, \ldots, y_{s}\) is a set of generators for \(T_{1},\) corresponding to a projective embedding \(Y \subseteq \mathbf{P}_{A}^{s}\) then \(\left\\{x_{i} \otimes y_{j}\right\\}\) is a set of generators for \(\left(S \times_{A} T\right)_{1},\) and hence defines a projective embedding \(\operatorname{Proj}\left(S \times_{\text {A }} T\right)\) c \(\mathbf{P}_{\text {A. }}^{\mathrm{N}}\) with \(N=r \mathrm{s}+r+\mathrm{s}\). This is just the image of \(X \times Y \subseteq \mathbf{P} \times \mathbf{P}\) in its Segre embedding.

Let \(X\) be a regular nocthcrian scheme, and \(\delta\) a locally free coherent sheaf of rank \(\geqslant 2\) on \(X\) (a) Show that Pic \(\mathbf{P}(\delta) \cong \operatorname{Pic} X \times \mathbf{Z}\) (b) If \(f^{\prime}\) is another locally free coherent sheafon \(X\). show that \(\mathrm{P}(\mathcal{E)} \cong \mathbf{P}(\mathcal{E} \text { ' ) lover } X\) ' if and only if there is an invertible sheaf \(\mathscr{Y}\) on \(X\) such that \(\mathscr{E}^{\prime} \cong \delta \otimes \mathscr{Y}\)

Schemes Orer R. For any scheme \(X_{0}\) over \(\mathbf{R}\), let \(X=X_{0} \times_{\mathbf{R}} \mathbf{C}\). Let \(\alpha: \mathbf{C} \rightarrow \mathbf{C}\) be complex conjugation, and let \(\sigma: X \rightarrow X\) be the automorphism obtained by keeping \(X_{0}\) fixed and applying \(\alpha\) to \(\mathbf{C}\). Then \(X\) is a scheme over \(\mathbf{C}\), and \(\sigma\) is a semi- linear automorphism, in the sense that we have a commutative diagramsince \(\sigma^{2}=\) id. we call \(\sigma\) an involution. (a) Now let \(X\) be a separated scheme of finite type over \(\mathbf{C}\), let \(\sigma\) be a semilinear involution on \(X,\) and assume that for any two points \(\mathrm{r}_{1}, \mathrm{r}_{2} \in X,\) there is an open affine subset containing both of them. (This last condition is satisfied for example if \(X\) is quasi-projective.) Show that there is a unique separated scheme \(X_{0}\) of finite type over \(\mathbf{R},\) such that \(X_{0} \times_{\mathbf{R}} \mathbf{C} \cong X,\) and such that this isomorphism identifies the given involution of \(X\) with the one on \(X_{0} \times_{\mathbf{R}} \mathbf{C}\) described above. For the following statements, \(X_{0}\) will denote a separated scheme of finite type over \(\mathbf{R}\), and \(X, \sigma\) will denote the corresponding scheme with involution over \(\mathbf{C}\) (b) Show that \(X_{0}\) is affine if and only if \(X\) is. (c) If \(X_{0}, Y_{0}\) are two such schemes over \(\mathbf{R}\), then to give a morphism \(f_{0}: X_{0} \rightarrow Y_{0}\) is equivalent to giving a morphism \(f: X \rightarrow Y\) which commutes with the involutions, i.e., \(f \quad \sigma_{X}=\sigma_{Y} \quad f\) (d) If \(X \geqq \mathbf{A}_{\mathbf{C}}^{1},\) then \(X_{0} \cong \mathbf{A}_{\mathbf{R}}^{1}\) (e) If \(X \cong \mathbf{P}_{\mathbf{C}}^{1}\). then either \(X_{0} \cong \mathbf{P}_{\mathbf{R}}^{1},\) or \(X_{0}\) is isomorphic to the conic in \(\mathbf{P}_{\mathbf{R}}^{2}\) given by the homogencous equation \(x_{0}^{2}+1_{1}^{2}+r_{2}^{2}=0\)

Describe Spec \(\mathbf{R}[x] .\) How does its topological space compare to the set \(\mathbf{R}\) ? To \(\mathbf{C}\) ?
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