/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 9 singular Curves. Here we give an... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Extension of Coherent Sheaves. We will prove the following theorem in several steps: Let \(X\) be a noetherian scheme, let \(U\) be an open subset, and let \(\mathscr{F}\) be a coherent sheaf on \(U\). Then there is a coherent sheaf \(\mathscr{F}^{\prime}\) on \(X\) such that \(\left.\mathscr{F}^{\prime}\right|_{v} \cong \mathscr{F}\) (a) On a noetherian affine scheme, every quasi-coherent sheaf is the union of its coherent subsheaves. We say a sheaf \(\mathscr{F}\) is the union of its subsheaves \(\mathscr{F}\) if for every open set \(U\), the group \(\mathscr{F}(U)\) is the union of the subgroups ?\((U)\) (b) Let \(X\) be an affine noetherian scheme, \(U\) an open subset, and \(\mathscr{F}\) coherent on \(U .\) Then there exists a coherent sheaf \(\mathscr{F}^{\prime}\) on \(X\) with \(\left.\mathscr{F}^{\prime}\right|_{v} \cong \mathscr{F} .\) [Hint: Let \(\left.i: U \rightarrow X \text { be the inclusion map. Show that } i_{*} \mathscr{F} \text { is quasi-coherent, then use }(a) .\right]\) (c) With \(X, U, \mathscr{F}\) as in (b), suppose furthermore we are given a quasi-coherent sheaf \(\mathscr{G}\) on \(X\) such that \(\left.\mathscr{F} \subseteq \mathscr{G}\right|_{v} .\) Show that we can find \(\mathscr{F}^{\prime}\) a coherent subsheaf of \(\mathscr{G},\) with \(\left.\mathscr{F}^{\prime}\right|_{v} \cong \mathscr{F}\). [Hint: Use the same method, but replace \(i_{*} \mathscr{F}\) by \(\left.\rho^{-1}\left(i_{*} \mathscr{F}\right) \text { , where } \rho \text { is the natural } \operatorname{map} \mathscr{G} \rightarrow i_{*}\left(\left.\mathscr{G}\right|_{U}\right) .\right]\) (d) Now let \(X\) be any noetherian scheme, \(U\) an open subset, \(\mathscr{F}\) a coherent sheaf on \(U,\) and \(\mathscr{G}\) a quasi-coherent sheaf on \(X\) such that \(\left.\mathscr{F} \subseteq \mathscr{G}\right|_{V} .\) Show that there is a coherent subsheaf \(\mathscr{F}^{\prime} \subseteq \mathscr{G}\) on \(X\) with \(\left.\mathscr{F}^{\prime}\right|_{v} \cong \mathscr{F}\). Taking \(\mathscr{I}=i_{*} \mathscr{F}\) proves the result announced at the beginning. [Hint: Cover \(X\) with open affines, and extend over one of them at a time. (e) As an extra corollary, show that on a noetherian scheme, any quasi- coherent sheaf \(\mathscr{F}\) is the union of its coherent subsheaves. [Hint: If \(s\) is a section of \(\mathscr{F}\) over an open set \(U,\) apply (d) to the subsheaf of \(\left.\mathscr{F}\right|_{v}\) generated by s.]

Let \(X\) be a noetherian scheme, and let \(\mathscr{F}\) be a coherent sheaf. (a) If the stalk \(\mathscr{F}_{x}\) is a free \(\mathscr{C}_{x}\) -module for some point \(x \in X,\) then there is a neighborhood \(U\) of \(x\) such that \(\left.\mathscr{F}\right|_{v}\) is free. (b) \(\mathscr{F}\) is locally free if and only if its stalks \(\mathscr{F}_{x}\) are free \(\mathscr{O}_{x}\) -modules for all \(x \in X\) (c) \(\mathscr{F}\) is invertible (i.e., locally free of rank 1 ) if and only if there is a coherent sheaf \(\mathscr{G}\) such that \(\mathscr{F} \otimes \mathscr{G} \cong \mathscr{O}_{X} .\) (This justifies the terminology invertible: it means that \(\mathscr{F}\) is an invertible element of the monoid of coherent sheaves under the operation \(\otimes .\)

Extending a Sheaf by Zero. Let \(X\) be a topological space, let \(Z\) be a closed subset. let \(i: Z \rightarrow X\) be the inclusion, let \(U=X-Z\) be the complementary open subset and let \(j: U \rightarrow X\) be its inclusion. (a) Let \(\mathscr{J}\) be a sheaf on \(Z\). Show that the stalk \(\left(i_{*}, \overline{\mathscr{H}}\right)_{p}\) of the direct image sheaf on \(X\) is \(\mathscr{F}_{P}\) if \(P \in Z, 0\) if \(P \notin Z\). Hence we call \(i_{*}\). \(\bar{y}\) the sheaf obtained by extending of \(i_{*} \overline{\mathscr{H}},\) and say "consider \(\mathscr{F}\) as a sheaf on \(X\)," when we mean "consider \(i_{*}\). (b) Now let \(\overline{\mathscr{H}}\) be a sheaf on \(U\). Let \(j\), \((\overrightarrow{\mathscr{H}})\) be the sheaf on \(X\) associated to the presheaf \(V \mapsto \mathscr{F}(V)\) if \(V \subseteq U, V \mapsto 0\) otherwise. Show that the stalk \((j,(\mathscr{F}))_{P}\) is equal to \(\overline{\mathscr{I}}_{p}\) if \(P \in U, 0\) if \(P \notin U\), and show that \(j\), \(\overline{\mathscr{H}}\) is the only sheafon \(X\) which has this property, and whose restriction to \(U\) is \(\mathscr{F}\). We call \(j\). F. Fhe sheaf obtained by extending \(\mathscr{F}\) by zero outside \(U\) (c) Now let \(\mathscr{F}\) be a sheaf on \(X\). Show that there is an exact sequence of sheaves on \(X\) $$0 \rightarrow j \cdot\left(\left.\overline{\mathscr{H}}\right|_{c}\right) \rightarrow \overline{\mathscr{H}} \rightarrow i_{*}\left(\left.\mathscr{F}\right|_{Z}\right) \rightarrow 0$$

A topological space is quasi-compact if every open cover has a finite subcover. (a) Show that a topological space is noetherian (I, \(\$ 1)\) if and only if every open subset is quasi-compact. (b) If \(X\) is an affine scheme. show that \(\operatorname{sp}(X)\) is quasi- compact. but not in general noetherian. We say a scheme \(X\) is quati-ciompact if \(\operatorname{sp}(X)\) is. (c) If \(A\) is a noetherian ring. show that spiSpec 1 ) is a nocthcrian topological space. (d) Give an example to show that sp(Spec \(A\) ) can be noetherian even when \(A\) is not.

Examples of Valuation Rings. Let \(k\) be an algebraically closed field. (a) If \(K\) is a function field of dimension 1 over \(k(I, \$ 6),\) then every valuation ring of \(K / k\) (except for \(K\) itself) is discrete. Thus the set of all of them is just the abstract nonsingular curve \(C_{K}\) of \((\mathrm{I}, \$ 6)\) (b) If \(K / k\) is a function field of dimension two, there are several different kinds of valuations. Suppose that \(X\) is a complete nonsingular surface with function field \(K\) (1) If \(Y\) is an irreducible curve on \(X\), with generic point \(x_{1},\) then the local ring \(R=C_{x_{1}, x}\) is a discrete valuation ring of \(K k\) with center at the (nonclosed) point \(x_{1}\) on \(X\) (2) If \(f: X^{\prime} \rightarrow X\) is a birational morphism, and if \(Y^{\prime}\) is an irreducible curve in \(X^{\prime}\) whose image in \(X\) is a single closed point \(x_{0},\) then the local ring \(R\) of the generic point of \(Y^{\prime}\) on \(X^{\prime}\) is a discrete valuation ring of \(K k\) with center at the closed point \(x_{0}\) on \(X\) (3) Let \(r_{0} \in X\) be a closed point. Let \(f: X_{1} \rightarrow X\) be the blowing-up of \(x_{0}\) (I. \(\$ 4)\) and let \(E_{1}=f^{-1}\left(r_{0}\right)\) be the exceptional curve. Choose a closed point \(x_{1} \in E_{1},\) let \(f_{2}: X_{2} \rightarrow X_{1}\) be the blowing-up of \(x_{1},\) and let \(E_{2}=\) \(f_{2}^{-1}\left(x_{1}\right)\) be the exceptional curve. Repeat. In this manner we obtain a sequence of varieties \(X\), with closed points \(x_{i}\) chosen on them, and for each \(i,\) the local ring \(C_{1,1,1}, x_{1},\) dominates \(C_{x_{1}, x_{1}},\) Let \(R_{0}=\bigcup_{1=0}^{x} C_{x_{1}, x_{1}}\) Then \(R_{0}\) is a local ring, so it is dominated by some valuation ring \(R\) of \(K / k\) by \((\mathrm{I}, 6.1 \mathrm{A}) .\) Show that \(R\) is a valuation ring of \(K / k\). and that it has center \(x_{0}\) on \(X .\) When is \(R\) a discrete valuation ring? Note. We will see later (V.Ex. 5.6) that in fact the \(R_{0}\) of (3) is already a valuation ring itself, so \(R_{0}=R\). Furthermore, every valuation ring of \(K, k\) (except for \(K\) itself) is one of the three kinds just described.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.