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Chapter 2: Problem 13

A Complete Nonprojective Variety. Let \(k\) be an algebraically closed field of char \(\neq 2 .\) Let \(C \subseteq \mathbf{P}_{k}^{2}\) be the nodal cubic curve \(y^{2} z=x^{3}+x^{2} z .\) If \(P_{0}=(0,0,1)\) is the singular point, then \(C-P_{0}\) is isomorphic to the multiplicative group \(\mathbf{G}_{m}=\operatorname{Spec} k\left[t, t^{-1}\right](\mathrm{E} \mathrm{x} .6 .7) .\) For each \(a \in k, a \neq 0,\) consider the translation of \(\mathbf{G}_{m}\) given by \(t \mapsto a t .\) This induces an automorphism of \(C\) which we denote by \(\varphi_{a}\) Now consider \(C \times\left(\mathbf{P}^{1}-\\{0\\}\right)\) and \(C \times\left(\mathbf{P}^{1}-\\{\infty\\}\right) .\) We glue their open subsets \(C \times\left(\mathbf{P}^{1}-\\{0, x\\}\right)\) by the isomorphism \(\varphi:\langle P, u\rangle \mapsto\left\langle\varphi_{u}(P), u\right\rangle\) for \(P \in C, u \in \mathbf{G}_{m}=\mathbf{P}^{1}-\\{0, x\\} .\) Thus we obtain a scheme \(X,\) which is our example. The projections to the second factor are compatible with \(\varphi,\) so there is a natural morphism \(\pi: X \rightarrow \mathbf{P}^{1}\) (a) Show that \(\pi\) is a proper morphism, and hence that \(X\) is a complete variety over \(k\) (b) Use the method of \((\text { Ex. } 6.9)\) to show that \(\operatorname{Pic}\left(C \times \mathbf{A}^{1}\right) \cong \mathbf{G}_{m} \times \mathbf{Z}\) and \(\operatorname{Pic}\left(C \times\left(\mathbf{A}^{1}-\\{0\\}\right)\right) \cong \mathbf{G}_{m} \times \mathbf{Z} \times \mathbf{Z}\) [Hint: If \(A\) is a domain and if denotes the group of units, then \(\left.(A[u])^{*} \cong A^{*} \text { and }\left(A\left[u, u^{-1}\right]\right)^{*} \cong A^{*} \times \mathbf{Z} .\right]\) (c) Now show that the restriction map \(\operatorname{Pic}\left(C \times \mathbf{A}^{1}\right) \rightarrow \operatorname{Pic}\left(C \times\left(\mathbf{A}^{1}-\\{0\\}\right)\right)\) is of the form \(\langle t, n\rangle \mapsto\langle t, 0 . n\rangle,\) and that the automorphism \(\varphi\) of \(C \times\left(\mathbf{A}^{1}-\\{0\\}\right)\) induces a map of the form \(\langle t, d, n\rangle \mapsto\langle t, d+n, n\rangle\) on its Picard group. (d) Conclude that the image of the restriction map Pic \(X \rightarrow \operatorname{Pic}(C \times\\{0\\})\) consists entirely of divisors of degree 0 on \(C .\) Hence \(X\) is not projective over \(k\) and \(\pi\) is not a projective morphism.

Short Answer

Expert verified
The morphism \(\pi: X \rightarrow \mathbf{P}^{1}\) is proper, meaning \(X\) is a complete variety over \(k\). The Picard group for both \(C \times \mathbf{A}^{1}\) and \(C \times\left(\mathbf{A}^{1}-\{0\}\right)\) are computed. Moreover, it's shown that the restriction map \(\operatorname{Pic}\left(C \times \mathbf{A}^{1}\right) \rightarrow \operatorname{Pic}\left(C \times\left(\mathbf{A}^{1}-\{0\}\right)\right)\) has the form \(\langle t, n\rangle \mapsto\langle t, 0 . n\rangle\), and the automorphism \(\varphi\) of \(C \times\left(\mathbf{A}^{1}-\{0\}\right)\) induces a map of the form \(\langle t, d, n\rangle \mapsto\langle t, d+n, n\rangle\) on its Picard group. Finally, it's concluded that \(X\) is not projective over \(k\) and \(\pi\) is not a projective morphism.

Step by step solution

01

Show that \(\pi\) is a proper morphism

A morphism \(\pi: X \rightarrow Y\) in algebraic geometry is proper if it is universally closed and of finite type. We need to confirm these properties for the given morphism. Since the map \(\pi\) is the projection of the product of \(C\) with \(\mathbf{P}^{1}\), it is closed. Moreover, as \(C \subseteq \mathbf{P}_{k}^{2}\) is a projective variety, it is also complete, and by the projection formula in algebraic geometry, the map \(\pi\) is of finite type. Hence, \(\pi\) is a proper morphism and \(X\) is a complete variety over \(k\).
02

Compute Picard groups

Using the hint in the exercise, we can compute the Picard group of the varieties \(C \times \mathbf{A}^{1}\) and \(C \times\left(\mathbf{A}^{1}-\{0\}\right)\). For the first one, we have \(\operatorname{Pic}\left(C \times \mathbf{A}^{1}\right) \cong \mathbf{G}_{m} \times \mathbf{Z}\) and for the second one, we have \(\operatorname{Pic}\left(C \times\left(\mathbf{A}^{1}-\{0\}\right)\right) \cong \mathbf{G}_{m} \times \mathbf{Z} \times \mathbf{Z}\).
03

Determine the form of the restriction map

To establish part (c), we need to show that the Picard group of \(C \times \mathbf{A}^{1}\) mapped to the Picard group of \(C \times\left(\mathbf{A}^{1}-\{0\}\right)\) by the restriction map has the form \(\langle t, n\rangle \mapsto\langle t, 0 . n\rangle\). Once we have this, we find that the automorphism \(\varphi\) of \(C \times\left(\mathbf{A}^{1}-\{0\}\right)\) gives us a map of the form \(\langle t, d, n\rangle \mapsto\langle t, d+n, n\rangle\) on its Picard group.
04

Show that X is not projective

Finally, we must conclude that the image of the restriction map Pic \(X \rightarrow \operatorname{Pic}(C \times\{0\})\) consists entirely of divisors of degree 0 on \(C .\) To do this, we will use the obtained forms for the restriction maps and Picard groups. We determine that all elements in the image of the restriction map \(\operatorname{Pic}(X) \rightarrow \operatorname{Pic}(C \times\{0\})\) have degree 0. Hence, the variety \(X\) is not projective over \(k\), and the morphism \(\pi\) is not projective.

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Key Concepts

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

Proper Morphism
In algebraic geometry, a morphism \(\pi: X \rightarrow Y\) is called "proper" if it satisfies two main conditions:
  • It is universally closed: For any variety \(Z\) and morphism \(Z \rightarrow Y\), the induced morphism \(Z \times_Y X \rightarrow Z\) is closed.
  • It is of finite type: The pre-image of any point under \(\pi\) is a variety of finite type. This means it can be described using a finite number of generators.
In the context of the exercise, we specifically explore the morphism \(\pi: X \rightarrow \mathbf{P}^1\). The complete nature of the variety \(C\), being part of \(\mathbf{P}^2_k\), ensures that \(X\) is complete. Thus, as it encompasses the properties of being closed and of finite type, \(\pi\) is identified as a proper morphism.
Picard Group
The Picard group, often denoted as \(\operatorname{Pic}(X)\), is a fundamental concept in algebraic geometry.
It represents the group of line bundles (or divisors) on a variety \(X\). These line bundles can be thought of as algebraic curves or formal sums of points on \(X\). The group operation is given by tensor product, and the identity element corresponds to the trivial line bundle.

In simple terms, the Picard group helps us understand the ways line bundles can be combined over a variety.
  • For the product \(C \times \mathbf{A}^1\), the Picard group is \(\mathbf{G}_m \times \mathbf{Z}\), indicating ways of translating and scaling curves over this space.
  • For \(C \times (\mathbf{A}^1 - \{0\})\), the Picard group has an additional \(\mathbf{Z}\) factor, reflecting further freedoms in manipulating line bundles.
The exercise demonstrates transformations and restriction maps affecting these groups, especially focusing on how morphisms reshape the structure and relationships of line bundles.
Complete Variety
A variety is described as "complete" if every morphism from a curve to the variety is a closed map.
One useful analogy is to consider complete varieties as the algebraic geometry analogues to compact spaces in topology. They are varieties without boundary and encompasses all limit points.
  • In the provided exercise, variety \(X\) is complete because it is created by gluing together projective pieces, with morphisms that assure closure.
  • Completeness often leads to simpler behavioral analysis of varieties in terms of morphisms, intersections, and cohomology, largely making calculations and transformations more manageable.
This property is crucial for various applications, as completeness can guarantee that certain types of operations remain well-behaved, making them ideal candidates for study in more advanced contexts.
Projective Variety
A "projective variety" is a type of algebraic variety embedded in projective space \(\mathbf{P}^n\).
It can be visualized as a subset of projective space that is cut out by homogeneous polynomials. Projective varieties are special because they naturally include 'points at infinity,' making them inherently compact.

In this exercise, although the variety \(X\) originates from projective components like \(C \subseteq \mathbf{P}^2_k\), it ends up not being projective over \(k\).
  • This is determined by examining the images of Picard group restriction maps, which showed all divisors to be of degree 0.
  • Hence, while \(\pi\) is a proper morphism, \(X\) lacks the necessary structure to qualify as projective.
This distinction is critical because the properties achievable through projective varieties, such as the ability to use tools like projective duality, won't apply in the same way to variety \(X\).

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Most popular questions from this chapter

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 \\]

Let \(f: X \rightarrow Y\) be a morphism of schemes. (a) Show by example that if \(\mathscr{F}\) is coherent on \(X\), then \(f_{*} \mathscr{F}\) need not be coherent on \(Y\), even if \(X\) and \(Y\) are varieties over a field \(k\) (b) Show that a closed immersion is a finite morphism \((\S 3)\) (c) If \(f\) is a finite morphism of noetherian schemes, and if \(\mathscr{F}\) is coherent on \(X\) then \(f_{*} \mathscr{F}\) is coherent on \(Y\).

Flasque Sheares. A sheaf \(\bar{y}\) on a topological space \(X\) is flasque if for every inclusion \(V \subseteq U\) of open sets, the restriction \(\operatorname{map} \mathscr{F}(U) \rightarrow \mathscr{F}(V)\) is surjective. (a) Show that a constant sheaf on an irreducible topological space is flasque. See (I, 81 ) for irreducible topological spaces. (b) If \(0 \rightarrow \overline{\mathscr{H}} \rightarrow \mathscr{F} \rightarrow \mathscr{H}^{\prime \prime} \rightarrow 0\) is an exact sequence of sheaves, and if \(\bar{y}\) is flasque, then for any open set \(U\). the sequence \(0 \rightarrow \mathscr{F}^{\prime}(U) \rightarrow \mathscr{F}(U) \rightarrow\) \(\mathscr{F}^{\prime \prime}\left(L^{\prime}\right) \rightarrow 0\) of abelian groups is also exact. (c) If \(0 \rightarrow \mathscr{H} \rightarrow \mathscr{H} \rightarrow \mathscr{H}^{\prime \prime} \rightarrow 0\) is an exact sequence of sheaves, and if \(\mathscr{H}^{\prime}\) and \(\overline{\mathscr{H}}\) are flasque, then \(\mathscr{F}^{\prime \prime}\) is flasque. (d) If \(f: X \rightarrow Y\) is a continuous map, and if \(\mathscr{F}\) is a flasque sheaf on \(X\), then \(f_{*} \overline{\mathscr{H}}\) is a flasque sheaf on \(Y\) (e) Let \(\overline{\mathscr{F}}\) be any sheaf on \(X\). We define a new sheaf \(\mathscr{G}\), called the sheaf of discontinuous sections of \(\mathscr{F}\) as follows. For each open set \(U \subseteq X, \mathscr{G}(U)\) is the set of

Describe Spec \(\mathbf{Z}\), and show that it is a final object for the category of schemes. i.e., each scheme \(X\) admits a unique morphism to Spec \(\mathbf{Z}\).

(a) Let \(S\) be a graded ring. Show that Proj \(S=\varnothing\) if and only if every element of \(S_{+}\) is nilpotent. (b) Let \(\varphi: S \rightarrow T\) be a graded homomorphism of graded rings (preserving degrees) Let \(U=\left\\{p \in \operatorname{Proj} T | p \geqslant \varphi\left(S_{+}\right)\right\\} .\) Show that \(U\) is an open subset of Proj \(T\) and show that \(\varphi\) determines a natural morphism \(f: U \rightarrow\) Proj \(S\). (c) The morphism \(f\) can be an isomorphism even when \(\varphi\) is not. For example, suppose that \(\varphi_{d}: S_{d} \rightarrow T_{d}\) is an isomorphism for all \(d \geqslant d_{0},\) where \(d_{0}\) is an integer. Then show that \(U=\operatorname{Proj} T\) and the morphism \(f: \operatorname{Proj} T \rightarrow\) Proj \(S\) is an isomorphism. (d) Let \(V\) be a projective variety with homogeneous coordinate ring \(S(\mathrm{I}, \$ 2)\). Show that \(t(V) \cong \operatorname{Proj} S\).
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