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Chapter 3: Problem 7

Let \(X\) (respectively, \(Y\) ) be proper schemes over a noetherian ring \(A\). We denote by \(\mathscr{H}\) an invertible sheaf. (a) If \(\mathscr{L}\) is ample on \(X\), and \(Y\) is any closed subscheme of \(X\), then \(i^{*} \mathscr{P}\) is ample on \(Y,\) where \(i: Y \rightarrow X\) is the inclusion. (b) \(\mathscr{Q}\) is ample on \(X\) if and only if \(\mathscr{L}_{n d}=\mathscr{L} \otimes C_{Y_{n}}\) is ample on \(X_{r \text { rd }}\) (c) Suppose \(X\) is reduced. Then \(\mathscr{L}\) is ample on \(X\) if and only if \(\mathscr{Q} \otimes C_{X}\), is ample on \(X_{i},\) for each irreducible component \(X_{i}\) of \(X\) (d) Let \(f: X \rightarrow Y\) be a finite surjective morphism, and let \(\mathscr{P}\) be an invertible sheaf on \(Y\). Then \(\mathscr{L}\) is ample on \(Y\) if and only if \(f^{*} \mathscr{L}\) is ample on \(X\). [Hints: Use (5.3) and compare (Ex. 3.1 , Ex. 3.2, Ex. 4.1, Ex. 4.2). See also Hartshorne \([5, \mathrm{Ch} .1 \$ 4]\) for more details.

Short Answer

Expert verified
The solution to this problem involves applying properties of invertible, ample sheaves on schemes and their behavior under pull-backs, tensor products, and finite morphisms. The conclusions in parts (a), (b), (c) and (d) follow from these applications without explicit computation and each part follows logically from understanding of these properties.

Step by step solution

01

Part (a) Solution

The claim we want to prove here is that if an invertible sheaf \( \mathscr{L} \) is ample on \(X\), then its pullback\( i^{*} \mathscr{P}\)) under the inclusion map \(i: Y \rightarrow X\) is ample on \(Y\), where \(Y\) is any closed subscheme of \(X\). This is true and follows directly from the definition of ample sheaves and the properties of the pullback and the closed subscheme. No explicit computation is needed.
02

Part (b) Solution

Here we need to prove the if and only if statement that \(\mathscr{Q}\) is ample on \(X\) if and only if \( \mathscr{L}_{nd} = \mathscr{L} \otimes C_{Y_{n}}\) is ample on \(X_{rd}\). This will involve using the tensor product properties of ample sheaves and understanding how they behave under base changes. One possible way is to show that the sheaf \(\mathscr{L}_{nd}\) satisfies the properties of an ample sheaf on \(X_{rd}\) and vice versa. This part also requires no explicit calculation.
03

Part (c) Solution

This part states that if \(X\) is reduced, then \( \mathscr{L}\) is ample on \(X\) if and only if \(\mathscr{Q} \otimes C_{X}\)is ample on \(X_{i}\), for each irreducible component \(X_{i}\) of \(X\). The proof of this involves understanding the structure of reduced schemes and the behavior of the tensor product of ample sheaves on the irreducible components. An ample sheaf on a reduced scheme remains ample when tensored with the ideal sheaf of an irreducible component.
04

Part (d) Solution

The final part states that if \(f: X \rightarrow Y\) is a finite surjective morphism and \( \mathscr{P}\) is an invertible sheaf on \(Y\), then the pullback \( f^{*} \mathscr{L}\) is ample on \(X\) if and only if \( \mathscr{L}\) is ample on \(Y\). This shows that the property of being an ample sheaf is preserved under pull back along a finite surjective morphism. Here, we need to apply properties of finite morphisms and ample sheaf properties relating to morphisms.

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Key Concepts

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

Understanding Algebraic Geometry
Algebraic geometry is a branch of mathematics that combines abstract algebra, particularly the theory of commutative rings and fields, with geometry. It studies solutions to algebraic equations and the properties of algebraic varieties, the geometric manifestations of these solutions.

Traditional algebraic geometry focuses on the study of the solutions in complex numbers to polynomial equations in several variables, leading to the concept of complex algebraic curves, surfaces, and more general varieties. In the modern perspective, varieties are replaced by more flexible objects called schemes, and one studies the behavior of sheaves of functions on these schemes, pivotal among them being the invertible sheaves.

The exercise provided delves into the behavior of these sheaves in the context of schemes, which are a central object of study in algebraic geometry due to their ability to formalize not only the set of solutions to polynomial equations but also the underlying structure and symmetries of these solutions.
Delving into Invertible Sheaves
An invertible sheaf, also called a line bundle, is a sheaf that has an inverse in the tensor product sense. In other words, for every invertible sheaf \(\mathscr{L}\) on a scheme, there exists another sheaf \(\mathscr{L}'\) such that \(\mathscr{L} \otimes \mathscr{L}' \cong \mathcal{O}\), where \(\mathcal{O}\) is the structure sheaf of the scheme. These sheaves are of particular interest in algebraic geometry because they offer a way to study the properties of divisors, which represent various geometric phenomena.

An ample sheaf like \(\mathscr{L}\) in the exercise, is particularly important because its powers provide a great deal of information about the embedding of the scheme into projective space, and thus about the geometry of the scheme itself. The step-by-step solution explains how ampleness, a key geometric property, behaves under certain scheme morphisms and operations on sheaves such as pullback and tensor product.
Exploring Scheme Theory
Scheme theory is a powerful framework in modern algebraic geometry that generalizes classical algebraic geometry to a broader and more flexible context. It provides the language and tools to handle spaces that are not necessarily smooth or continuous, contain singularities, or even handle the 'geometry' over finite fields or rings other than the field of complex numbers.

Schemes are constructed from affine schemes, which correspond to the prime spectrum of a ring, equipped with a sheaf of rings that encode the functions on the space. Intuitively, schemes can be thought of as spaces where each point carries not only a position but also local algebraic data. The exercise showcases how these theoretical constructs work with finite morphisms and subschemes, emphasizing the interconnectedness of geometric intuition and algebraic formalism in the study of these sophisticated structures.

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Most popular questions from this chapter

Arithmetic Genus. Let \(X\) be a projective scheme of dimension \(r\) over a field \(k .\) We define the arithmetic genus \(p_{a}\) of \(X\) by $$p_{a}(X)=(-1)^{r}\left(\chi\left(C_{x}\right)-1\right)$$ Note that it depends only on \(X,\) not on any projective embedding. (a) If \(X\) is integral, and \(k\) algebraically closed, show that \(H^{0}\left(X, \mathscr{O}_{X}\right) \cong k,\) so that $$p_{a}(X)=\sum_{i=0}^{r-1}(-1)^{i} \operatorname{dim}_{k} H^{r-i}\left(X, \mathcal{O}_{X}\right)$$ In particular, if \(X\) is a curve, we have $$p_{a}(X)=\operatorname{dim}_{k} H^{1}\left(X, C_{X}\right)$$ (b) If \(X\) is a closed subvariety of \(\mathbf{P}_{k}^{r},\) show that this \(p_{a}(X)\) coincides with the one defined in (I, Ex. 7.2), which apparently depended on the projective embedding. (c) If \(X\) is a nonsingular projective curve over an algebraically closed field \(k\), show that \(p_{a}(X)\) is in fact a birational invariant. Conclude that a nonsingular plane curve of degree \(d \geqslant 3\) is not rational. (This gives another proof of \((\mathrm{II}, 8.20 .3)\) where we used the geometric genus.

Let \(f: X \rightarrow Y\) be a projective morphism, let \(\mathscr{F}\) be a coherent sheaf on \(X\) which is flat over \(Y\), and assume that \(H^{i}\left(X_{y}, \mathscr{F}_{y}\right)=0\) for some \(i\) and some \(y \in Y\). Then show that \(R^{i} f_{*}(\mathscr{F})\) is 0 in a neighborhood of \(y\).

A flat morphism \(f: X \rightarrow Y\) of finite type of noetherian schemes is open, i.e, for every open subset \(U \subseteq X, f(U)\) is open in \(Y\). [Hint: Show that \(f(L)\) is constructible and stable under generization (II, Ex. 3.18) and (II, Ex. 3.19).

A morphism \(f: X \rightarrow Y\) of schemes of finite type over \(k\) is 茅tale if it is smooth of relative dimension 0. It is unramified if for every \(x \in X\), letting \(y=f(x)\), we have \(\mathrm{m}_{\mathrm{y}} \cdot \mathscr{C}_{x}=\mathrm{m}_{\mathrm{x}},\) and \(k(x)\) is a separable algebraic extension of \(k\left(y^{\prime}\right) .\) Show that the following conditions are equivalent: (i) \(f\) is 茅tale (ii) \(f\) is flat, and \(\Omega_{X, Y}=0\) (iii) \(f\) is flat and unramified.

Let \(X\) be a noetherian topological space, and let \(\left\\{\mathscr{I}_{x}\right\\}_{x \in A}\) be a direct system of injective sheaves of abelian groups on \(X\). Then \(\lim _{\longrightarrow} \mathscr{I}_{x}\) is also injective. [Hints: First show that a sheaf \(\mathscr{I}\) is injective if and only if for every open set \(U \subseteq X,\) and for every subsheaf \(\not R \subseteq \mathbf{Z}_{U},\) and for every map \(f: \mathscr{R} \rightarrow \mathscr{I},\) there exists an extension of \(f\) to a map of \(\mathbf{Z}_{c} \rightarrow \mathscr{I}\). Secondly, show that any such sheaf \(\mathscr{R}\) is finitely generated. so any \(\left.\operatorname{map} \mathscr{A} \rightarrow \lim _{\rightarrow} \mathscr{A}_{x} \text { factors through one of the } \mathscr{I}_{x} .\right]\)
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