91Ó°ÊÓ

Prove that every one-dimensional proper scheme \(X\) over an algebraically closed field \(k\) is projective (a) If \(X\) is irreducible and nonsingular, then \(X\) is projective by (II, 6.7 ) (b) If \(X\) is integral, let \(\tilde{X}\) be its normalization (II, Ex. 3.8). Show that \(\tilde{X}\) is complete and nonsingular, hence projective by (a). Let \(f: \tilde{X} \rightarrow X\) be the projection. Let \(\mathscr{Q}\) be a very ample invertible sheaf on \(\tilde{X}\). Show there is an effective divisor \(D=\sum P_{i}\) on \(\tilde{X}\) with \(\mathscr{L}(D) \cong \mathscr{L},\) and such that \(f\left(P_{i}\right)\) is a nonsingular point of \(X,\) for each \(i .\) Conclude that there is an invertible sheaf \(\mathscr{L}_{0}\) on \(X\) with \(f^{*} \mathscr{L}_{0} \cong\) \(\mathscr{L} .\) Then use (Ex. \(5.7 \mathrm{d}\) ), (II, 7.6) and (II, 5.16.1) to show that \(X\) is projective. (c) If \(X\) is reduced, but not necessarily irreducible, let \(X_{1}, \ldots . . X_{r}\) be the irre. ducible components of \(X\). Use (Ex. 4.5 ) to show Pic \(X \rightarrow \oplus\) Pic \(X\) is sur jective. Then use (Ex. 5.7c) to show \(X\) is projective (d) Finally, if \(X\) is any one-dimensional proper scheme over \(k\), use ( 2.7 ) and (Ex. 4.6 ) to show that Pic \(X \rightarrow\) Pic \(X_{n d}\) is surjective. Then use (Ex. \(5.7 \mathrm{b}\) ) to show \(X\) is projective.

Step by step solution

01

Prove projectivity for irreducible and nonsingular schema

If \(X\) is irreducible and nonsingular, then apply (II,6.7) to show that \(X\) is projective.

02

Show projectivity for integral schema

Let \(\tilde{X}\) be the normalization of \(X\) (by referencing II, Ex. 3.8). Show that \(\tilde{X}\) is complete (wholly defined), and nonsingular (its local rings are discrete valuation rings), thus projective by the previous step. Let \(f: \tilde{X} \rightarrow X\) be the projection, and let \(\mathscr{Q}\) be a very ample invertible sheaf on \(\tilde{X}\). Show there is an effective divisor, \(D=\sum P_{i}\), on \(\tilde{X}\) such that \(\mathscr{L}(D) \cong \mathscr{L}\), and \(f\left(P_{i}\right)\) is a nonsingular point of \(X,\) for each \(i .\) Deduce that \(\mathscr{L}_{0}\) is an invertible sheaf on \(X\) with \(f^{*} \mathscr{L}_{0} \cong\) \(\mathscr{L}\). Then apply (Ex. \(5.7 \mathrm{d}\)), (II, 7.6), and (II, 5.16.1) to show that \(X\) is projective.

03

Verify projectivity for reduced schema

Let \(X_{1}, \ldots ., X_{r}\) be the irreducible components of \(X\). Using (Ex. 4.5 ) it can be shown that Pic \(X\) \(\rightarrow \oplus\) Pic \(X\) is surjective. Then apply (Ex. 5.7c) to conclude that \(X\) is projective when it is reduced, but not necessarily irreducible.

04

Affirm projectivity for a general one-dimensional proper scheme

If \(X\) is any one-dimensional proper scheme over \(k\), use (2.7) and (Ex. 4.6 ) to show that Pic \(X \rightarrow\) Pic \(X_{n d}\) is surjective. Then apply (Ex. \(5.7 \mathrm{b}\)) to affirm that \(X\) is projective.

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Key Concepts

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

Algebraically Closed Field

When delving into the world of algebraic geometry, an algebraically closed field is a fundamental concept. Simply put, a field is considered algebraically closed if every non-constant polynomial in one variable has at least one root in that field. For instance, the field of complex numbers is algebraically closed because every polynomial equation has a solution in the complex numbers, as stated by the Fundamental Theorem of Algebra.

In the context of the projective scheme discussed in the exercise, the field k being algebraically closed ensures that certain algebraic properties hold, such as the existence of roots for polynomials, which is crucial when analyzing the properties of schemes defined over k. This concept sets a foundation for discussing the geometric structure of schemes and their various morphisms, which rely on the underlying properties of the field.

Normalization of Schemes

The notion of the normalization of schemes is somewhat analogous to the process of simplifying fractions in basic arithmetic; it is a process of making a scheme 'better-behaved' algebraically. Normalization typically involves taking a non-normal scheme, which may have singularities or other irregularities, and constructing a new scheme where these issues are resolved.

The normalized scheme, denoted as \( \tilde{X} \), is complete, meaning it has all its points 'filled in,' and nonsingular, implying that it does not contain any 'sharp corners' or 'crossing paths'. This process is crucial because nonsingular, complete schemes are easier to work with and often have better geometric and algebraic properties, such as being projective as shown in the exercise. Normalization is a key step in resolving singularities and studying the intrinsic properties of algebraic spaces.

Very Ample Invertible Sheaf

An essential tool in the algebraic geometer's toolkit is the very ample invertible sheaf. It is connected to the idea of embedding a scheme into projective space. A sheaf, in this context, is a mathematical object that systematically keeps track of algebraic information over a scheme. It's a bit like a function that assigns a set of values to every open subset of our scheme, following some glueing properties.

When we say a sheaf is invertible, it means that it has an inverse operation with respect to tensor product, similar to how an invertible matrix has an inverse with respect to matrix multiplication. A very ample sheaf allows us to find an embedding of our scheme into a projective space, which is ideally where we would like to have our scheme sit for various analytical benefits. In the exercise, the very ample invertible sheaf is used to establish the existence of an effective divisor, leading to the proof that the scheme is projective.

Picard Group

Last but not least is the Picard group. This concept may sound highly abstract when first encountered, but it's essentially like a ledger recording different ways to twist or tweak the scheme's geometry without changing its underlying space. Mathematically, the Picard group of a scheme X is the group of isomorphism classes of invertible sheaves, with the group operation being the tensor product.

Why is this group important? Because it describes various features of the scheme such as its divisors, and helps in classifying line bundles over X. A powerful aspect of the Picard group is its role in deducing the projectivity of a scheme, as seen in the exercise where the surjectivity of certain Picard group maps confirms the projectivity of the scheme. In algebraic geometry, the Picard group serves as a bridge between the purely algebraic data of line bundles and the rich geometry of the underlying space.