Let \(k\) be a field. In this remark, by a curve over \(k\) we mean a separated finite type \(k\)-scheme of dimension \(1\), i.e., every local ring of a closed point has dimension \(1\).

Theorem. Let \(C\) be a curve. Then \(C\) is a quasi-projective \(k\)-scheme, i.e., \(C\) is isomorphic to a locally closed subscheme of some projective space.

We will not discuss the complete proof of the theorem here, but only sketch some of the steps.

(I) Assume that \(C\) is normal, i.e., all local rings of \(C\) at closed points are discrete valuation rings. The key point is then that the projective space \(\mathbb {P}^n_k\) satisfies the valuative criterion of properness:

Theorem 1.2

(Valuative criterion of properness, noetherian version, [ GW1 ] Theorem 15.9)

Let \(S\) be a noetherian scheme and let \(f\colon X\to Y\) be a morphism of finite type. We consider commutative diagrams of the form

where \(R\) is a discrete valuation ring with field of fractions \(K\) and the vertical arrow on the left is the canonical inclusion.

The following are equivalent:

1. The morphism \(f\) is proper (resp., separated).

2. In every diagram as above there exists a unique (resp., at most one) morphism \(\operatorname{Spec}R\to X\) making the resulting diagram commutative.

We have proved in AG2 that the structure morphism \(\mathbb {P}^n_k\to \operatorname{Spec}k\) is proper. It follows that it has the property stated in the criterion. However, since we did not prove the valuative criterion of properness, in class we verified directly that it holds for \(\mathbb {P}^n_k\) over \(k\). This is easy using the description of \(S\)-valued points of projective space; for \(S\) the spectrum of a local ring, one basically obtains a description in terms of homogeneous coordinates (cf.  [ GW1 ] Exer. 4.6 for a precise statement). Proving (ii) \(\Rightarrow \) (ii) in the valuative criterion would now give a different proof of the fact that projective space is proper over the base.

Oct. 11,
2023

Now let \(U\subseteq C\) be open affine and choose an immersion \(f\colon U\hookrightarrow \mathbb A^n_k\hookrightarrow \mathbb {P}^n_k\). For \(x\in C\setminus U\), the above shows that we can extend the morphism \(\operatorname{Spec}K(C)\to \mathbb {P}^n_k\) given by \(f\) to a morphism \(\operatorname{Spec}{\mathscr O}_{C, x}\to \mathbb {P}^n_k\). This morphism can be extended to some open neighborhood \(V\) of \(x\) (view \({\mathscr O}_{C,x}\) as the localization of \(\Gamma (V, {\mathscr O}_C)\) with respect to some prime ideal; the image of \(\operatorname{Spec}{\mathscr O}_{C, x}\) is contained in one of the standard open charts of \(\mathbb {P}^n_k\), which we can write as \(k[X_1,\dots , X_n]\); in the images of the \(X_i\) in \({\mathscr O}_{C, x}\) only finitely many denominators are involved, hence on the ring level the homomorphism factors through the localization with respect to a suitable element \(s\); for the scheme morphism this means that it extends to \(D(s)\subseteq V\); see  [ GW1 ] Prop. 10.52). Since \(C\) is reduced and \(\mathbb {P}^n_k\) is separated, and the morphisms \(U\to \mathbb {P}^n_k\) and \(V\to \mathbb {P}^n_k\) coincide on \(\operatorname{Spec}K(C)\), which is dense, they actually coincide on \(U\cap V\) (AG2, Problem [ GW1 ] Cor. 9.9) and we can glue them. Repeating this, if necessary, we can extend the morphism \(U\to \mathbb {P}^n_k\) to a morphism \(C\to \mathbb {P}^n_k\).

At this point, the choice of an affine open \(U\) and an immersion \(U\to \mathbb {P}^n_k\) gives us a morphism \(C\to \mathbb {P}^n_k\). However, in general this will not be an immersion. To finish the proof of the theorem for normal curves, we proceed as follows. Let \(C = \bigcup _{i=1}^m U-i\) be an affine open cover. For each \(U_i\), as above we obtain a morphism \(f_i\colon C\to \mathbb {P}^{n_i}_k\) such that \(f_{i|U_i}\) is an immersion. This gives us a morphism \(C\to \prod _i \mathbb {P}^{n_i}_k\) 8where the product is the fiber product over \(\operatorname{Spec}k\)), and composing this morphism with the Segre embedding \(\prod _i \mathbb {P}^{n_i}_k \to \mathbb {P}^N_k\) (where \(N\) depends on the \(n_i\) as dictated by the Segre embedding, a closed embedding; see  [ GW1 ] Section (4.14)) we obtain a morphism \(f\colon C\to \mathbb {P}^N_k\) such that for every \(i\), the restriction \(f_{|U_i}\) is an immersion. We can then conclude by the following lemma.

Proof

The issue here is to show that under the given assumptions, \(f\) is injective.

We sketch the proof in case \(X\) is irreducible and all \(f_{|U_i}\) are open immersions, which is the case relevant for us. By replacing \(Y\) with the reduced closure of the image of \(f\), we reduce to the case that \(Y\) is integral and that \(f\) is dominant (i.e., the image of \(f\) is dense in \(Y\)). It follows that for every \(x\in X\), the ring homomorphism \({\mathscr O}_{Y, f(x)} \to {\mathscr O}_{X, x}\) is an isomorphism and thus in particular flat.

That all these ring homomorphisms are flat is usually expressed by saying that \(f\) is flat. This property is stable under base change. We will use that whenever \(\varphi \colon A\to B\) is a flat local ring homomorphism of local rings, then the induced morphism \(\operatorname{Spec}(B)\to \operatorname{Spec}(A)\) is surjective; this is a commutative algebra result which sometimes goes by the name going down for flat morphisms, see e.g.  [ GW1 ] Example B.18, [ M2 ] Theorem 9.5. A maximal point of a scheme is a point \(x\) such that there exists no point \(x'\ne x\) with \(x\in \overline{\{ x'\} }\), i.e., \(x\) is a generic point of an irreducible component. As a consequence of the above discussion we see that under a flat morphism \(X\to Y\) of schemes, every maximal point of \(X\) is mapped to a maximal point of \(Y\):

Coming back to the specific situation at hand, to show that \(f\) is injective, we will show that the diagonal morphism \(\Delta \colon X\to X\times _YX\) is surjective. The injectivity of \(f\) is an easy consequence of this. Since by assumption \(f\) is separated, \(\Delta \) is a closed immersion, thus it is enough to show that all maximal points of \(X\times _YX\) are in its image.

So let \(\zeta \in X\times _YX\) be maximal. Since the projections \(X\times _YX\to X\) and \(f\) are flat morphisms, \(\zeta \) maps to the unique maximal point of \(X\) under the projections, and to the unique maximal point in \(Y\) under the composition with \(f\). Looking at local rings, we obtain a commutative square

But since the maximal point of \(X\) lies in each of the subsets \(U_i\), our assumptions imply that \(K(X) = K(Y)\). This implies that the morphism \(\operatorname{Spec}{\mathscr O}_{X\times _YX, \zeta }\to X\times _YX\) factors through \(\operatorname{Spec}K(X)\otimes _{K(Y)}K(X) = \operatorname{Spec}K(X)\) and hence through \(\Delta \), which shows that \(\zeta \) is in the image of \(\Delta \).

(II) With a bit more work, this strategy can be extended to cover all reduced curves, see  [ GW1 ] Theorem 15.18.

(III) For the general case, we use the description of \(S\)-valued points of projective space in terms of line bundles. We search for a line bundle \({\mathscr M}\) on the given curve \(C\) together with a surjection \({\mathscr O}_C^{n+1}\to {\mathscr L}\) such that the corresponding morphism \(C\to \mathbb {P}^n_k\) is an immersion.

Let \(C_{\rm red}\) be the underlying reduced subscheme of \(C\) and by \(\iota \colon C_{\rm red}\to C\) the corresponding closed immersion. By step (II) we know, that on \(C_{\rm red}\) a line bundle \({\mathscr L}'\) (together with a surjection \({\mathscr O}_{C_{\rm red}}^{n'}\to {\mathscr L}'\)) with the desired property exists. The crucial step then, is to show that there exists a line bundle \({\mathscr L}\) on \(C\) with \(\iota ^*{\mathscr L}\cong {\mathscr L}'\). One can then show that from such an \({\mathscr L}\) one can construct \({\mathscr M}\) as desired (in fact, this is also an application of cohomological methods, namely “Serre’s criterion for ampleness”); we skip this step here.

To proceed, we need the following cohomological description of the Picard group of a scheme.

The existence of \({\mathscr L}\) follows from the following lemmas together with the Grothendieck vanishing theorem.

See  [ GW2 ] Theorem 26.16 for references to the missing pieces.

Let \(k\) be a field, and let \(X\) be a separated \(k\)-scheme of finite type of dimension \(2\) (i.e., all local rings at closed points have dimension \(2\)). We call \(X\) a surface.

Theorem. If \(X\) is regular (i.e., all local rings of \(X\) are regular local rings), then \(X\) can be embedded into some projective space over \(k\) as a locally closed subscheme.

To prove the theorem, by “Nagata’s compactification theorem” (itself a difficult theorem, see  [ GW1 ] Section (12.15) for its statement and references) one can restrict to the case that \(X\) is a proper \(k\)-scheme.

For \(X\) proper, a key ingredient is the Lemma of Enriques-Severi-Zariski that we have seen at the end of the Algebraic Geometry 2 class. See  [ GW2 ] Theorem 25.151 for a proof of the theorem (which requires quite a few other ingredients, many of them also relying on cohomology).

The regularity assumption cannot be dropped in the theorem. In higher dimension, the corresponding statement fails even for regular \(k\)-schemes.