Prove that every one-dimensional proper scheme \(X\) over an algebraically closed field \(k\) is projective.
If \(X\) is irreducible and nonsingular, then \(X\) is projective by (II, 6.7).
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 \(f:\tilde X\to X\) be the projection and let \(\mcl\) be a very ample invertible sheaf on \(\tilde X\). Show that there is an effective divisor \(D=\sum P_i\) on \(\tilde X\) with \(\OO_{\tilde X}(D)\cong\mcl\), such that \(f(P_i)\) is a nonsingular point of \(X\) for every \(i\).
Conclude that there is an invertible sheaf \(\mcl_0\) on \(X\) with \(f^*\mcl_0\cong\mcl\). Then use (Ex. 5.7d), (II, 7.6) and (II, 5.16.1) to show that \(X\) is projective.
If \(X\) is reduced, but not necessarily irreducible, let \(X_1,\ldots,X_r\) be its irreducible components with their reduced structures. Use (Ex. 4.5) to show that \(\Pic X\to\bigoplus_i\Pic X_i\) is surjective. Then use (Ex. 5.7c) to show that \(X\) is projective.
For an arbitrary one-dimensional proper scheme \(X\) over \(k\), use (2.7) and (Ex. 4.6) to show that \(\Pic X\to\Pic X_{\mathrm{red}}\) is surjective. Then use (Ex. 5.7b) to show that \(X\) is projective.
Solution
For a proper scheme of finite type over \(k\), existence of an ample invertible sheaf implies projectivity. Indeed, a positive power is very ample by [@Har10a, Theorem II.7.6] and gives an immersion into a finite-dimensional projective space. This immersion is proper, since its source is proper and its target is separated over \(k\), and a proper immersion is a closed immersion [@Har10a, Remark II.5.16.1]. We will construct such an ample sheaf in each case.
1. A proper nonsingular integral curve over \(k\) is projective, proving (a).
This is the equivalence between completeness and projectivity for nonsingular curves [@Har10a, Proposition II.6.7]. Properness is completeness for the curve, so the stated hypothesis applies.
2. For integral \(X\), its normalization is a nonsingular projective curve, and a very ample sheaf on it is represented by a reduced effective divisor supported over the nonsingular locus of \(X\).
Normalization of an integral curve of finite type over a field is finite and birational [@Har10a, Exercise II.3.8]. Thus \(f:\tilde X\to X\) is finite surjective, and \(\tilde X\) is proper over \(k\). Its one-dimensional normal local rings are DVRs, so it is nonsingular [@Har10a, Theorem I.6.2A]. Step 1 makes it projective.
The nonsingular locus of \(X\) is open and dense by P-AGH281DIFFNONCLOSED, part (d). Its complement is a finite set of closed points of the integral curve, and finiteness of \(f\) makes its inverse image finite as well. Choose a projective embedding of \(\tilde X\) with pullback of \(\OO(1)\) equal to the given very ample \(\mcl\). The hyperplanes avoiding that finite inverse image form a nonempty open subset of the dual projective space. Bertini’s theorem gives another dense open subset whose intersections with the nonsingular curve are reduced zero-dimensional divisors [@Har10a, Theorem II.8.18]. Their intersection has a \(k\)-point because \(k\) is algebraically closed. The resulting hyperplane section is \(D=\sum P_i\) with distinct closed points and \(\OO_{\tilde X}(D)\cong\mcl\). Its support avoids the inverse image of the singular locus, as required in (b).
3. Every proper integral curve \(X\) is projective, completing (b).
The normalization is an isomorphism over the nonsingular locus, since the local rings there are normal. Thus the points \(f(P_i)\) from step 2 are distinct nonsingular points of \(X\) and define an effective Cartier divisor \(D_0=\sum f(P_i)\) on \(X\). Each has a local equation given by a uniformizer, and away from these points the divisor is zero. Set \(\mcl_0=\OO_X(D_0)\). Pullback of its local equations through the normalization gives \(f^*D_0=D\), with multiplicity one because \(f\) is an isomorphism there. Hence \(f^*\mcl_0\cong\OO_{\tilde X}(D)\cong\mcl\). The very ample \(\mcl\) is ample, and finite-surjection descent in P-AGH357AMPLENESS, part (d), makes \(\mcl_0\) ample on \(X\). Properness and the ample-sheaf criterion preceding step 1 give projectivity.
4. For a reduced proper scheme \(X\) of dimension at most one, restriction \(\Pic X\to\bigoplus_i\Pic X_i\) is surjective.
Induct on the finite number of irreducible components. The one-component and empty cases are immediate. Separate one component \(Z\) from the reduced union \(W\) of the others, and let \(I,J\) be their ideals in \(X\). Reducedness and \(Z\cup W=X\) give \(I\cap J=0\). Thus the structure sheaf is the fibre product \[ \OO_X\cong\OO_Z\times_{\OO_T}\OO_W, \qquad T=Z\cap W, \] where all three sheaves on closed subschemes are pushed forward to \(X\) and \(T\) has its scheme-theoretic intersection structure. The fibre-product identity is the ring identity \(B\cong B/I\times_{B/(I+J)}B/J\) for \(I\cap J=0\).
The closed subset \(T\) is zero-dimensional or empty, since distinct components of a curve have no common generic point. It need not be reduced. There is an exact sequence of abelian sheaves \[ 1\to\OO_X^\times\to\OO_Z^\times\times\OO_W^\times \xrightarrow{(u,v)\mapsto u|_T/v|_T}\OO_T^\times\to1. \] The kernel assertion follows from the fibre-product identity, including invertibility because compatible inverse pairs also glue. For surjectivity, at a point of \(T\) the quotient map of local rings \(\OO_{Z,x}\to\OO_{T,x}\) is surjective and lifts units: a lift of a unit lies outside the maximal ideal and is a unit. Lift on the \(Z\) side and choose \(1\) on the \(W\) side. Away from \(T\) the last stalk is trivial, so the sheaf sequence is exact everywhere.
The noetherian space underlying \(T\) has dimension zero, so \(H^1(T,\OO_T^\times)=0\) by [@Har10a, Theorem III.2.7]. Closed direct image preserves cohomology, and P-AGH345PICH1 identifies degree-one unit cohomology with Picard groups. The long exact sequence therefore makes \(\Pic X\to\Pic Z\oplus\Pic W\) surjective. Composing with the induction surjection for \(W\) proves the stated map onto all component Picard groups.
5. Every reduced one-dimensional proper scheme is projective, proving (c).
Every one-dimensional integral component is projective by step 3 and has an ample invertible sheaf. A zero-dimensional integral component is a single reduced point with residue field finite over \(k\), hence is \(\Spec k\) and is projective; its structure sheaf is ample. Step 4 allows the chosen invertible sheaves on all the components to be the restrictions of one invertible sheaf \(L\) on \(X\). Ampleness on reduced components detects ampleness by P-AGH357AMPLENESS, part (c), so \(L\) is ample. The proper scheme \(X\) is therefore projective. This also handles any isolated zero-dimensional components of the original one-dimensional scheme.
6. For arbitrary \(X\) in (d), the map \(\Pic X\to\Pic X_{\mathrm{red}}\) is surjective and \(X\) is projective.
2.Choose \(e\ge1\) with \(N^e=0\), and write \(X_j=(|X|,\OO_X/N^j)\) for \(1\le j\le e\).Then \(X_1=X_{\mathrm{red}}\) and \(X_e=X\).The ideal \(N^j/N^{j+1}\) of \(X_{j+1}\) has square zero, since \(2j\ge j+1\).The sequence in P-AGH346SQUAREZEROPIC gives\[ \Pic X_{j+1}\longrightarrow\Pic X_j \longrightarrow H^2(X_{j+1},N^j/N^{j+1}). \]The last group is zero by Grothendieck vanishing: the underlying space has dimension one [@Har10a, Theorem III.2.7].Thus every transition on Picard groups is surjective, and composing them proves \(\Pic X\to\Pic X_{\mathrm{red}}\) is surjective.
Step 5 makes the reduction projective, so choose an ample invertible sheaf on it and lift its Picard class to an invertible sheaf \(L\) on \(X\). By P-AGH357AMPLENESS, part (b), \(L\) is ample. The proper ample-sheaf criterion now proves that \(X\) is projective, including all of its nilpotent structure.
7. Q.E.D.
Step 1 proves (a), steps <1>2–<1>3 prove (b), steps <1>4–<1>5 prove (c), and step 6 proves (d) and the full theorem.