b) Let X be the scheme obtained by glueing U1 and U2 along the open subsets U1 r {0

Duke University – Spring 2017 – MATH 627 April 13, 2017

Problem set 3 Olivier Debarre Due Tuesday May 2, 2017

Problem 1. Letkbe a field. We consider two copiesU₁ := Spec(k[T₁])andU₂ := Spec(k[T₂])of the affine lineA¹_k.

a) Compute the Picard groups ofA¹_kandA¹_kr{0}(Hint: you may use without proof the fact that ifAis a unique factorization domain, the Picard group ofSpec(A)is trivial).

Proof. SinceA¹_k = Spec(k[T])andA¹_kr{0}:= Spec(k[T, T⁻¹])and both ringsk[T]andk[T, T⁻¹] are unique factorization domains, their Picard groups are trivial.

b) Let X be the scheme obtained by glueing U₁ and U₂ along the open subsets U₁ r {0} = Spec(k[T₁, T₁⁻¹])and U₂ r{0} = Spec(k[T₂, T₂⁻¹])by the isomorphismk[T₁, T₁⁻¹]→^∼ k[T₂, T₂⁻¹] ofk-algebras sendingT₁toT₂⁻¹. Which scheme isX?

Asnwer. The schemeX is the projective lineP¹_k.

c) Compute the Picard group of X (Hint: explain that you may use Leray’s theorem to compute H¹(X,OX^∗)).

Proof. The scheme X is covered by the affine subsetsU₁ andU₂. Moreover, H^q(U_i,OX^∗) = 0 for q ≥ 2becauseU_i has dimension 1, and forq = 1 becausePic(U_i) = Pic(A¹_k) = 0by question a).

Similarly, H^q(U₁ ∩U₂,OX^∗) = 0forq > 0for the same reasons. We may therefore apply Leray’s theorem to computeH¹(X,OX^∗)as the cokernel of the map

Γ(U₁,OX^∗)×Γ(U₂,OX^∗) −→ Γ(U₁ ∩U₂,OX^∗) (s₁, s₂) 7−→ s₁/s₂.

SinceΓ(U_i,OX^∗) =k[T_i]^∗ =k^∗andΓ(U₁∩U₂,OX^∗) =k[T, T⁻¹]^∗ =k^∗× hTi 'k^∗×Z, we obtain Pic(X)'H¹(X,OX^∗)'Z.

d) Find the global sections of each invertible sheaf onX.

Proof. LetLm be the invertible sheaf onX corresponding to (1, T₁^m) ∈ k^∗ × hT₁i. The sections of Lm on X correspond to pairs (P₁, P₂), where P_i ∈ Γ(U_i,Lm) ' Γ(U_i,OUi) ' k[T_i], with P₁(T₁)/P₂(T₁⁻¹) = T₁^m. It follows thatΓ(X,Lm)is isomorphic to the space of polynomials ink[T] of degree≤m: ifm≥0, the sections are the pairs(P(T₁), T₂^mP(T₂⁻¹)), wheredeg(P)≤m.

e) Let Y be the scheme obtained by glueing U₁ and U₂ as in b), but using now the isomorphism k[T₁, T₁⁻¹]→^∼ k[T₂, T₂⁻¹] that sends T₁ to T₂. Compute the Picard group of Y (Hint: proceed as in c)).

Proof. The same proof as in c) givesPic(Y)'H¹(Y,OY^∗)'Z.

f) Find the global sections of each invertible sheaf onY.

Proof. Let Lm be the invertible sheaf on Y corresponding to (1, T₁^m) ∈ k^∗ × hT1i. The sections of Lm on Y correspond to pairs (P₁, P₂), where P_i ∈ Γ(U_i,Lm) ' Γ(U_i,OUi) ' k[T_i], with P₁(T₁)/P₂(T₁) = T₁^m. It follows that Γ(Y,Lm) ' k[T] for all m: if m ≥ 0, the sections are the pairs(T₁^mP(T₁), P(T₂)); ifm ≤ 0, the sections are the pairs(P(T₁), T₂^−mP(T₂)). In all cases, Γ(Y,Lm)is isomorphic tok[T].

g) Prove that there are no ample invertible sheaves onY.

Proof. If m > 0, the global sections of Lm all vanish at the origin on U₁; ifm < 0, the global sections of Lm all vanish at the origin onU2. Hence Lm is never generated by global sections if m6= 0. It follows thatLmis not ample for anym(apply the definition of ampleness withF =L1).

Problem 2. Prove that the schemeYn :=Aⁿ_kr{0}is not an affine scheme for anyn ≥2(Hint:use Leray’s theorem to computeH¹(Y₂,OY2)).

Proof.The schemeY2is covered by the affine open subsetsU1 :=A¹_k×k(A¹_kr{0}) = Spec(k[T1, T2, T₂⁻¹]) andU₂ := (A¹_kr{0})×_kA¹_k) = Spec(k[T₁, T₁⁻¹, T₂]). SinceOY2 is a coherent sheaf, we can com- puteH¹(Y₂,OY2)using Leray’s theorem as the cokernel of the map

Γ(U₁,OY^∗2)×Γ(U₂,OY^∗2) −→ Γ(U₁∩U₂,OY^∗2) (P₁, P₂) 7−→ P₁−P₂.

We have

Γ(U₁,OY^∗2) = k[T₁, T₂, T₂⁻¹], Γ(U₂,OY^∗2) = k[T₁, T₁⁻¹, T₂], Γ(U₁∩U₂,OY^∗2) = k[T₁, T₁⁻¹, T₂, T₂⁻¹],

henceH¹(Y2,OY2)is an infinite-dimensionalk-vector space with basis(T₁^mT₂ⁿ)m,n<0. In particular, Y₂ is not affine. SinceY₂ is a closed subscheme ofY_n for alln ≥ 2and a closed subscheme of an affine scheme is affine,Y_nis also not an affine scheme for alln≥2.

Problem 3.LetX be a projective scheme over a field and letL andM be invertible sheaves onX.

a) IfL is generated by global sections andM is very ample, the invertible sheafL ⊗M is very ample (Hint: use a Segre embedding).

Proof. Since L is generated by global sections, there exists a morphism u: X → P^m_k such that u^∗OP^m_k(1) = L. SinceM is very ample, there exists a closed embeddingv: X ,→ Pⁿ_k such that v^∗OPⁿ_k(1) = M. The morphism (u, v) :X → P^m_k ×Pⁿ_kis then also a closed embedding (because its composition with the second projection is) and so is the composition

w: X −−−→^(u,v) P^m_k ×Pⁿ_k−−−→^Segre P(m+1)(n+1)−1

k .

Sincew^∗O_P(m+1)(n+1)−1 k

(1) =L ⊗M, this proves thatL ⊗M is very ample.

b) IfM is ample, the invertible sheaf L ⊗M^⊗r is very ample forallsufficiently large integersr (Hint:we proved in class thatL ⊗M^⊗ris ample for some integerr >0).

Proof. SinceM is ample, there exists an integer r₀ such that L ⊗M^⊗r is generated by global sections for all r ≥ r₀, and there exists an integer s₀ such that M^⊗s0 is very ample. For any r≥r0+s0, the invertible sheafL ⊗M^⊗r =L ⊗M^⊗(r−s0)⊗M^⊗s0 is then very ample by a).

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