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 lineA1k.
a) Compute the Picard groups ofA1kandA1kr{0}(Hint: you may use without proof the fact that ifAis a unique factorization domain, the Picard group ofSpec(A)is 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?
c) Compute the Picard group of X (Hint: explain that you may use Leray’s theorem to compute H1(X,OX∗)).
d) Find the global sections of each invertible sheaf onX.
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)).
f) Find the global sections of each invertible sheaf onY. g) Prove that there are no ample invertible sheaves onY.
Problem 2. Prove that the schemeYn :=Ankr{0}is not an affine scheme for anyn ≥2(Hint:use Leray’s theorem to computeH1(Y2,OY2)).
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).
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).
