Math 256B. Homework 9 Due Wednesday 3 April

1. Hartshorne III Ex. 5.8.

2. Let X be a projective scheme over a field k, and let ℒ be a very ample line sheaf on

X (over k). Show that there exists an integer m_{0} such that the map

H^{0}(X, ℒ^{⊗m})^{⊗n} ⟶ H^{0}(X, ℒ^{⊗mn})

is surjective for all m ≥ m_{0} and all n ∈ ℤ_{>0} . [Hint: Look at the case X = ℙ_{k}^{r} first.]

3. Hartshorne II Ex. 6.1. Also say what the isomorphism is.