Schemes

Exercises 3

Schemes – examples are important

Please also argue that your examples are as required. Feel free to quote results from Hartshorne or EGA.

1. Give an example of a morphism of integral schemes f : X → Y such that the induced maps O_{Y,f(x)} → O_{X,x} are surjective for all x ∈ X, but f is not a closed immersion.

2. Give examples of graded rings S such that (a) Proj(S) is affine and nonempty, and

(b) Proj(S) is integral, nonempty but not isomorphic to ℙ_{A}^{n} for any n ≥ 0, any ring A.

3. Give an example of a nonconstant morphism of schemes ℙ1_{ℂ} → ℙ5_{ℂ} over Spec(ℂ).

4. Give an example of an isomorphism of schemes ℙ1_{ℂ} → Proj(ℂ[X_{0}, X_{1}, X_{2}]/(X_{0}^{2} + X_{1}^{2} + X_{2}^{2})).

5. Give an example of a morphism of schemes f : X → 𝔸1_{ℂ} = Spec(ℂ[T]) such that the (scheme theoretic) fibre of f over t ∈ 𝔸1_{ℂ} is (a) isomorphic to ℙ1_{ℂ} when t is a closed point not equal to 0, and (b) not isomorphic to ℙ1_{ℂ} when t = 0.

Remark. This can be done in many, many ways. Here are some additional restraints you can impose: Can you do it with fibre at t = 0 projective? Can you do it with special fibre irreducible and projective? Can you do it with special fibre integral and projective? Can you do it with fibre at t = 0 smooth and projective? What about similar questions when you replace ℙ1_{ℂ} with another variety over ℂ?