Toric Varieties Seminar
Final Problem Set
1. Describe all open sets of C using the Zariski topology.
2. Show that the Zariski topology on C2 is not the same thing as the product topology (using the Zariski
topology on each factor separately).
3. Is the ring of infinitely many polynomials x1 , x2 , . . . with coefficients in C Noetherian?
4. Is complex conjugation C → C a ring homomorphism? What is its kernel?
5. For a complex number a ∈ C, let ϕa : C[x] → C denote the evaluation map f 7→ f (a).
(i) Calculate the kernel ma = ker ϕa .
(ii) Show that ma is a maximal ideal.
(iii)∗ Show that any maximal ideal of C[x] is of this form.
6. Let S be the sub semi-group of Z>0 generated by 2, 3.
(i) Find the affine toric variety SpecC[S].
(ii) Is this toric variety normal?
(iii)∗ More generally, show that an affine toric variety of the form SpecC[S] for a sub semi-group S of Zn is
normal if and only if the semi-group is saturated. Recall that S is called saturated if whenever m ∈ Z
is an integer and v ∈ Zn satisfy mv ∈ S, then v ∈ S.
7. Let N be a lattice generated by e1 , e2 and let σ be the cone generated by e1 , e1 + 2e2 .
(i) Identify the affine toric variety Uσ .
(ii) Let τ be the face of σ generated by e1 . Verify that Uτ is a principal open subset of Uσ , in the sense
that Uτ = Uσ \ V (f ) for some regular function f on Uσ .
(iii) Is Uσ isomorphic to C2 ?
8. Find a lattice N and a fan ∆ of NR such that the corresponding toric variety X(∆) satisfies
(i) X(∆) = C2
(ii) X(∆) = (C∗ )2
(iii) X(∆) = C × C∗
(iv) X(∆) = CP2
(v) X(∆) = V (xy − zw) ⊂ C4 (Hint: V (xy − zw) = Spec(C[x, y, yz, xz]))
(vi) X(∆) = V (uw − v 2 ) ⊂ C3 (Hint: V (uw − v 2 ) = Spec(C[x2 , xy, y 2 ]))
(vii) X(∆) = CP1 × CP1 .
9. Say whether each of the varieties from the previous question is
(a) non-singular (or smooth)
(b) normal (Remember that non-singular implies normal)
(c) compact
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