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By I.G. Macdonald

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Prove that |x y z | > |xyz|. ) (d) Conclude that either C(Q) = ∅ or else C(Q) is an infinite set. (e) ** Characterize, in terms of a, b, c, d, whether C(Q) contains any points. 10. Let C be a smooth curve. The support of a divisor D = ∗ ¯ be a function such that div(f ) and D of points P ∈ C for which nP = 0. Let f ∈ K(C) have disjoint supports. Then it makes sense to define f (P )nP . f (D) = P ∈C Let φ : C1 → C2 be a nonconstant map of smooth curves. Prove that the following two equalities are valid in the sense that if both sides are well-defined, then they are equal.

Xn )q However, since K is perfect, we know that every element of K is a q th power, so K[X0 , . . , Xn ] q = K[X0q , . . , Xnq ]. Thus the set of quotients f (Xiq )/g(Xiq ) and the set of quotients f (Xi )q /g(Xi )q give the exact same subfield of K(C). (b) Immediate from (a). (c) Taking a finite extension of K if necessary, we may assume that there is a smooth point P ∈ K(C). 1). 4) says that K(C) is separable over K(t). Consider the tower of fields K(C) ❏ ✡ ❏ purely separable ✡ ❏ inseparable ✡ q ✡ K(C) (t) ❏ ❅ ❏ ✡ ❅ ❏ ✡ K(C)q K(t) It follows that K(C) = K(C)q (t), so from (a), deg φ = K(C)q (t) : K(C)q .

X0 ∂Xn Thus for hypersurfaces in projective space, we can check for smoothness using homogeneous coordinates. (b) Let n ≥ 1, and let W ⊂ Pn be a smooth algebraic set, each of whose component varieties has dimension n − 1. Prove that W is a variety. (Hint. 12. (a) Let V /K be an affine variety. Prove that ¯ ] : f σ = f for all σ ∈ GK/K . K[V ] = f ∈ K[V ¯ ¯ (Hint. One inclusion is clear. For the other, choose some polynomial F ∈ K[X] with σ → I(V ) defined by σ → F − F is a F ≡ f (mod I(V )). Show that the map GK/K ¯ 1-cocycle; see (B §2).

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Algebraic geometry : introduction to schemes by I.G. Macdonald

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