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By H. A. Nielsen

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Extra info for Algebraic varieties [Lecture notes]

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2 there is y ∈ Y such that dim Ty Y = dim Y . Now mY,y = mX,y /f OX,y so dim Ty X ≤ dim Y + 1 = dim X and y is nonsingular on X. 5. Let X be a normal irreducible variety and let f : U → k be a regular function on an open subset U ⊂ X. If for every component Z of X − U dim Z ≤ dim X − 2 then f is the restriction to U of a regular function on X. Proof. Assume X affine and f ∈ / k[X]. k[X] is noetherian so choose the ideal P = {g ∈ k[X]|ghf ∈ k[X]} maximal in the family over h ∈ k[X], hf ∈ / k[X].

If D is a divisor with l(D) > 0 and l(K − D) > 0 then 1 l(D) ≤ 1 + deg(D) 2 Proof. By the hypothesis assume D ≥ 0 and K − D ≥ 0 and such that l(D − x) < l(D) for all x. 4. If n ≥ 2g then for every x ∈ X there is f ∈ k(X) with "pole of order n at x", that is vx (f ) = −n, vy (f ) ≥ 0, y = x Proof. 2. 6. Zeuthen-Hurwitz formula Let f : X → Y be a nonconstant morphism of curves. 1. f is surjective and by composition there is defined a finite field extension k(Y ) ⊆ k(X). The degree of f is deg(f ) = dimk(Y ) k(X) For x ∈ X the ramification is ex = vx (f ∗ (u)) where u is a local parameter at f (x).

1. The morphism A1 → A2 , x → y = (x21 , x31 ) is a bijection onto the affine variety V (Y13 − Y22 ) ⊂ A2 . The induced ring homomorphism of coordinate rings k[Y1 , Y2 ]/(Y13 − Y22 ) → k[X1 ] is not surjective, so the morphism above is not an isomorphism. Moreover the varieties A1 and V (Y13 − Y22 ) are not isomorphic by any morphism, since the ring k[X12 , X13 ] is not normal. 2. The morphism P1 → P2 , x → y = (x20 , x0 x1 , x21 ) is an isomorphism onto the projective variety V (Y0 Y2 − Y12 ) ⊂ P2 .

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