By H. A. Nielsen

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3. 7. Let f : X → Y be a finite morphism with Y irreducible. The following are equivalent. (1) f is flat. (2) The rank dimk Ox /f ∗ (my ) is independent of points y = f (x). Proof. 3. CHAPTER III Algebraic curves In this note a curve is an irreducible nonsingular projective variety of dimension 1 over a fixed algebraically closed ground field k. The general theory for nonsingular and normal varieties, chapter II, section 9-11, provides several useful facts. 4. 5. 8. (4) By normalization any finitely generated field K of trdegk K = 1 is the field of rational functions K k(X) for a curve X unique up to isomorphism.

2 regular functions f1 , . . , fn such that k[X] is integral over the polynomial ring k[f1 , . . , fn ] and k(f1 , . . , fn ) ⊂ k(X) is separable. 1. 11. Let X be an irreducible variety. Then for x ∈ X dim Tx X ≥ dim X Proof. 12 reduces to an affine variety X = V (f1 , . . , fm ) ⊂ An . The subset of points x where dim Tx X = min{dim Tz X|z ∈ X} is given by nonvanishing minors of the Jacobi matrix ∂fi /∂Xj and is therefore open. 3. 10. 1. An irreducible variety X is nonsingular,(smooth, regular) at x if dim Tx X = dim X else singular, (nonsmooth,nonregular).

The curve given by X03 X1 + X13 X2 + X0 X23 = 0 has genus g = 3 and | Aut(X)| = 84(g − 1) = 168 18. 1. The set of linear subspaces of fixed dimension m in Pn is bijectively described by a n+1 projective subset of P(m+1)−1 called the Grassmann variety Gn,m . 2. Choose a basis of the m + 1 dimensional subspace of k n+1 as row vectors, giving a (m + 1) × (n + 1)-matrix. jm ) ∈ P(m+1)−1 A change of basis multiply all minors with the determinant of the base change matrix. 3. The Grassmann variety is the projective subset given by the Plücker relations homogeneous of degree 2.