10 1. Aﬃne and Projective Varieties closure in the Zariski topology of a principal open set is the whole aﬃne space. Hence, as An C is irreducible, we obtain that principal open sets are irreducible. If V is closed in An C , each polynomial f(X) ∈ C[X1,...,Xn] defines a C-valued function on V . But different polynomials may define the same function. It is clear that we have a 1-1 correspondence between the distinct polynomial functions on V and the residue class ring C[X1,...,Xn]/I(V ). We denote this ring by C[V ] and call it the coordinate ring of V . It is a finitely generated algebra over C and is reduced (i.e. without nonzero nilpotent elements) because I(V ) is a radical ideal. Remark 1.1.16. If V is an aﬃne variety in An C , we can consider in V the Zariski topology induced by the topology in An C . The closed sets in this topology can be defined as V(I) := {P ∈ V : f(P ) = 0, ∀f ∈ I} for an ideal I of C[V ]. If V is irreducible, equivalently if I(V ) is a prime ideal, C[V ] is an integral domain. We can then consider its field of fractions C(V ), which is called function field of V . Elements f ∈ C(V ) are called rational functions on V . Any rational function can be written f = g/h, with g, h ∈ C[V ]. In general, this representation is not unique. We can only give f a well-defined value at a point P if there is a representation f = g/h, with h(P ) = 0. In this case we say that the rational function f is regular at P . The domain of definition of f is defined to be the set dom(f) = {P ∈ V : f is regular at P }. Example 1.1.17. We consider V := V(Y 2 −X3+X) ⊂ A2 C and P = (0, 0) ∈ V . The function X/Y is regular at P as it can be written as Y/(X2 − 1) in C(V ). Proposition 1.1.18. Let V be an irreducible variety. For a rational func- tion f ∈ C(V ), the following statements hold a) dom(f) is open and dense in V . b) dom(f) = V ⇔ f ∈ C[V ]. c) If h ∈ C[V ] and Vh := {P ∈ V : h(P ) = 0}, then dom(f) ⊃ Vh ⇔ f ∈ C[V ][1/h]. Proof. a) For f ∈ C(V ), we consider the ideal of denominators Jf = {h ∈ C[V ] : hf ∈ C[V ]} ⊂ C[V ].

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