1.3. Projective varieties 23 Let V be a closed subset of An. Identify An with one of the aﬃne open subsets Uj of Pn, say U0. Then the closure V of V in Pn is called the projective closure of V . In particular, Pn is the projective closure of An. The homogeneous ideal of V is then {β(g)|g ∈ I(V )}. Since V = U0 ∩ V , V is open in V . Thus V is irreducible if and only if V is so. Example 1.3.12. We consider the Fermat conic V = V(X2+Y 2 −Z2) ⊂ P2 C . Its three aﬃne pieces are - V0 = V(1 + Y 2 − Z2), which is a hyperbola and has two points at infinity, namely V ∩ H0 = {(0 : 1 : 1), (0 : 1 : −1)}. - V1 = V(X2 + 1 − Z2), which is again a hyperbola and has two points at infinity, namely V ∩ H1 = {(1 : 0 : 1), (1 : 0 : −1)}. - V2 = V(X2 + Y 2 − 1), which is a circle and has two points at infinity, namely V ∩ H2 = {(1 : i : 0), (1 : −i : 0)}, called the circular points at infinity, since they are the intersection of any circle in the aﬃne plane with the line at infinity. Example 1.3.13. Let V = V(Z − Y 2 , T − Y 2 ) in A3 C identified with U0. Its projective closure is V = V ∪ {(0 : 0 : z : t) ∈ P3}. The homogeneous ideal of V is not generated by the homogenizations XZ − Y 2 and XT − Y 2 of the generators of the ideal of V as Z − T is a homogeneous polynomial in I(V ) which is not in XZ − Y 2 , XT − Y 2 . Example 1.3.14. Let V = V(Y 2 − X3 − a X − b) in A2 C identified with U2. Its projective closure is V = V ∪ {(0 : 1 : 0)} hence V has only one point at infinity. We shall now see how to define functions on projective varieties. If f, g ∈ C[X0,X1,...,Xn], then F := f/g can be seen as a function on Pn (defined at the points where g does not vanish) only if f and g are homogeneous polynomials of the same degree, in which case we refer to F as a rational function of degree 0. If g(P ) = 0 for some point P ∈ Pn, we say that F is regular at P . Note that if a rational function of degree 0 is regular in some point, then it is regular in a neighborhood of this point. If U is a subset of a projective variety V ⊂ Pn, a function F : U → C is regular if for any point P in U there exists an open neighborhood U of P and homogeneous polynomials f, g ∈ C[X0,X1,...,Xn] of the same degree such that F = f/g in U and g(P ) = 0. If U is an open set, we write OV (U) for the set of all regular functions on U. Let U be an open subset of Pn contained in some of the aﬃne open sets Ui = {(x0 : x1 : · · · : xn) ∈ Pn : xi = 0}. Then U is also open in Ui, which is canonically identified with An. We claim that OPn(U) = OAn(U). We take, for example, i = 0. If F ∈ OPn(U), for each P ∈ U

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