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Summary of Projective Varieties Definition of Projective Algebraic Definition of projective algebraic varieties Definition An ideal I / K [X0; X1;:::; Xn] is called homogeneous if and only if it can Summary of Projective Varieties be generated by homogeneous elements. This does not mean that all the elements of I are homogeneous. Dr Gabor´ Megyesi Definition School of Mathematics Let I / K [X0; X1;:::; Xn] be a homogeneous ideal. The projective The University of Manchester algebraic variety defined by I is the set n 22 March, 2019 V(I) = f(X0 : X1 : ::: : Xn) 2 P j F(X0; X1;:::; Xn) = 0; 8F 2 I; F homogeneousg: G. Megyesi Projective Varieties 1 / 12 G. Megyesi Projective Varieties 2 / 12 Another characterisation of homogeneous ideals Constructing projective varieties Definition Proposition 4.2 Let F 2 K [X0; X1;:::; Xn] be a polynomial. The degree i homogeneous (Cf. Proposition 1.3) part of F, denoted by F[i], is the sum of all the terms of degree i in F. If (i) Let V1 = V(I1),V2 = V(I2),...,Vk = V(Ik ) be projective algebraic n i < 0 or i > deg F, F[i] is defined to be 0. varieties in P . Then Lemma 4.1 V1 [ V2 [ ::: [ Vk = V(I1 \ I2 \ ::: \ Ik ) = V(I1I2 ::: Ik ) The ideal I / K [X0; X1;:::; Xn] is homogeneous if and only if for any is also a projective algebraic variety. 2 n F I, all the homogeneous parts of F are also elements of I. (ii) Let Vα = V(Iα), α 2 A, be projective algebraic varieties in P . Then \ X Vα = V Iα α2A α2A is also a projective algebraic variety. There is no direct analogue of Proposition 1.3 (iii) for projective + varieties because Pm × Pn is very different from Pm n. G. Megyesi Projective Varieties 3 / 12 G. Megyesi Projective Varieties 4 / 12 Nullstellensatz Irreducibility Definition Definition The homogeneous ideal of a set Z ⊆ Pn is the ideal A projective algebraic variety V is reducible if and only if it can be I(Z ) / K [X0; X1;:::; Xn] generated by the set written as V = V1 [ V2, where V1, V2 are also projective algebraic varieties, V1 6= V 6= V2. If V is not reducible, it is called irreducible. fF 2 K [X0; X1;:::; Xn] j F homogeneous, F(X0;:::; Xn) = 0 8(X0 : ::: : Xn) 2 Zg: Proposition 4.4 (Cf. Theorem 1.8) Theorem 4.3 Every projective algebraic variety V can be decomposed into a union (Projective Nullstellensatz, cf. Theorem 1.7) V = V1 [ V2 [ ::: [ Vk such that every Vi , 1 ≤ i ≤ k, is an irreducible Let K be an algebraically closed field and let J / K [X0; X1;:::; Xn]. projective algebraic variety and Vi 6⊆ Vj for i 6= j. The decomposition is p(i) V(J) = ; if and only if J = K [X0; X1;:::; Xn] or unique up to the ordering of the components. The Vi , 1 ≤ i ≤ k, are J = hX0; X1;:::;p Xni. p called the irreducible components of V . (ii) I(V(J)) = J unless J = hX0; X1;:::; Xni. G. Megyesi Projective Varieties 5 / 12 G. Megyesi Projective Varieties 6 / 12 Characterisation of irreducibility Tangent spaces, dimension, singularity Proposition 4.5 Given a point a P of a projective variety V , there exists an affine piece Vi containing P. The tangent space TP V can be defined as the A homogeneous ideal I / K [X0; X1;:::; Xn] is prime if and only if for any projective closure of TP Vi , and then it can be used to define dim V and homogeneous polynomials F, G 2 K [X0; X1;:::; Xn], FG 2 I implies Sing V . F 2 I or G 2 I. dim V and Sing V can be calculated by using the Jacobian similarly to the affine case without having to consider affine pieces. Proposition 4.6 (Cf. Proposition 1.9) A projective algebraic variety V is irreducible if and only if I(V ) is prime. G. Megyesi Projective Varieties 7 / 12 G. Megyesi Projective Varieties 8 / 12 Homogeneous co-ordinate rings and function fields Rational maps and morphism Definition Definition The homogeneous co-ordinate ring of a projective variety V ⊆ Pn is Let V ⊆ Pm(K ) and W ⊆ Pn(K ) be projective algebraic varieties and K [V ] = K [X0; X1;:::; Xn]=I(V ). If V is irreducible, the function field of let V be irreducible. A rational map Φ: V 99K W is a partial function V , K (V ), is the set of equivalence classes of fractions F=G, where F, defined by an equivalence class of (n + 1)-tuples of homogeneous G 2 K [V ], G 6= 0, F, G are homogeneous of the same degree, and two elements of K [V ] of the same degree, fractions F1=G1, F2=G2 are equivalent if and only if F1G2 = F2G1 in (Φ0 :Φ1 : ::: :Φn) ∼ (Ψ0 :Ψ1 : ::: :Ψn) if and only if Φi Ψj = Φj Ψi for K [V ]. every i, j, 0 ≤ i; j ≤ n. Φ: V 99K W is defined at P 2 V if and only if Φ can be represented by Unlike in the affine case, the elements of K [V ] are not functions on V , (Φ0 :Φ1 : ::: :Φn) such that Φi (P) 6= 0 for some i, 0 ≤ i ≤ n, in this they cannot be evaluated at points of V . case we require that Φ(P) = (Φ0(P):Φ1(P): ::: :Φn(P)) 2 W . The elements of K (V ) are partial functions on V , just like in the affine case. Definition A rational map Φ: V 99K W is a morphism, denoted by Φ: V ! W , if and only if it is defined at every point of V . G. Megyesi Projective Varieties 9 / 12 G. Megyesi Projective Varieties 10 / 12 Theorem 4.7 Corollary 4.8 m n (Cf. Theorem 2.4) Let V ⊆ P (K ) and W ⊆ P (K ) be irreducible (Cf. Corollary 2.5) Φ: V 99K W is a birational equvalence if and only if ∗ projective algebraic varieties. Let (Y0 : Y1 : Y2 : ::: : Yn) homogeneous Φ : K (W ) ! K (V ) is an isomorphism of fields. co-ordinates on Pn and assume that W does not lie in the hyperplane V and W are birationally equivalent if and only if K (V ) and K (W ) are Y0 = 0. There exists a bijection between dominant rational maps isomorphic as extensions of K . ∼ φ : V 99K W and field homomorphisms α : K (W ) ! K (V ) preserving A variety V is rational if and only if K (V ) = K (t1; t2;:::; tk ) for some k. K given by the following constructions: for a rational map Φ: V 99K W, define Φ∗ : K (W ) ! K (V ), Φ∗(f ) = f ◦ Φ, and for a homomorphism There is no analogue of Theorem 2.1 and Corollary 2.2 for projective α : K (W ) ! K (V ) preserving K , define the corresponding rational varieties, there is no correspondence between morphisms of projective map by (1 : α(Y1=Y0); α(Y2=Y0); : : : ; α(Yn=Y0)). varieties and ring homomorphisms between the homogeneous co-ordinate rings. G. Megyesi Projective Varieties 11 / 12 G. Megyesi Projective Varieties 12 / 12.
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