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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 Let I / K [X0, X1,..., Xn] be a homogeneous ideal. The projective The University of Manchester algebraic variety defined by I is the

n 22 March, 2019 V(I) = {(X0 : X1 : ... : Xn) ∈ P | F(X0, X1,..., Xn) = 0, ∀F ∈ I, F homogeneous}.

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 ∈ K [X0, X1,..., Xn] be a . 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. ∈ n F I, all the homogeneous parts of F are also elements of I. (ii) Let Vα = V(Iα), α ∈ A, be projective algebraic varieties in P . Then \ X  Vα = V Iα α∈A α∈A

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. {F ∈ K [X0, X1,..., Xn] |

F homogeneous, F(X0,..., Xn) = 0 ∀(X0 : ... : Xn) ∈ Z}. 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 √(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,...,√ Xni. √ called the irreducible components of V . (ii) I(V(J)) = J unless J = hX0, X1,..., Xni.

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Characterisation of irreducibility Tangent spaces, , singularity

Proposition 4.5 Given a point a P of a V , there exists an affine piece Vi containing P. The 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 F, G ∈ K [X0, X1,..., Xn], FG ∈ I implies Sing V . F ∈ I or G ∈ 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 fields Rational maps and morphism

Definition Definition The homogeneous co-ordinate 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 ∈ 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 ∈ 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)) ∈ 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.

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