Chapter 4

Question 1

The n-dimensional projective space over a field K, denoted byP^n(K) or by P^n if the field is understood

Answer 1

is the set of equivalence classes
of K^n+1 \ {(0, 0, . . . , 0)} under the equivalence relation (x0, x1, . . . , xn) ∼ (λx0, λx1, . . . , λxn) for any λ ∈ K \ {0}.

Question 2

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

Answer 2

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

Question 3

The ideal I / K[X0, X1, . . . , Xn] is homogeneous

Answer 3

if and only if for any F ∈ I, all the homogeneous parts of F are also elements of I.

Question 4

V(J) = ∅

Answer 4

if and only if J = K[X0, X1, . . . , Xn] or √J = [X0, X1, . . . , Xn].

Question 5

I(V(J)) = √J

Answer 5

√J = [X0, X1, . . . , Xn]

Question 6

A homogeneous ideal I / K[X0, X1, . . . , Xn] is prime

Answer 6

if and only if for any homogeneous polynomials F, G ∈ K[X0, X1, . . . , Xn], F G ∈ I
implies F ∈ I or G ∈ I.

Question 7

A projective algebraic variety V is irreducible

Answer 7

if and only if I(V ) is prime

Question 8

A projective transformation

Answer 8

is a rationalmap Φ : Pn - Pn for which there exists an invertible (n+ 1)×(n+ 1) matrix A such that Φ(X0 : X1. . : Xn) = (Y0 : Y1 : . . . : Yn),

Question 9

Two sets V, W ⊆ Pn are projectively equivalent

Answer 9

if and only if
there exists a projective transformation Φ : P
n- Pn such that Φ(V ) = W.

Question 10

Let z1,z2, z3, z4 ∈ K ∪ {∞}, with at least 3 of them distinct. The
cross ratio of z1,z2, z3, z4 is defined as

Answer 10

-

Question 11

Let z1, z2, z3, z4 ∈ K ∪ {∞} with at least 3 of them

distinct

Answer 11

Then (z1, z2; z3, z4) = (ϕ(z1), ϕ(z2); ϕ(z3), ϕ(z4)) for any projective
transformation

Question 12

a) Let z1, z2, z3 , w1, w2, w3 be elements K ∪ {∞} such that z1, z2, z3 are pairwise distinct and w1, w2, w3 are also pairwise
distinct.

Answer 12

Then there exists a projective transformation ϕ : P1 → P1 such
that ϕ(zi) = wi for i = 1, 2, 3. In particular, any two sets of 3 distinct points in P1 are projectively equivalent.

Question 13

Any isomorphism P1 (C) → P1 (C)

Answer 13

is a projective transformation