Definitions likely to come up!

Question 1

What is an affine algebraic variety?

Answer 1

Let K be a field and I be an ideal of K.

V(I) = {(a1,a2,…an) ∈ K^n | f(a1,a2,….,an) = 0, for all f ∈ I}

Question 2

K is a field and let f1,…fm ∈ K[x1,…,xn] and I is an ideal of K generated by f1,….,fm. This implies what?

Answer 2

V(I) = {(a1,..,an) ∈ K^n | fsubi(a1,…,an) = 0 for all i ∈ [1,m]}

Question 3

Let X ⊆ A^n. The ideal of X is defined as..?

Answer 3

I(X)={f ∈ K[x1, x2, . . . , xn] | f(a1, a2, . . . , an) = 0, ∀(a1, a2, . . . ,an) ∈ X}

Question 4

If J is an ideal of K[x1, x2, . . . , xn], then… ?

Answer 4

J ⊆ I(V(J)), where V(J) is the affine algebraic variety

Question 5

Let R be a (commutative) ring. Let I be an ideal of R ideal. The radical of I, denoted by √I or rad I, is… ?

Answer 5

√I = {x ∈ R | ∃n such that x^n ∈ I}, I is called a radical ideal if and only if √I = I

Question 6

An affine algebraic variety V is reducible IFF?

Answer 6

If and only if it can be written as V = V1 ∪ V2 where V1 not equal to V which is not equal to V2. V1 and V2 are aav’s. If V is not reducible it is irreducible.

Question 7

Let R be a commutative ring. An ideal I of R is called a prime ideal if and only if … ?

Answer 7

if and only if ab ∈ I implies a ∈ I or b ∈ I.

Question 8

An affine algebraic variety W is irreducible iff?

Answer 8

if and only if I(W) is prime.

Question 9

Let V ⊆ A^n(K) be an affine algebraic variety. Its co-ordinate ring is defined?

Answer 9

K[V ] = K[x1, x2, . . . , xn]/I(V ).

Question 10

Let V ⊆ A^m(K) and W ⊆ A^n(K) be affine algebraic varieties. A morphism ϕ : V → W is a function of the form?

Answer 10

ϕ(P) = (ϕ1(P), ϕ2(P), . . . , ϕn(P)) with ϕi ∈ K[V ] for each i, 1 ≤ i ≤ n

Question 11

The morphism ϕ : V → W is called an isomorphism if and only if?

Answer 11

If it has an inverse morphism, i. e., if there exists a morphism ψ : W → V such that ψ ◦ ϕ = idV and ϕ ◦ ψ = idW. V and W are called isomorphic if and only if there exists an isomorphism between them.

Question 12

Let V be an irreducible affine algebraic variety. The function field of V , denoted by K(V ), is?

Answer 12

Is the set of fractions f/g f, g ∈ K[V ], g not equal to 0, considering two such fractions f1/g1 and f2/g2 equal if and only if f1g2 = f2g1 in K[V ]. The elements of this set are known as rational functions.

Question 13

Let V ⊆ A^m(K) and W ⊆ A^n(K) be affine algebraic varieties and let V be irreducible. A rational map ϕ : V –> W is a function defined ?

Answer 13

A rational map ϕ : V –> W is a function defined on a non-empty subset of V given by ϕ(P) = (ϕ1(P), ϕ2(P), . . . , ϕn(P)) with ϕi ∈ K(V ) for each i, 1 ≤ i ≤ n, and such that if ϕ(P) is defined at P, i. e., if ϕi(P) is defined for each i, 1 ≤ i ≤ n, then ϕ(P) ∈ W.

Question 14

A rational map ϕ : V –> W is called a birational equivalence if and only if?

Answer 14

There exists a rational map ψ : W –> V such that ψ ◦ ϕ = idV and ϕ◦ψ = idW. (In this case ϕ and ψ are necessarily dominant.) V and W are BIRATIONALLY EQUIVALENT if and only if there exists a birational equivalence between them. A variety is rational if and only if it is birationally equivalent to A^n for some n.

Question 15

Let P ∈ V be a point and let v ∈ K^n be a non-zero vector. Let l = {P +tv | t ∈ K} be the line through P with direction vector v. Explain what is meant by saying that l is tangent to V at P.

Answer 15

l is said to be tangent to V at P if and only if for every f ∈ I(V), f(P + tv) has a zero of multiplicity at least 2 at t = 0, or is identically 0.

Question 16

Define the tangent space T_pV to V at P.

Answer 16

The tangent space to V at P, denoted by T_pV is the union of the tangent lines to V at P and the point P.

Question 17

What is the dimension of V where V is an aav (in terms of tangent spaces)?

Answer 17

Let V be an irreducible affine algebraic variety. If V does not equal ∅, there exists a proper subvariety of V (possibly ∅) such that dim T_pV is constant outside this subvariety. (In other words, dim T_pV is constant on a non-empty Zariski open subset of V .) This common value is called the dimension of V , denoted by dim V .

Question 18

What is the set of singular points of V? Sometimes denoted the SINGULAR LOCUS or singV?

Answer 18

The points where dim T_pV > dim V

Question 19

Let V be an arbitrary (not necessarily irreducible) affine algebraic variety. The dimension of V , dim V , is ?

Answer 19

The maximum of the dimensions of the irreducible components of V .

Question 20

Let P ∈ V, P is called non-singular if and only if?

Answer 20

dim T_pV is equal to the local dimension

of V at P.

Question 21

P ∈ V, P is called singular if and only if?

Answer 21

dim TP V is greater than the local dimension of V at P. The set of singular points of V is denoted by Sing V , just as in the irreducible case.

Question 22

The n-dimensional projective space over a field K is defined how?

Answer 22

Denoted by P^n(K) or by P^n if the field is understood, is the set of equivalence classes of Kn+1 \ {(0, 0, . . . , 0)} under the equivalence relation (x0, x1, . . . , xn) ∼(λx0, λx1, . . . , λxn) for any λ ∈ K \ {0}. (The points of P^n correspond to lines through the origin in K^(n+1).)

Question 23

What are the homogeneous

co-ordinates on P^n?

Answer 23

The equivalence class of a point (X0, X1, . . . , Xn) ∈ Kn+1 \ {(0, 0, . . . , 0)} is denoted by (X0 : X1 : . . . : Xn). X0, X1, . . . X^n are called homogeneous co-ordinates on P^n.

Question 24

A projective transformation (also called a homography) is ?

Answer 24

A projective transformation (also called a homography) is a rational map Φ : P^n –> P^n for which there exists an invertible (n+ 1)×(n+ 1) matrix A such that Φ(X0 : X1 : . . . : Xn) = (Y0 : Y1 : . . . : Yn), where : (Y0,Y1,…,Yn)^T = A * (X0,X1,…,Xn)^T.

Question 25

What is it to say that two sets V, W ⊆ P^n are projectively equivalent?

Answer 25

Two sets V, W ⊆ P^n are projectively equivalent if and only if there exists a projective transformation Φ : P^n –> P^n such that Φ(V) = W. Projective equivalence is an equivalence relation.

Question 26

Describe without proof how one can define the addition operation on the points
of E to obtain a commutative group in which O is the zero element.

Answer 26

If we let A,B ∈ E. Take the line AB (the tangent line to A if A=B) and let Q be the 3rd point of intersection with E. A + B is the 3rd point of intersection of the line OQ with E. If a line is tangent to E, the ‘3rd point’ is defined using the intersection multiplicities e.g. if A =/ B and the line AB is tangent to E at A then Q=A.

Question 27

What is an elliptic curve ?

Answer 27

An elliptic curve is a projective variety isomorphic to a nonsingular curve of degree 3 in P^2 together with a distinguished point O ∈ E.