Projective Spaces and Varieties

Projective Spaces and Varieties

Dr G ´abor Megyesi

School of Mathematics The University of Manchester

20 March, 2020

G. Megyesi Projective Spaces and Varieties 1 / 11

Projective spaces

Definition

The n-dimensional projective space over a field K , denoted by P

(K ) or by P

if the field is understood, is the set of equivalence classes of K

\ {(0, 0, . . . , 0)} under the equivalence relation

(x

, x

, . . . , x

) ∼ (λx

, λx

, . . . , λx

) for any λ ∈ K \ {0}.

(The points of P

correspond to lines through the origin in K

.) Definition

The equivalence class of a point

(X

, X

, . . . , X

) ∈ K

\ {(0, 0, . . . , 0)} is denoted by (X

: X

: . . . : X

).

X

, X

, . . . X

are called homogeneous co-ordinates on P

. Note: There are n + 1 homogeneous co-ordinates on P

and the homogeneous co-ordinates of a point are not unique, e. g., ( 1 : 2 : 3) = (−1 : −2 : −3) = (2 : 4 : 6) ∈ P

.

G. Megyesi Projective Spaces and Varieties 2 / 11

Projective spaces

Definition

The n-dimensional projective space over a field K , denoted by P

(K ) or by P

if the field is understood, is the set of equivalence classes of K

\ {(0, 0, . . . , 0)} under the equivalence relation

(x

, x

, . . . , x

) ∼ (λx

, λx

, . . . , λx

) for any λ ∈ K \ {0}.

(The points of P

correspond to lines through the origin in K

.) Definition

The equivalence class of a point

(X

, X

, . . . , X

) ∈ K

\ {(0, 0, . . . , 0)} is denoted by (X

: X

: . . . : X

).

X

, X

, . . . X

are called homogeneous co-ordinates on P

. Note: There are n + 1 homogeneous co-ordinates on P

and the homogeneous co-ordinates of a point are not unique, e. g., ( 1 : 2 : 3) = (−1 : −2 : −3) = (2 : 4 : 6) ∈ P

.

G. Megyesi Projective Spaces and Varieties 2 / 11

Projective spaces

Definition

The n-dimensional projective space over a field K , denoted by P

(K ) or by P

if the field is understood, is the set of equivalence classes of K

\ {(0, 0, . . . , 0)} under the equivalence relation

(x

, x

, . . . , x

) ∼ (λx

, λx

, . . . , λx

) for any λ ∈ K \ {0}.

(The points of P

correspond to lines through the origin in K

.)

Definition

The equivalence class of a point

(X

, X

, . . . , X

) ∈ K

\ {(0, 0, . . . , 0)} is denoted by (X

: X

: . . . : X

).

X

, X

, . . . X

are called homogeneous co-ordinates on P

. Note: There are n + 1 homogeneous co-ordinates on P

and the homogeneous co-ordinates of a point are not unique, e. g., ( 1 : 2 : 3) = (−1 : −2 : −3) = (2 : 4 : 6) ∈ P

.

G. Megyesi Projective Spaces and Varieties 2 / 11

Projective spaces

Definition

The n-dimensional projective space over a field K , denoted by P

(K ) or by P

if the field is understood, is the set of equivalence classes of K

\ {(0, 0, . . . , 0)} under the equivalence relation

(x

, x

, . . . , x

) ∼ (λx

, λx

, . . . , λx

) for any λ ∈ K \ {0}.

(The points of P

correspond to lines through the origin in K

.) Definition

The equivalence class of a point

(X

, X

, . . . , X

) ∈ K

\ {(0, 0, . . . , 0)} is denoted by (X

: X

: . . . : X

).

X

, X

, . . . X

are called homogeneous co-ordinates on P

.

Note: There are n + 1 homogeneous co-ordinates on P

and the homogeneous co-ordinates of a point are not unique, e. g., ( 1 : 2 : 3) = (−1 : −2 : −3) = (2 : 4 : 6) ∈ P

.

G. Megyesi Projective Spaces and Varieties 2 / 11

Projective spaces

Definition

The n-dimensional projective space over a field K , denoted by P

(K ) or by P

if the field is understood, is the set of equivalence classes of K

\ {(0, 0, . . . , 0)} under the equivalence relation

(x

, x

, . . . , x

) ∼ (λx

, λx

, . . . , λx

) for any λ ∈ K \ {0}.

(The points of P

correspond to lines through the origin in K

.) Definition

The equivalence class of a point

(X

, X

, . . . , X

) ∈ K

\ {(0, 0, . . . , 0)} is denoted by (X

: X

: . . . : X

).

X

, X

, . . . X

are called homogeneous co-ordinates on P

. Note: There are n + 1 homogeneous co-ordinates on P

and the homogeneous co-ordinates of a point are not unique, e. g., (1 : 2 : 3) = (−1 : −2 : −3) = (2 : 4 : 6) ∈ P

.

G. Megyesi Projective Spaces and Varieties 2 / 11

Projective spaces

Let U

= {(X

: X

: . . . : X

) ∈ P

| X

6= 0}.

(X

: X

: . . . : X

) = (1 : X

/X

: . . . : X

/X

) in U

and X

/X

, X

/X

, . . . , X

/X

can take arbitrary values in K , so the points of U

are in bijection with the points of A

.

The set P

\ U

= {(X

: X

: . . . : X

) ∈ P

| X

= 0} is clearly a copy of P

.

Therefore P

= A

∪ P

as a set.

If n = 1, then P

= A

∪ P

and P

is just a single point, so we often identify P

with K ∪ {∞}.

If n = 2, then P

= A

∪ P

. The extra points correspond to equivalence classes of parallel lines, where those parallel lines intersect.

G. Megyesi Projective Spaces and Varieties 3 / 11

Projective spaces

Let U

= {(X

: X

: . . . : X

) ∈ P

| X

6= 0}.

(X

: X

: . . . : X

) = (1 : X

/X

: . . . : X

/X

) in U

and X

/X

, X

/X

, . . . , X

/X

can take arbitrary values in K , so the points of U

are in bijection with the points of A

.

The set P

\ U

= {(X

: X

: . . . : X

) ∈ P

| X

= 0} is clearly a copy of P

.

Therefore P

= A

∪ P

as a set.

If n = 1, then P

= A

∪ P

and P

is just a single point, so we often identify P

with K ∪ {∞}.

If n = 2, then P

= A

∪ P

. The extra points correspond to equivalence classes of parallel lines, where those parallel lines intersect.

G. Megyesi Projective Spaces and Varieties 3 / 11

Projective spaces

Let U

= {(X

: X

: . . . : X

) ∈ P

| X

6= 0}.

(X

: X

: . . . : X

) = (1 : X

/X

: . . . : X

/X

) in U

and X

/X

, X

/X

, . . . , X

/X

can take arbitrary values in K , so the points of U

are in bijection with the points of A

.

The set P

\ U

= {(X

: X

: . . . : X

) ∈ P

| X

= 0} is clearly a copy of P

.

Therefore P

= A

∪ P

as a set.

If n = 1, then P

= A

∪ P

and P

is just a single point, so we often identify P

with K ∪ {∞}.

If n = 2, then P

= A

∪ P

. The extra points correspond to equivalence classes of parallel lines, where those parallel lines intersect.

G. Megyesi Projective Spaces and Varieties 3 / 11

Projective spaces

Let U

= {(X

: X

: . . . : X

) ∈ P

| X

6= 0}.

(X

: X

: . . . : X

) = (1 : X

/X

: . . . : X

/X

) in U

and X

/X

, X

/X

, . . . , X

/X

can take arbitrary values in K , so the points of U

are in bijection with the points of A

.

The set P

\ U

= {(X

: X

: . . . : X

) ∈ P

| X

= 0} is clearly a copy of P

.

Therefore P

= A

∪ P

as a set.

If n = 1, then P

= A

∪ P

and P

is just a single point, so we often identify P

with K ∪ {∞}.

If n = 2, then P

= A

∪ P

. The extra points correspond to equivalence classes of parallel lines, where those parallel lines intersect.

G. Megyesi Projective Spaces and Varieties 3 / 11

Projective spaces

Let U

= {(X

: X

: . . . : X

) ∈ P

| X

6= 0}.

(X

: X

: . . . : X

) = (1 : X

/X

: . . . : X

/X

) in U

and X

/X

, X

/X

, . . . , X

/X

can take arbitrary values in K , so the points of U

are in bijection with the points of A

.

The set P

\ U

= {(X

: X

: . . . : X

) ∈ P

| X

= 0} is clearly a copy of P

.

Therefore P

= A

∪ P

as a set.

If n = 1, then P

= A

∪ P

and P

is just a single point, so we often identify P

with K ∪ {∞}.

If n = 2, then P

= A

∪ P

. The extra points correspond to equivalence classes of parallel lines, where those parallel lines intersect.

G. Megyesi Projective Spaces and Varieties 3 / 11

Projective spaces

Let U

= {(X

: X

: . . . : X

) ∈ P

| X

6= 0}.

(X

: X

: . . . : X

) = (1 : X

/X

: . . . : X

/X

) in U

and X

/X

, X

/X

, . . . , X

/X

can take arbitrary values in K , so the points of U

are in bijection with the points of A

.

The set P

\ U

= {(X

: X

: . . . : X

) ∈ P

| X

= 0} is clearly a copy of P

.

Therefore P

= A

∪ P

as a set.

If n = 1, then P

= A

∪ P

and P

is just a single point, so we often identify P

with K ∪ {∞}.

If n = 2, then P

= A

∪ P

. The extra points correspond to equivalence classes of parallel lines, where those parallel lines intersect.

G. Megyesi Projective Spaces and Varieties 3 / 11

Projective spaces

Let U

= {(X

: X

: . . . : X

) ∈ P

| X

6= 0}.

(X

: X

: . . . : X

) = (1 : X

/X

: . . . : X

/X

) in U

and X

/X

, X

/X

, . . . , X

/X

can take arbitrary values in K , so the points of U

are in bijection with the points of A

.

The set P

\ U

= {(X

: X

: . . . : X

) ∈ P

| X

= 0} is clearly a copy of P

.

Therefore P

= A

∪ P

as a set.

If n = 1, then P

= A

∪ P

and P

is just a single point, so we often identify P

with K ∪ {∞}.

If n = 2, then P

= A

∪ P

. The extra points correspond to equivalence classes of parallel lines, where those parallel lines intersect.

G. Megyesi Projective Spaces and Varieties 3 / 11

Homogeneous polynomials and varieties

Definition

A polynomial is called homogeneous if and only if all of its terms have the same degree.

Definition

An ideal I / K [X

, X

, . . . , X

] is called homogeneous if and only if it can be generated by homogeneous elements.

This does not mean that all the elements of I are homogeneous. The generators of a homogeneous ideal can have different degrees.

G. Megyesi Projective Spaces and Varieties 4 / 11

Homogeneous polynomials and varieties

Definition

A polynomial is called homogeneous if and only if all of its terms have the same degree.

Definition

An ideal I / K [X

, X

, . . . , X

] is called homogeneous if and only if it can be generated by homogeneous elements.

This does not mean that all the elements of I are homogeneous. The generators of a homogeneous ideal can have different degrees.

G. Megyesi Projective Spaces and Varieties 4 / 11

Homogeneous polynomials and varieties

Definition

A polynomial is called homogeneous if and only if all of its terms have the same degree.

Definition

An ideal I / K [X

, X

, . . . , X

] is called homogeneous if and only if it can be generated by homogeneous elements.

This does not mean that all the elements of I are homogeneous. The generators of a homogeneous ideal can have different degrees.

G. Megyesi Projective Spaces and Varieties 4 / 11

Homogeneous polynomials and varieties

Definition

A polynomial is called homogeneous if and only if all of its terms have the same degree.

Definition

An ideal I / K [X

, X

, . . . , X

] is called homogeneous if and only if it can be generated by homogeneous elements.

This does not mean that all the elements of I are homogeneous.

The generators of a homogeneous ideal can have different degrees.

G. Megyesi Projective Spaces and Varieties 4 / 11

Homogeneous polynomials and varieties

Definition

A polynomial is called homogeneous if and only if all of its terms have the same degree.

Definition

An ideal I / K [X

, X

, . . . , X

] is called homogeneous if and only if it can be generated by homogeneous elements.

This does not mean that all the elements of I are homogeneous.

The generators of a homogeneous ideal can have different degrees.

G. Megyesi Projective Spaces and Varieties 4 / 11

Definition of projective algebraic varieties

Definition

Let I / K [X

, X

, . . . , X

] be a homogeneous ideal. The projective algebraic variety defined by I is the set

V(I) = {(X

: X

: . . . : X

) ∈ P

|

F (X

, X

, . . . , X

) = 0, ∀F ∈ I, F homogeneous}.

Projective algebraic varieties can be thought of as affine algebraic varieties with some extra points added, which correspond to asymptotic directions of the affine algebraic variety.

G. Megyesi Projective Spaces and Varieties 5 / 11

Definition of projective algebraic varieties

Definition

Let I / K [X

, X

, . . . , X

] be a homogeneous ideal. The projective algebraic variety defined by I is the set

V(I) = {(X

: X

: . . . : X

) ∈ P

|

F (X

, X

, . . . , X

) = 0, ∀F ∈ I, F homogeneous}.

Projective algebraic varieties can be thought of as affine algebraic varieties with some extra points added, which correspond to asymptotic directions of the affine algebraic variety.

G. Megyesi Projective Spaces and Varieties 5 / 11

Another characterisation of homogeneous ideals

Definition

Let F ∈ K [X

, X

, . . . , X

] be a polynomial. The degree i homogeneous part of F , denoted by F

, is the sum of all the terms of degree i in F . If i < 0 or i > deg F , F

is defined to be 0.

Lemma 4.1

The ideal I / K [X

, X

, . . . , X

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

G. Megyesi Projective Spaces and Varieties 6 / 11

Another characterisation of homogeneous ideals

Definition

Let F ∈ K [X

, X

, . . . , X

] be a polynomial. The degree i homogeneous part of F , denoted by F

, is the sum of all the terms of degree i in F . If i < 0 or i > deg F , F

is defined to be 0.

Lemma 4.1

The ideal I / K [X

, X

, . . . , X

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

G. Megyesi Projective Spaces and Varieties 6 / 11

Constructing projective varieties

Proposition 4.2 (Cf. Proposition 1.3)

(i) Let V

= V(I

), V

= V(I

), . . . , V

= V(I

) be projective algebraic varieties in P

. Then

V

∪ V

∪ . . . ∪ V

= V(I

∩ I

∩ . . . ∩ I

) = V(I

I

. . . I

) is also a projective algebraic variety.

(ii) Let V

= V(I

), α ∈ A, be projective algebraic varieties in P

. Then

\

α∈A

V

= V X

α∈A

I

is also a projective algebraic variety.

There is no direct analogue of Proposition 1.3 (iii) for projective varieties because P

× P

is very different from P

.

G. Megyesi Projective Spaces and Varieties 7 / 11

Constructing projective varieties

Proposition 4.2 (Cf. Proposition 1.3)

(i) Let V

= V(I

), V

= V(I

), . . . , V

= V(I

) be projective algebraic varieties in P

. Then

V

∪ V

∪ . . . ∪ V

= V(I

∩ I

∩ . . . ∩ I

) = V(I

I

. . . I

) is also a projective algebraic variety.

(ii) Let V

= V(I

), α ∈ A, be projective algebraic varieties in P

. Then

\

α∈A

V

= V X

α∈A

I

is also a projective algebraic variety.

There is no direct analogue of Proposition 1.3 (iii) for projective varieties because P

× P

is very different from P

.

G. Megyesi Projective Spaces and Varieties 7 / 11

Nullstellensatz

Definition

The homogeneous ideal of a set Z ⊆ P

is the ideal I(Z ) / K [X

, X

, . . . , X

] generated by the set

{F ∈ K [X

, X

, . . . , X

] |

F homogeneous, F (Z

, . . . , Z

) = 0 ∀(Z

: . . . : Z

) ∈ Z }.

Theorem 4.3

(Projective Nullstellensatz, cf. Theorem 1.7)

Let K be an algebraically closed field and let J / K [X

, X

, . . . , X

]. (i) V(J) = ∅ if and only if J = K [X

, X

, . . . , X

] or

√ J = hX

, X

, . . . , X

i. (ii) I(V(J)) = √

J unless √

J = hX

, X

, . . . , X

i.

G. Megyesi Projective Spaces and Varieties 8 / 11

Nullstellensatz

Definition

The homogeneous ideal of a set Z ⊆ P

is the ideal I(Z ) / K [X

, X

, . . . , X

] generated by the set

{F ∈ K [X

, X

, . . . , X

] |

F homogeneous, F (Z

, . . . , Z

) = 0 ∀(Z

: . . . : Z

) ∈ Z }.

Theorem 4.3

(Projective Nullstellensatz, cf. Theorem 1.7)

Let K be an algebraically closed field and let J / K [X

, X

, . . . , X

].

(i) V(J) = ∅ if and only if J = K [X

, X

, . . . , X

] or

√ J = hX

, X

, . . . , X

i.

(ii) I(V(J)) = √

J unless √

J = hX

, X

, . . . , X

i.

G. Megyesi Projective Spaces and Varieties 8 / 11

Irreducibility

Definition

A projective algebraic variety V is reducible if and only if it can be written as V = V

∪ V

, where V

, V

are also projective algebraic varieties, V

6= V 6= V

. If V is not reducible, it is called irreducible.

Proposition 4.4 (Cf. Theorem 1.8)

Every projective algebraic variety V can be decomposed into a union V = V

∪ V

∪ . . . ∪ V

such that every V

, 1 ≤ i ≤ k , is an irreducible projective algebraic variety and V

6⊆ V

for i 6= j. The decomposition is unique up to the ordering of the components. The V

, 1 ≤ i ≤ k , are called the irreducible components of V .

G. Megyesi Projective Spaces and Varieties 9 / 11

Irreducibility

Definition

A projective algebraic variety V is reducible if and only if it can be written as V = V

∪ V

, where V

, V

are also projective algebraic varieties, V

6= V 6= V

. If V is not reducible, it is called irreducible.

Proposition 4.4 (Cf. Theorem 1.8)

Every projective algebraic variety V can be decomposed into a union V = V

∪ V

∪ . . . ∪ V

such that every V

, 1 ≤ i ≤ k , is an irreducible projective algebraic variety and V

6⊆ V

for i 6= j. The decomposition is unique up to the ordering of the components. The V

, 1 ≤ i ≤ k , are called the irreducible components of V .

G. Megyesi Projective Spaces and Varieties 9 / 11

Characterisation of irreducibility

Proposition 4.5

A homogeneous ideal I / K [X

, X

, . . . , X

] is prime if and only if for any homogeneous polynomials F , G ∈ K [X

, X

, . . . , X

], FG ∈ I implies F ∈ I or G ∈ I.

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 Spaces and Varieties 10 / 11

Characterisation of irreducibility

Proposition 4.5

A homogeneous ideal I / K [X

, X

, . . . , X

] is prime if and only if for any homogeneous polynomials F , G ∈ K [X

, X

, . . . , X

], FG ∈ I implies F ∈ I or G ∈ I.

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 Spaces and Varieties 10 / 11

Tangent spaces, dimension, singularity

Given a point a P of a projective variety V , there exists an affine piece V

containing P. The tangent space T

V can be defined as the

projective closure of T

V

, and then it can be used to define dim V and Sing V .

dim V and Sing V can be calculated by using the Jacobian similarly to the affine case without having to consider affine pieces.

G. Megyesi Projective Spaces and Varieties 11 / 11

Tangent spaces, dimension, singularity

Given a point a P of a projective variety V , there exists an affine piece V

containing P. The tangent space T

V can be defined as the

projective closure of T

V

, and then it can be used to define dim V and Sing V .

dim V and Sing V can be calculated by using the Jacobian similarly to the affine case without having to consider affine pieces.

G. Megyesi Projective Spaces and Varieties 11 / 11