4 Projective space and projective varieties

4 Projective space and projective varieties

One of the motivations for projective geometry is to understand perspective.

It is well-known phenomenon that parallel lines such as railway tracks ap- pear to meet at a point, called the vanishing point. It was not until the Renaissance that painters were able to give a geometrically correct represen- tation of objects. The painting below, The delivery of keys by Pietro Perug- ino in the Sistine Chapel (source https://en.wikipedia.org/wiki/File:

Entrega_de_las_llaves_a_San_Pedro_(Perugino).jpg) is a good exam- ple of the use of perspective, the parallel lines on the ground converge towards a point near the centre of the painting.

Projective geometry gives a nicer theory than affine geometry in many ways, for example, two lines in the projective plane always meet, or more generally two curves defined by degree m and degree n equations, resp., have mn intersection points counted with multiplicity.

Projective geometry also has important practical applications, images are

combined by applying projective transformations in photo stitching, which

is used in panoramic photos, Google Street View and virtual reality.

4.1 Basic properties of projective space and projective varieties

The definition of projective space is also motivated by the idea that if a camera is at the origin of the co-ordinate system, all points of a ray through the origin will be mapped to the same point of the image, so points of the image correspond to lines through the origin.

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

.)

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

.

Example: (1 : 2 : 3), (2 : 4 : 6) and (−1 : −2 : −3) are the same point in P

, this is a peculiarity of projective space that the co-ordinates of a point are not unique.

Two ways of looking at projective space

1. 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

U

= {(X

: X

) ∈ P

| X

6= 0} = { 1 : X

X

| X

∈ K, X

6= 0} = {(1 : x) | x ∈ K}, so we can identify U

with A

via (X

: X

) ↔ X

/X

∈ A

.

P

\ U

= {(X

: X

) ∈ P

| X

= 0} = {(0 : X

) | X

∈ K \ {0}} = {(0 : 1)}, because (0 : X

) = (0 : 1) for any K \ {0}. This is the decomposition, P

= A

∪ P

, as P

consists of just a single point.

We can identify A

with K and denote the single point of P

by ∞, this way

we can identify P

with K ∪ {∞}.

If n = 2, then

U

= {(X

: X

: X

) ∈ P

| X

6= 0} = { 1 : X

X

: X

X

| X

, X

∈ K, X

6= 0}

= {(1 : x : y) | x, y ∈ K}

so we can identify U

with A

via (X

: X

: X

) ↔ (X

/X

, X

/X

) ∈ A

. The complement of U

is

P

\U

= {(X

: X

: X

) ∈ P

| X

= 0} = {(0 : X

: X

) | (X

, X

) ∈ K

\{(0, 0)}}, this is a copy of P

.

To understand these points, consider the parallel lines x+y = 0 and x+y−2 = 0 in A

= U

. As x = X

/X

and y = X

/X

, we can rewrite the equations in terms of homogeneous co-ordinates as X

X

+ X

X

= 0 and X

X

+ X

X

− 2 = 0.

After multiplying by X

we get X

+ X

= 0 and X

+ X

− 2X

= 0. The solutions of this system of linear equations are X

= 0, X

= −X

, which correspond to the point (0 : 1 : −1) ∈ P

. Therefore these parallel lines in A

= U

intersect in P

\ U

. Any other line parallel to them also contains (0 : 1 : −1).

x+y=0 x+y-2=0 H0:1:-1L

-4 -2 2 4x

-4 -2 2 4 y

Similarly, all lines with direction vector (X

, X

) intersect at (0 : X

: X

) ∈

P

\ U

.

This is true in general for arbitrary n, the points of P

\U

= P

correspond to directions of lines in A

or equivalence classes of parallel lines. (0 : X

: . . . : X

) is the point where all the lines with direction vector (X

, X

, . . . , X

) in A

meet. (The single point of P

corresponds to the only line in A

.) For this reason, these points are often called points at infinity, but this does not mean that they are intrinsically different, the disctinction depends on the choice of co-ordinates.

The youtube video https://www.youtube.com/watch?v=q3turHmOWq4 (also linked to from the Animations and videos section of the course website) contains a good explanation of the projective plane.

2. The other approach is is to consider the sets

U

= {(X

: X

: . . . : X

) ∈ P

| X

6= 0},

0 ≤ i ≤ n. Each one is a copy of A

and their union is P

, since there is no point (0 : 0 : . . . : 0). These n + 1 copies are glued together by identifying their points via rational maps ϕ

: U

99K U

so that P ∈ U

is the same point of P

as ϕ

(P ) ∈ U

.

For example, if n = 1, let t = X

/X

be the co-ordinate on U

, and u = X

/X

the co-ordinate on U

. We have the rational maps ϕ

: U

99K U

, t 7→ 1/t, and ϕ

: U

99K U

, u 7→ 1/u. A point t ∈ U

is the same point in P

as ϕ

(t) ∈ U

, whenever ϕ

(t) is defined and similarly, a point u ∈ U

is the same point in P

as ϕ

(u) ∈ U

, whenever ϕ

(u) is defined.

If n = 2, then let x = X

/X

, y = X

/X

be the co-ordinates on U

, u = X

/X

, v = X

/X

the co-ordinates on U

and s = X

/X

, t = X

/X

the co-ordinates on U

. Now we have u = 1/x and v = y/x, which give the rational map ϕ

: U

99K U

, (x, y) 7→ (1/x, y/x), so that (x, y) ∈ U

and ϕ

(x, y) ∈ U

are the same point in P

whenever the latter is defined.

The point (2 : 3 : 4) ∈ P

is the same as (1 : 3/2 : 2), (2/3 : 1 : 4/3) or (1/2 : 3/4 : 1), therefore (3/2, 2) ∈ U

, (2/3, 4/3) ∈ U

and (1/2, 3/4) ∈ U

correspond to the same point of P

and they get identified in the gluing process.

In general, if F ∈ K[X

, X

, . . . , X

] and P = (X

: X

: . . . : X

) ∈ P

, F (P ) cannot be defined, since F (λX

, λX

, . . . , λX

) will give different values for different values of λ ∈ K \ {0}. However, if F is homogeneous of degree d, then F (λX

, λX

, . . . , λX

) = λ

F (X

, X

, . . . , X

), so we can tell whether F (P ) = 0 or not, and this is enough to define projective algebraic varieties.

Definition. A polynomial is called homegeneous if and only if all of its terms

have the same degree.

E. g., x

+ 2y

z − 3xz

is homogeneous of degree 3, while x

− y

is not homogeneous.

Definition. An ideal I / K[X

, X

, . . . , X

] is called homogeneous if and only if it can be generated by homogeneous elements. (Warning: This does not mean that all elements of the ideal are homogeneous polynomials.)

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}.

Two ways of looking at projective varieties

1. Let’s consider the variety V = V(hX

X

− X

i) ⊂ P

. If X

6= 0, we can divide X

X

− X

= 0 by X

to get (X

/X

)(X

/X

) − 1 = 0, which can be written as xy − 1 = 0 in terms of the affine co-ordinates x = X

/X

and y = X

/X

on U

= A

, so V

= V ∩ U

is a hyperbola. The points V \ V

are the points of V with X

= 0. By substituting X

= 0 into X

X

− X

= 0 we obtain X

X

= 0, therefore the points at infinity are (0 : 1 : 0), which corresponds to lines parallel to the x-axis, and (0 : 0 : 1), which corresponds to lines parallel to the y-axis. The asymptotes of the hyperbola pass through these points at infinity. V consists of the hyperbola with two points at infinity corresponding to the asymptotes.

xy-1=0 H0:0:1L

H0:1:0L

-4 -2 2 4x

-4 -2 2 4 y

Starting with the equation xy − 1 = 0 we can recover X

X

− X

= 0 by

substituting x = X

/X

and y = X

/X

into xy − 1 = 0 and multiplying it

by X

.

Let’s now consider the variety V = V(hX

− X

X

i) ⊂ P

. If X

6= 0, we can divide X

− X

X

= 0 by X

to get (X

/X

)

− (X

/X

) = 0, which can be written as x

− y = 0 in terms of the affine co-ordinates x = X

/X

and y = X

/X

on U

= A

, so V

= V ∩ U

is a parabola. The points V \ V

are the points of V with X

= 0. By substituting X

= 0 into X

− X

X

= 0 we obtain X

= 0, therefore the only points at infinity is (0 : 0 : 1), which corresponds to lines parallel to the y-axis. Unlike the hyperbola, the parabola has no asympotes, but for large x and y, its tangent direction approaches the direction of the y-axis.

x²-y=0 (0:0:1)

-3 -2 -1 1 2 3 x

2 4 6 8 y

Starting with the equation x

− y = 0 we can recover X

− X

X

= 0 by substituting x = X

/X

and y = X

/X

into x

− y = 0 and multiplying it by X

.

Given a homogeneous polynomial F ∈ K[X

, X

, . . . , X

], F [X

, X

, . . . , X

]

X

= F [1, x

, x

, . . . , x

]

where x

= X

/X

, 1 ≤ i ≤ n, is called the dehomogenisation of F with respect to X

. If we dehomogenise all homogeneous elements of an homo- geneous ideal J / K[X

, X

, . . . , X

], we obtain an ideal defining the affine algebraic variety V

= V ∩ U

⊆ A

, called the affine piece X

6= 0 of V = V(J ) ⊆ P

.

If we start with a polynomial f (x

, x

, . . . , x

),

X

f (X

/X

, X

/X

, . . . , X

/X

) ∈ K[X

, X

, . . . , X

]

is a homogeneous polynomial, the homogenisation of f . Homogenisation can also be done at the level of ideals, the projective algebraic variety in P

defined by the ideal generated by the homogenisation of the elements of an ideal J / K[x

, x

, . . . , x

] is called the projective closure of V(J ) ⊆ A

. The affine piece X

6= 0 of the projective closure is exactly V(J), while the points at infinity correspond to asymptotic directions of V(J ) as we have seen in the example.

Homogenisation and dehomogenisation are one-sided inverses of each other.

Homogenisation followed by dehomogenisation always yields the same poly- nomial, and similarly for varieties, taking the projective closure of an affine variety and then the affine piece X

6= 0 gives the original variety.

Dehomogenising and then homogenising a homogeneous polynomial will give the polynomial divided by the highest power of X

dividing it. (For example, the dehomogenisation of X

X

is x

, whose homogenisation is just X

.) For varieties this means that taking projective closure of the affine piece X

6= 0 of a projective algebraic variety will give the union of irreducible components of the original variety not contained in the hyperplane X

= 0.

(Irreducibility for projective varieties will be defined later, but it is completely analogous to the affine case.)

If a projective variety V has no irreducible components contained in the hyperplane X

= 0, then V is the projective closure the affine piece V

= V ∩ U

and the points of V correspond to the points of V

and the asympotic directions of V

.

2. The other approach is to consider all the affine pieces V

= V ∩ U

, 0 ≤ i ≤ n, of a projective variety V . These are affine varieties and they are glued together by identifying the points via the rational maps ϕ

: U

99K U

defined previously. For example, by homogenising the equation y

−x

−x

= 0 of the nodal cubic in A

, we obtain X

X

−X

−X

X

= 0. The projective curve defined by this equation is the projective closure V , whose affine piece V

= V ∩ U

is original affine nodal cubic curve. Substituting X

= 0 into the equation gives X

= 0, therefore the only point with X

= 0 is (0 : 0 : 1), the asymptotic direction of the y-axis.

x and y can be expressed as x = X

/X

, y = X

/X

in terms of the homo-

geneous co-ordinates. The affine co-ordinates on the other affine pieces are

u = X

/X

, v = X

/X

on U

and s = X

/X

, t = X

/X

on U

. Deho-

mogenising X

X

− X

− X

X

= 0 with respect to X

gives uv

− 1 − u = 0,

dehomogenising it with respect to X

gives s − t

− st

= 0. Therefore the

projective curve has the affine pieces shown below.

y

− x

− x

= 0 uv

− 1 − u = 0 s − t

− st

= 0

-10 -5 5 10 x

-10 -5 5 10 y

-10 -5 5 10 u

-10 -5 5 10 v

-10 -5 5 10 s

-10 -5 5 10 t

They are glued together by the maps ϕ

: U

99K U

, (x, y) 7→ (1/x, y/x) and ϕ

: U

99K U

, (x, y) 7→ (1/y, x/y).

Definition. Let F ∈ K[X

, X

, . . . , X

] be a polynomial. The degree i ho- mogeneous 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.

It is clear from this definition that F = F

+ F

+ . . . + F

.

Example: Let F = X

+X

X

−X

+X

X

X

∈ K[X

, X

, X

], then F

= 0, F

= X

, F

= X

X

− X

and F

= X

X

X

.

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.

Proof. Assume that I is homogeneous, let F

, F

, . . . , F

be a set of ho- mogeneous generators for I. Let F ∈ I, then F =

r

P

i=1

F

G

for some G

∈ K[X

, X

, . . . , X

], 1 ≤ i ≤ r. Then F

=

r

P

i=1

F

G

∈ I for every d, 0 ≤ d ≤ deg F , so all the homogeneous parts of F are also elements of I.

Assume now that F ∈ I implies that F

, F

, . . . , F

are also in I. Let F

, F

, . . . , F

be an arbitrary set of generators for I. We claim that the set {F

| 1 ≤ i ≤ r, 0 ≤ j ≤ deg F

} generates I. On one hand F

=

deg Fi

P

j=0

F

is contained in the ideal generated by this set for each i, 1 ≤ i ≤ r, so this ideal contains I. On the other hand, each of the generators is in I, so the ideal generated by them is a subset of I. Therefore the ideal generated by this set is exactly I and since these generators are homogeneous, I is a homogeneous ideal.

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.

Proof. Just imitate the proof of Proposition 1.3 (i) and (ii).

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

× P

is very different from P

.

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 (X

, . . . , X

) = 0 ∀(X

: . . . : X

) ∈ Z}.

Theorem 4.3 (Projective Nullstellensatz, cf. Theorem 1.7)

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

, X

, . . . , X

] be a homogeneous ideal.

(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.

Idea of proof. For any homogeneous ideal J / K[X

, X

, . . . , X

] we can consider the projective algebraic variety V(J ) defined by in P

and also the affine algebraic variety defined by J in A

, called affine cone on V(J ).

This gives an almost bijective correspondence between projective algebraic varieties in P

and certain affine algebraic varieties in A

, the only failure of bijectivity is that K[X

, X

, . . . , X

] and hX

, X

, . . . , X

i both define the empty set as a projective variety.

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 .

Proof. Completely analogous to the proof of Theorem 1.8.

Lemma 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

], F G ∈ I implies F ∈ I or G ∈ I.

Proof. If I is prime then F G ∈ I implies F ∈ I or G ∈ I for all F, G ∈ K[X

, X

, . . . , X

] by the definition of a prime ideal.

Let’s assume I is a homogeneous ideal which is not prime. Then there exist polynomials P / ∈ I, Q / ∈ I such that P Q ∈ I. Let d ≥ 0 be the minimal integer such that P

∈ I. There exists such a d since if P /

∈ I for every j ≥ 0, then P =

deg P

P

j=0

P

∈ I, too. Similarly, let e ≥ 0 be the minimal integer such that Q

∈ I. Then (P Q) /

=

d+e

P

j=0

P

Q

. If j < d, then P

∈ I, so P

Q

∈ I, too. If j > d, then then Q

∈ I as d + e − j < e, so P

Q

∈ I, too. Therefore all the terms in the sum except for P

Q

are in I, (P Q)

∈ I by Lemma 4.1 since P Q ∈ I and I is homogeneous.

Hence P

Q

∈ I, too, but P

∈ I and Q /

∈ I. Therefore F = P /

and G = Q

are homogeneous polynomials with the property that F, G / ∈ I but F G ∈ I.

Proposition 4.6 (Cf. Proposition 1.9) A projective algebraic variety V is irreducible if and only if I(V ) is prime.

Proof. Imitate the proof of Proposition 1.9 and use Lemma 4.5.

4.2 Tangent spaces, dimension and singularities

There are two possible approaches.

1. One can define tangent lines and imitate the development of the affine theory. The line through the points P 6= Q consists of the points λP + µQ, where (λ : µ) ∈ P

, and this line is tangent to V at P if and only if for any homogeneous polynomial F ∈ I(V ), F (λP + µQ) contains no degree 1 term in µ.

2. Let V ⊆ P

be an irreducible projective algebraic variety, and let P ∈ V . Let (X

: X

: X

: . . . , X

) be homogeneous co-ordinates on P

. Choose i such that P is not contained in the hyperplane X

= 0. Let V

= {(X

: X

: . . . : X

) ∈ V | X

6= 0} be the affine piece X

6= 0 of V . The tangent space T

V is defined as the projective closure of T

V

, the local dimension of V at P is the local dimension of V

at P , and the P is a singular point of V if and only if it is a singular point of V

. (It needs to be proved that these definitions are independent of the choice of i, which can be done by an improved version of Theorem 3.5.)

Tangent spaces, dimension and singular points of projective varieties can be calculated by using the Jacobian matrix in the same way as in the affine case.

Example: Find the singular points, if any, of the curve C defined by the equation Y

Z − X

− X

Z = 0 in P

(C).

Let F = Y

Z − X

− X

Z, then F

= 3X

− 2XZ, F

= 2Y Z and F

= Y

− X

. The rank of the Jacobian J is 0 where all three partial derivatives vanish, and 1 elsewhere.

The points P ∈ C where rank J

= 0 are the solutions of F = F

= F

=

F

= 0. F

= 0 implies Y = 0 or Z = 0. If Y = 0, then F

= 0 implies

X = 0, and we get the point (0 : 0 : 1), which is indeed on C. (In projective

space, (0 : 0 : 1) = (0 : 0 : Z) for any Z 6= 0.) If Z = 0, then F

= 0 implies

X = 0, and then F

= 0 implies Y = 0, but (0 : 0 : 0) is not a point in P

.

Therefore rank J = 0 and dim T

C = 2 at (0 : 0 : 1), and rank J = 1 and

dim T

C = 1 at all other points of C. Hence dim C = 1 and (0 : 0 : 1) is the

only singular point of C. If we take the affine piece Z 6= 0 of C, we get the

familiar nodal cubic curve y

− x

− x

= 0, which has a singular point at

(0, 0), corresponding the singular point of the projective curve.