Projective Geometry Mathematical Sphere KTH

Projective Geometry Mathematical Sphere KTH

Projective GeometryMathematical Sphere KTH

Sandra Di Rocco

February 18, 2013


Table of contents

Introduction

The space of lines

Projective geometry

homogenous polynomials

algebraic closed fields

Bezout Theorem

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Introduction

Geometry

A frequent idea of the geometry around us is:

EUCLIDEAN GEOMETRY

Rn, distance, angles, volumes, vectors.

Do we need more and why?


Introduction

I Euclidean geometry is not ”symmetric”In R2 or R3 for example you know that:

through any two distinct points passes a UNIQUE line (1)

It is not true that every two lines meet in a point (2)

I Already in the 16-the century (Renaissance) artists started thestudy of PROSPECTIVE DRAWINGThe distance when projecting a 3D image to a 2D spacemight differ from the reality

I In application it is often not the euclidean space but ratherthe space of lines that plays a role.


The space of lines

Examples

Projections: CAT Scan:

Projective geometry provides the means to describe analytically thespace of lines.


The space of lines

The projective space

DefinitionLet V be an (n + 1)-dimensional vector space over a field F. Theprojective space P(V ) = Pn

F is the set of all 1-dimensionalsub-vector spaces (lines through origin) of V .The dimension of Pn

F is n.

Example

Consider F = R. PnR Classes of antipodal points on

Sn = {~v ∈ Rn+1, |x | = 1}.P1R P2

R


The space of lines

homogeneous coordinates

I a point [v ] = [~v ] ∈ Pn, v 6= 0 represents a whole line{λ~v , λ ∈ F}.If e1, . . . , en+1 is a basis for V the coordinates of vectors in Vinduce what we will call homogeneous coordinates on P(V ) :

DefinitionLet v = (x0, . . . , xn) ∈ V then the point [v ] ∈ P(V )hashomogeneous coordinates:

[v ] = (x0 : . . . : xn)

I Note that (x0 : . . . : xn) = (λx0 : . . . : λxn) for λ 6= 0.Execrice 1 Show that Pn

F∼= Fn ∪ Pn−1

F , where Pn−1F can be

identified with the locus where x0 = 0 and it is called thehyperplane at infinity and Fn is the affine part.P1R∼= R ∪ {∞}.


The space of lines

Projective geometry

I Linear spaces A linear space of P(V ) is the projective spaceof a subspace: P(U) ⊆ P(V ), where U 6 V .A linear space of dimension 1 is a projective line.

I Exercise 2 Through any two distinct points in P(V ) passes aunique projective line.

TheoremIn a projective plane any two distinct lines intersect in a point.


The space of lines

Projective geometry

I Projective transformations Given an invertible lineartransformation τ : U → V notice that lines are taken to lines.It does induces a well defined function

τ : P(U)→ P(V )

[u] 7→ [τ(u)]

Example

A Mobius transformation z → az+bcz+d is the affine part of a

projective transformation:

τ : P1C → P1

C defined by

(a bc d

), ac − bc 6= 0

Consider (z0 : z1)→ (az0 + bz1, cz0 + dz1) from the affine linez1 6= 0 to the affine line cz0 + d 6= 0.



Homogeneous polynomials

I Let T (x0, . . . , xn) =∑|I |6d) aI x

I be a polynomial of degree din n + 1 variables, where:I = (i0, . . . , in) ∈ Zn+1, |I | = i0 + . . .+ in, x

I = Πn0x

ijj .

I Can we define zeroes of a polynomial in PnF?

T (x) = 0⇔ T (λx) = 0 happens if and only if T (x) =∑|I |=d

aI xI .

This polynomials are called homogeneous polynomials. Wedenote by V (T ) = {x ∈ Pn,T (x) = 0}. If n = 2 V (T ) iscalled a projective curve.



Homogeneous polynomials

Example

Consider P(x , y , z) = yz − x2 in P2R = R2 ∪ {z = 0}. Then

V (P) ∩ R2 ∼= V (y − x2) and V (P) ∩ {z = 0} = {∞}.


algebraic closed fields

Complex numbers

I Recall that a field F is said to be algebraically closed if everypolynomial over F has all its roots in F.The fundamental theorem of algebra states that C isalgebraically closed.

I Exercise Let A(x,y) be an homogeneous polynomial of degreed in two variables over C. Show that:

A(x , y) = (a1x + b1y) · . . . · (adx + bdy), ci , di ∈ C.

I We deduce that for an homogeneous polynomial in threevariables of degree d :

P(x , y , z) = A(x , y)+zB(x , y , z) = Πd1 (aix +biy)+zB(x , y , z)


Bezout Theorem

Bezout theorem

I Let V (P) be a projective plane curve. The line at infinity,V (z), intersects the curve in:

V (P) ∩ L∞ = V (P) ∩ {z = 0} = V (Πd1 (aix + biy) =

{(bl ,−ak , 0), k = 1, . . . , d} = {pk}.

I We define the multiplicity of the point (at infinity) pk relativeto the curve V (P) asmult(pk) = the number of times the factor (akx + bky)appears in A(x , y).

I Notice that given a projective line L ⊂ P2C there is always a

projective transformation τ taking L to the line at infinity L∞.


Bezout Theorem

Bezout theorem

I Suppose that L is an arbitrary line in P2C and that

p ∈ V (P) ∩ L.

I Suppose moreover that L 6⊂ V (P).

I One can choose a projective transformation T such thatT (L) = L∞.

I We define the multiplicity of p relative to the curve V (P) tobe the multiplicity of the point T (P) = L∞ ∩ T (VP) relativeto the curve T (V (P)).Exercise 3 The multiplicity is well defined.


Bezout Theorem

Bezout theorem

Theorem (Bezout)

Suppose that P(x , y , z) is a complex homogeneous polynomial ofdegree d and let L be a line which is not contained in V (P). ThenL ∩ V (P) consists of d points counted with their multiplicity.

I The general case of Bezout theorem states that a projectivecurve of degree d1 and a projective curve of degree d2, havingno common components, intersect in exactly d1d2 pointscounted with multiplicity.It is valid over any algebraically closed field. The definition ofmultiplicity and the proof is much harder.


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References

If you want to know more on Projective Algebraic Geometry

I M. Reid, Undergraduate Algebraic Geometry. LondonMathematical Society Student Texts, Cambridge UniversityPress, 1988.

I Audun Holme, A Royal Road to Algebraic Geometry, Springer(2012)

I A. Gathman, Notes on Algebraic Geometry.http://www.mathematik.uni-kl.de/ gathmann/class/alggeom-2002/main.pdf

I W. Fulton, An Introduction to Algebraic Geometry.http://www.math.lsa.umich.edu/ wfulton/CurveBook.pdf

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Projective Geometry Mathematical Sphere KTH