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ALGEBRAIC

A 2003 Math session by Renzo

INTRODUCTION, MOTIVATION AND GOALS

We are going to make a rst few exploratory steps in the fascinating but

mysterious world of algebraic . is a branch of

that studies geometric problems with the help of algebra (like

the more geometrical oriented p eople would say) or algebraic problems with

the help of geometry (like the algebraic p eople would say). Whether one

opinion is more correct than the other is a minor p oint. What's really co ol

is that there is such a close connection b etween these apparently unrelated

elds of mathematics.

We are going to lo ok in particular in the world of curves, which have

the big advantage of b eing \drawable" and hence a little more intuitive and

less scary. Algebraic curves have b een studied for a very long time. The

ancient Greeks were already pretty handy with them and had gured out

a fair amount of things ab out curves that allowed them to solve famous

geometrical problems (such as the trisection of the or the doubling of

the cub e, but this is another story...). Curves are still an extremely interesting

ob ject of study nowadays. Even though a LOT of things are known ab out

them, there are many questions that are still op en. For example we are

still trying to understand the geometry of various mo duli spaces of curves

(which you can think of as \spaces of curves with given characteristics". For

example, all in the are identi ed uniquely by the co ordinates

of the center and by the radius. Hence we can think that there is a whole

three dimensional space of circles: you give me three numb ers, and I will

pro duce you a corresp onding circle; you give me a circle, I'll tell you the

three corresp onding numb ers).

Along the way we'll have a chance to intro duce a \new world", where

there are no such things as parallel lines. Again, as bizzarre as this may seem,

it's an extremely handy things to do. And it's no funky mo dern invention:

the theory of pro jective geometry was initially develop ed by architects and

painters during the Renaissance, and heavily used for prosp ectic studies and

all sort of applications.

Have I convinced you that we are embarking in a worthwile, even though

mayb e a little mind quenching adventure? Let's go then!!

DEFINITION: An Algebraic consists of the of p oints in

the plane that satisfy a p olynomial .

2

C = f(x; y ) 2 R jP (x; y ) = 0g

The Degree of the curve is just what we exp ect it to b e: the degree of the

p olynomial de ning the curve. EXAMPLES:

a many lines a 2

ax+by+c=0 (what is the equation?) y=x

Q1: Given two curves C and C with f = 0,f = 0, what is

1 2 1 2

the equation for the curve C [ C ?

1 2

DEFINITION: A curve C is said Irreducible if it do es not break up

into the union of \simpler" curves.

Q2: How do es this translate in terms of the equation for C ?

Q3: Is the equation for an unique?

PROBLEM...Our \dictionary" do esn't work so well...We want a go o d

corresp ondence b etween algebra (p olynomial equations) and geometry (curves).

A dictionary that uses the words banana, bro ccoli and balalaika to mean the

exact same thing is not all that go o d, is it?

FIX 1:We are using the WRONG numb ers. From now on our numb ers

will b e all the complex numb ers and not just the real ones! (Alas, we will

have to continue to draw curves in the same way b ecause we yet haven't

discovered 4-dimensional chalk!)

Notice, though, that now curves will b e \ ob jects of complex

1", which means they really are . Now mayb e you rememb er that we

know surfaces pretty well. We could even work out that all complex algebraic

curves are orientable, and hence must lo ok like a or a multiholed

doughnut (eventually pinched or punctured),

but that's another story....

Q4: What happ ens now to

2 2

x + y = 0

2 2

x + y = 1?

Q5: Is the x a complete x?

If you p onder a little bit you will nd out that the answer is \almost".

In fact we are still de ning the same curve if we multiply the equation by a

constant...but that's not such a big deal, and we'll see later on that we'll b e

able to x this to o by just slightly mo difying our \universe".

For now we have estabilished the following bijective corresp ondence:

 

al g ebr aic cur v es eq uations up to scal ing

$

NOW FOR SOME MORE FUN EXAMPLES:

Cubics: What are the graphs (of the real trace) of:

2 3

 y = x x;

2 3

 y = x ;

2 3 2

 y = x + x .

Pretty Flowers: Match the graphs with their equations:

2 2 3 2 2

 (x + y ) 4x y = 0;

2 2 2

 (x + y ) + 4xy = 0;

2 2 2 2 3

 (x + y ) 3x y y = 0;

More pretty owers...: Can you pro duce a sensible guess for the equation

of the following gure?

Now that we are starting to get a little more familiar with Algebraic

Curves, let's start asking ourselvs interesting questions, and let's start easy

to then get harder and harder:

1. How many p oints of intersection can 2 lines in the plane have?

2. How many p oints of intersection do a line and a conic have in the plane?

3. How many p oints of intersection do es a line have with a curve C if

degree d?

4. How many lines through one p oint?

5. how many lines through two p oints?

6. How many p oints do you need to pin down a unique conic?

7. Will any con guration of the right numb er of p oints pin down a unique

conic?

8. What's a quick way to determine the equation of the conic through

p

1 1

P = (1; 0), P = (0; 0), P = (0; 1), P = (1; 1), P = ( 2 ; )?

1 2 3 4 5

2 2

9. How many p oints pin down a unique curve of degree d?

2

THE \IDEAL" UNIVERSE : P

C

Take the usual plane and \add" to it a new (pro jective) line, that we will

2

call line at in nity. Points of this line represent directions of lines in C .

at infinity representing this direction

Notice that now every pair of lines meet at exactly one p oint!!!!

2

FORMALLY : We add an extra co ordinate, so that p oints in P will b e

C

identi ed by three complex numb ers

(a : b : c):

BUT (there's a big but!) we declare two p oints to b e the same p oint if the

co ordinates are prop ortional. So, for example

(1 : 2 : 3) = (2 : 4 : 6) = (23 : 46 : 69) = (k : 2k : 3k ):

Now we can see that

2 2

P = f(a : b : c); c 6= 0g [ f(a : b : 0)g = C [ l :

1

C

What happ ens to algebraic geometry now? We could a priori just take

2

p olynomials in three variables (for example P (x; y ; z ) : x + y + 3z + 4 = 0).

But obviously that's completely nuts!!

Q6: Why?

In general I have no hop e to make sense of the notion of the \value" of

2

a p olynomial at a p oint of P . But there are some sp ecial p olynomials such

C

that at least the zero-set is well de ned.

Q7: Can you guess which are they?

DEFINITION: A Homogoneous p olynomial is a p olynomial in which

all the have the same degree.

Q8: Gimme some examples.

Q9: Why are homogeneous p olynomials the \go o d ones" for our pur-

p oses?

DEFINITION: A Pro jective Curve is the zero set of a homogeneous

p olynomial. For example:

2 2 2

X + 2XY + Y + 3Z = 0:

Q9: Can we go back and forth b etween our previous notion of curves and

this current one?

The answer is homogenize and dehomogenize !!!!

For example:

2 2 2 2 2

x + y 1 = 0 $ X + Y Z = 0;

2 2

y x = 0 $ Y Z X = 0:

Q10: Are homogeneization and dehomogeneization always inverse op er-

ations to each other?

And now for some more problems and interesting stu to discover!

1. Find algebraically the intersection p oints in pro jective space of the

following pair of lines:

 y = x;

 y = x + 1.

2. Find algebraically the intersection p oints in pro jective space of the

following line and hyp erb ola:

 0 = x;

 y x = 1.

2

3. Make sense of the statement: \All conics in P are p otato es".

C

2

4. P gives us a real easy way to decide whether a (non pro jective) conic

C

is a parab ola, or hyp erb ola. What is it?

5. Decide which of the following conics is a parab ola, ellipse or hyp erb ola!

2 2

 x + 2xy + 1 + y + 2y = 0;

2 2

 x + 3xy + 2y + 2y + 2x + 6 = 0;

2 2

 x + xy + y + 4y + 3x 12 = 0.

A TINY BIT OF INTERSECTION THEORY

We want to de ne a notion of intersection of curves that resp ects the

following ideas:

1. If two curves intersect transversally we count exactly the numb er of

p oints of intersections.

1 1 1

1 1 1

Total intersection = 6

2. If one curve go es through a multiple p oint of the other curve, but

do esn't share a with any \branch", then the p oint of intersec-

tion counts for exactly the multeplicity of the multiple p oint.

2

1

3. If two curves intersect at a p oint and share the tangent line, then the

intersection multeplicity go es up!! ( We could make this notion precise,

but we won't, it would take just to o long!)

>1

Q11: We actually can nd a way to make p oint 3 precise if one of the curves

is just a line. Can you think of how?

2 y=x4 y=x y=x3

EXERCISE: Find jC \ C j in the following cases:

1 2

C 1 C 1

C 2

C 2

C 1 C 2

C 2 C1

Now here comes our big guy. The following theorem is quite straightfor-

ward to prove in the case one of the curves is a line (think ab out why...), but

requires some sophistication to b e proven in full glory in the following form:

BEZOUT'S THEOREM: If C and C are two pro jective curves of

1 2

degree d and d then

1 2

jC \ C j = d d :

1 2 1 2

With the help of Bezout we are nally ready to tackle a bunch of really

interesting questions and problems and to prove the conjectures we started

out from:

1. If a conic has a double p oint, then it splits into 2 lines.

2. A conic cannot have a triple p oint.

3. If a cubic has a triple p oint, then it's the union of three lines.

4. A cubic with two double p oints must b e the union of a line and a conic.

5. A cubic with three distinct double p oints must b e...?

6. An irreducible quartic cannot have 4 double p oints.

7. Pascal's Theorem: The opp osite sides of a hexagon inscrib ed in a

conic meet in collinear p oints.

8. Pappus' theorem: Given two lines l ; l , and chosen randomly three

1 2

p oints on each line (call them P ; P ; P ; Q ; Q ; Q ), then the p oints

1 2 3 1 2 3

P Q \ P Q ; R =

2 3 3 2 1

R = P Q \ P Q ;

2 1 3 3 1

P Q \ P Q R =

1 2 2 1 3

are collinear.