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- Algebraic Geometry

Marko Rajkovi´

c

supervisor: prof. Vladimir Berkovich

August 17, 2015

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

Algebraic Geometry - Establishing correspondences between geometric and algebraic

objects

Fundamental objects of study are algebraic varieties

Introduction

Studying systems of polynomial equations in several variables

and using abstract algebraic techniques for solving geometrical

problems about zeros of such systems

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

Fundamental objects of study are algebraic varieties

Introduction

Studying systems of polynomial equations in several variables

and using abstract algebraic techniques for solving geometrical

problems about zeros of such systems

Establishing correspondences between geometric and algebraic

objects

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

Introduction

Studying systems of polynomial equations in several variables

and using abstract algebraic techniques for solving geometrical

problems about zeros of such systems

Establishing correspondences between geometric and algebraic

objects

Fundamental objects of study are algebraic varieties

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

For

S ⊂ A = k[x1, . . . , xn] we define the zero set of S as:

Z (S ) := {P ∈ An; f (P) = 0 ∀f ∈ S }. Sets of this form are called

algebraic sets.

Examples of algebraic sets

An = Z (0)

∅ = Z (1)

(a1, . . . , an) = Z (x1 − a1, . . . , xn − an)

Arbitrary intersections and finite unions of algebraic sets are again

algebraic sets.

Affine Varieties

Definition

For an algebraically closed field k affine n-space over k is set

An := {(a1, . . . an); ai ∈ k for 1 ≤ i ≤ n}.

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

Examples of algebraic sets

An = Z (0)

∅ = Z (1)

(a1, . . . , an) = Z (x1 − a1, . . . , xn − an)

Arbitrary intersections and finite unions of algebraic sets are again

algebraic sets.

Affine Varieties

Definition

For an algebraically closed field k affine n-space over k is set

An := {(a1, . . . an); ai ∈ k for 1 ≤ i ≤ n}. For

S ⊂ A = k[x1, . . . , xn] we define the zero set of S as:

Z (S ) := {P ∈ An; f (P) = 0 ∀f ∈ S }. Sets of this form are called

algebraic sets.

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

∅ = Z (1)

(a1, . . . , an) = Z (x1 − a1, . . . , xn − an)

Arbitrary intersections and finite unions of algebraic sets are again

algebraic sets.

Affine Varieties

Definition

For an algebraically closed field k affine n-space over k is set

An := {(a1, . . . an); ai ∈ k for 1 ≤ i ≤ n}. For

S ⊂ A = k[x1, . . . , xn] we define the zero set of S as:

Z (S ) := {P ∈ An; f (P) = 0 ∀f ∈ S }. Sets of this form are called

algebraic sets.

Examples of algebraic sets

An = Z (0)

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

(a1, . . . , an) = Z (x1 − a1, . . . , xn − an)

Arbitrary intersections and finite unions of algebraic sets are again

algebraic sets.

Affine Varieties

Definition

For an algebraically closed field k affine n-space over k is set

An := {(a1, . . . an); ai ∈ k for 1 ≤ i ≤ n}. For

S ⊂ A = k[x1, . . . , xn] we define the zero set of S as:

Z (S ) := {P ∈ An; f (P) = 0 ∀f ∈ S }. Sets of this form are called

algebraic sets.

Examples of algebraic sets

An = Z (0)

∅ = Z (1)

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

Arbitrary intersections and finite unions of algebraic sets are again

algebraic sets.

Affine Varieties

Definition

For an algebraically closed field k affine n-space over k is set

An := {(a1, . . . an); ai ∈ k for 1 ≤ i ≤ n}. For

S ⊂ A = k[x1, . . . , xn] we define the zero set of S as:

Z (S ) := {P ∈ An; f (P) = 0 ∀f ∈ S }. Sets of this form are called

algebraic sets.

Examples of algebraic sets

An = Z (0)

∅ = Z (1)

(a1, . . . , an) = Z (x1 − a1, . . . , xn − an)

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

Affine Varieties

Definition

For an algebraically closed field k affine n-space over k is set

An := {(a1, . . . an); ai ∈ k for 1 ≤ i ≤ n}. For

S ⊂ A = k[x1, . . . , xn] we define the zero set of S as:

Z (S ) := {P ∈ An; f (P) = 0 ∀f ∈ S }. Sets of this form are called

algebraic sets.

Examples of algebraic sets

An = Z (0)

∅ = Z (1)

(a1, . . . , an) = Z (x1 − a1, . . . , xn − an)

Arbitrary intersections and finite unions of algebraic sets are again

algebraic sets.

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

Example

Algebraic (closed) sets in A1 are finite subsets (including empty

set) as sets of zeros of single non-zero polynomial and whole set

(corresponding to zero polynomial).

Definition

Zariski topology on An is the topology whose closed sets are the

algebraic sets. Any subset X of An will be equipped with the

topology induced by the Zariski topology on An. This is called the

Zariski topology on X .

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

Definition

Zariski topology on An is the topology whose closed sets are the

algebraic sets. Any subset X of An will be equipped with the

topology induced by the Zariski topology on An. This is called the

Zariski topology on X .

Example

Algebraic (closed) sets in A1 are finite subsets (including empty

set) as sets of zeros of single non-zero polynomial and whole set

(corresponding to zero polynomial).

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

Example

A1 is an irreducible affine variety since its only proper closed

subsets are finite and it is infinite. Generally, An is an irreducible

affine variety for every integer n.

Definition

A non-empty subset Y of topological space X is called irreducible

if it is not a union of two proper closed subsets.

An (irreducible) affine variety is an (irreducible) closed subset of

An with Zariski topology.

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

Definition

A non-empty subset Y of topological space X is called irreducible

if it is not a union of two proper closed subsets.

An (irreducible) affine variety is an (irreducible) closed subset of

An with Zariski topology.

Example

A1 is an irreducible affine variety since its only proper closed

subsets are finite and it is infinite. Generally, An is an irreducible

affine variety for every integer n.

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

Examples

I (An) = 0

I ((a1, . . . , an)) = (x1 − a1, . . . , xn − an)

Definition

For X ⊂ An we define the ideal of X as

I (X ) := {f ∈ A; f (P) = 0 ∀P ∈ X }

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

I ((a1, . . . , an)) = (x1 − a1, . . . , xn − an)

Definition

For X ⊂ An we define the ideal of X as

I (X ) := {f ∈ A; f (P) = 0 ∀P ∈ X }

Examples

I (An) = 0

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

Definition

For X ⊂ An we define the ideal of X as

I (X ) := {f ∈ A; f (P) = 0 ∀P ∈ X }

Examples

I (An) = 0

I ((a1, . . . , an)) = (x1 − a1, . . . , xn − an)

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

Corollary

There is a 1 : 1 correspondence

{points in An} ↔ {maximal ideals of k[x1, . . . , xn]}

given by

(a1, . . . , an) ↔ (x1 − a1, . . . , xn − an).

Theorem–Hilbert Nullstelensatz

For algebraically closed field k maximal ideals of k[x1, . . . , xn] are

exactly the ideals of the form (x1 − a1, . . . , xn − an) for some

ai ∈ k.

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

Theorem–Hilbert Nullstelensatz

For algebraically closed field k maximal ideals of k[x1, . . . , xn] are

exactly the ideals of the form (x1 − a1, . . . , xn − an) for some

ai ∈ k.

Corollary

There is a 1 : 1 correspondence

{points in An} ↔ {maximal ideals of k[x1, . . . , xn]}

given by

(a1, . . . , an) ↔ (x1 − a1, . . . , xn − an).

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

We call

A(Y ) := A/I (Y ) affine coordinate ring of Y .

Examples

An is irreducible since its ideal is zero ideal which is prime.

If f is irreducible polynomial in A = k[x1, . . . , xn] we get an

irreducible affine variety Y = Z (f ). For n = 2 we call it affine

curve of degree d , where d is degree of f. For n = 3 we have

surface and for n > 3 hypersurface.

Lemma and Definition

An algebraic set X ⊂ An is an irreducible affine variety if and only

if its ideal I (X ) ⊂ A = k[x1, . . . , xn] is a prime ideal.

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

Examples

An is irreducible since its ideal is zero ideal which is prime.

If f is irreducible polynomial in A = k[x1, . . . , xn] we get an

irreducible affine variety Y = Z (f ). For n = 2 we call it affine

curve of degree d , where d is degree of f. For n = 3 we have

surface and for n > 3 hypersurface.

Lemma and Definition

An algebraic set X ⊂ An is an irreducible affine variety if and only

if its ideal I (X ) ⊂ A = k[x1, . . . , xn] is a prime ideal. We call

A(Y ) := A/I (Y ) affine coordinate ring of Y .

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

If f is irreducible polynomial in A = k[x1, . . . , xn] we get an

irreducible affine variety Y = Z (f ). For n = 2 we call it affine

curve of degree d , where d is degree of f. For n = 3 we have

surface and for n > 3 hypersurface.

Lemma and Definition

An algebraic set X ⊂ An is an irreducible affine variety if and only

if its ideal I (X ) ⊂ A = k[x1, . . . , xn] is a prime ideal. We call

A(Y ) := A/I (Y ) affine coordinate ring of Y .

Examples

An is irreducible since its ideal is zero ideal which is prime.

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

Lemma and Definition

An algebraic set X ⊂ An is an irreducible affine variety if and only

if its ideal I (X ) ⊂ A = k[x1, . . . , xn] is a prime ideal. We call

A(Y ) := A/I (Y ) affine coordinate ring of Y .

Examples

An is irreducible since its ideal is zero ideal which is prime.

If f is irreducible polynomial in A = k[x1, . . . , xn] we get an

irreducible affine variety Y = Z (f ). For n = 2 we call it affine

curve of degree d , where d is degree of f. For n = 3 we have

surface and for n > 3 hypersurface.

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich

The Twisted Cubic Curve

Let Y = {(t, t2, t3); t ∈ k}. Then I (Y ) = (x 2 − y , x 3 − z) in

A = k[x , y , z].

A/I (Y ) = k[x , y , z]/(x 2 − y , x 3 − z) ∼

= k[x , x 2, x 3] ∼

= k[t]

which is an integral domain. Hence, I (Y ) is prime ideal and Y is

an affine variety.

Marko Rajkovi´

c supervisor: prof. Vladimir Berkovich