Algebraic variety

Algebraic varieties closely connected

Framework. In all this article K will indicate a Corps algebraically closed (for example $\ mathbb {C}$ ), N an entirety equal to or higher than one and $\ mathbb {} _k^n$ the space closely connected of dimension N has on K, i.e. the unit K ^N.

Definition. Is S part of the ring of the polynomials $k$ one calls variety associated with S and one notes V (S) the subset of $\ mathbb {has} _k^n$ :

$V (S) = \ {(x_1, \ ldots, x_n) \ in \ mathbb {has} _k^n, \ forall F \ in S, \ F (x_1, \ ldots, x_n) =0 \}$

I.e. the place of cancellation common to all the elements of S.

Remark. If I is the Idéal of $k$ generated by S, then V (I) =V (S). The Théorème of the zeros of Hilbert establishes a bijective correspondence between the algebraic varieties of $\ mathbb {has} _k^n$ and the ideal radiciels of $k$ . The points correspond to the ideal maximum, and the varieties corresponding to the ideal first are called irreducible varieties (attention: in the literature certain authors prefer to reserve the term of variety to the irreducible varieties and speak about algebraic units to indicate the nonirreducible varieties.)

Examples.

In the plan refines $\ mathbb {has} _k^2$ , the place of cancellation of a polynomial with two variables is an algebraic variety closely connected called algebraic Courbe and the degree of the polynomial is called degree curve. The lines are the algebraic curves closely connected of degree 1, the conical those of degree 2, the cubic those of degree 3 and so on.
In space refines $\ mathbb {has} _k^3$ the place of cancellation of a polynomial with three variables is an algebraic variety closely connected called algebraic surface . Just like for the curves one defines the degree of a surface, the plans are algebraic surfaces closely connected of degree 1, the quadric those of degree 2 etc
In a space refines, a finished union of points is an algebraic variety closely connected.

Projective algebraic varieties.

The projective algebraic geometry is a more comfortable framework when one wants to study the intersections between two varieties. The Théorème of Bezout is true only for projective varieties.

Framework. In this part $\ mathbb {P} _k^n$ indicates the projective Espace of dimension N on K, i.e. the unit K ^{N +1}− {0}/~, where ~ is the relation of equivalence identifying two points X and there if and only if X and is there on the same line passing by the origin. The projective space of dimension N is thus identified with the vectorial line whole of a K - vector space of dimension N +1.

Definition. Is S a whole of homogeneous polynomials of the ring $k$ . One calls variety associated with S and one notes V (S) the subset of $\ mathbb {P} _k^n$ :

$V (S) = \ {(x_0: \ ldots: x_n) \ in \ mathbb {P} _k^n, \ forall F \ in S, \ F (x_0, \ ldots, x_n) =0 \}$

$(x_0: \ ldots: x_n)$ is the Coordonnées homogeneous of a point of $\ mathbb {P} _k^n$ . Let us notice that the cancellation of the polynomial F in a point of K ^{N +1}− {0} depends only on its class modulo the relation ~. The unit V ( S ) is thus well defined.

Remark. If I is the homogeneous Idéal of $k$ generated by S then V (I) =V (S). Then, just like in the case of the algebraic varieties closely connected there exists a Théorème of the zeros of projective Hilbert which establishes a bijective correspondence between the projective algebraic varieties in $\ mathbb {P} _k^n$ and the radical homogeneous ideals distinct from the ideal generated by $(X_0, \ ldots, X_n)$ .

Topology of Zariski

Algebraic varieties closely connected (resp. projective) are usually provided with a topology known as of Zariski . If X indicates such a variety, part of X is declared closed for this topology if and only if it is form V ( S ) for a whole of polynomials (resp. homogeneous polynomials) S .

An open subset of a variety refines (resp. projective) is called variety quasi-closely connected (resp. quasi-projective ).

For all I =0,…, N , space refines $\ mathbb {has} _k^n$ can be identified with the open subset $U_ {I}$ of $\ mathbb {P} _k^n$ defined by X _I ≠ 0 via the application which with ( X ₁,… X _N) associates the point of the projective space which has as homogeneous coordinates ( X ₁,…, X _I, 1, X _{I +1}…, X _N). It is checked that this application induces a homeomorphism of space closely connected on its image. U _{{ I }}.

Morphisms of varieties

Random links: