Projective variety

In classical algebraic geometry, a branch of mathematics, a projective variety is a geometric object that can be described by homogeneous polynomials.

Definition

It should be a fixed algebraically closed field.

The -dimensional projective space over the field is defined as

For the equivalence relation

The equivalence class of the point is denoted by.

A homogeneous polynomial and a point condition is independent of the homogeneous coordinates of the selected.

A projective algebraic set is a subset of the projective space, the shape of the

Has for homogeneous polynomials in.

A projective variety is an irreducible projective algebraic set, ie polynomials to generate a prime ideal.

Examples

  • Is a projective variety by the Segre embedding
  • The fiber product of two projective varieties is a projective variety.
  • Hypersurfaces are zero sets of irreducible homogeneous polynomial. Each irreducible closed subset of codimension 1 is a hypersurface.
  • A smooth curve (ie curve without singularities ) is a projective variety if and only if it is complete. One example is, elliptic curves, which can be embed into. ( In general it can be embedded into any smooth complete curve. ) Smooth full curves of genus greater than 1 are called hyperelliptic curves when there is a finite morphism of degree 2 on the.
  • Abelian varieties possess a amples line bundle and are therefore projective. Examples are elliptic curves, Jacobi varieties and K3 surfaces.
  • Grassmann manifolds are projective varieties by means of Plücker embedding.
  • Flag manifolds are projective varieties.
  • Compact Riemann surfaces ( compact 1-dimensional complex manifolds ) are projective varieties. By the theorem of Torelli, they are uniquely determined by its Jacobian variety.
  • A compact 2-dimensional complex manifold with two algebraically independent meromorphic functions is a projective variety. ( Chow - Kodaira )
  • The Kodaira embedding theorem gives a criterion when a Kähler manifold is a projective variety.

Invariants

  • Hilbert Samuel polynomial of the homogeneous coordinates ring when the projective variety is defined by prime ideal homogeneous. From the Hilbert - Samuel polynomial, in particular, the dimension, the degree and arithmetic gender of the variety found.
  • The Picardgruppe ( the group of line bundles of Isomorphismenklassen ) and the Jacobi variety ( the core of ).
  • Algebraic Geometry
Lexicographical order Pullback (category theory) Hilbert series and Hilbert polynomial
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