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

Let it be a firmly chosen, algebraically closed body . ${\displaystyle K}$

The -dimensional projective space above the body is defined as ${\displaystyle n}$ ${\displaystyle K}$

${\displaystyle P^{n}:=(K^{n+1}\setminus \{(0,\ldots ,0)\})/\sim }$

for the equivalence relation

${\displaystyle (x_{0},\ldots ,x_{n})\sim (y_{0},\ldots ,y_{n})\Leftrightarrow \exists \lambda \in K\setminus \{0\}\colon x_{i}=\lambda y_{i},i=0,\ldots ,n}$.

The equivalence class of the point is denoted by. ${\displaystyle (x_{0},\ldots ,x_{n})}$ ${\displaystyle \left[x_{0}:\ldots :x_{n}\right]}$

For a homogeneous polynomial and a point , the condition is independent of the chosen homogeneous coordinates of . ${\displaystyle f\in K[X_{0},\ldots ,X_{n}]}$ ${\displaystyle x=[x_{0}:\ldots :x_{n}]}$ ${\displaystyle f(x_{0},\ldots ,x_{n})=0}$ ${\displaystyle x}$

A projective algebraic set is a subset of projective space that has the form

${\displaystyle \{x\in P^{n}\mid f_{1}(x)=\ldots =f_{k}(x)=0\}}$

for homogeneous polynomials in has. ${\displaystyle f_{1},\ldots ,f_{k}}$ ${\displaystyle K[X_{0},\ldots ,X_{n}]}$

A projective variety is an irreducible projective algebraic set; that is, the polynomials should generate a prime ideal in . ${\displaystyle f_{1},\ldots ,f_{k}}$ ${\displaystyle K[X_{0},\ldots ,X_{n}]}$

Examples

• ${\displaystyle P^{n}\times P^{m}}$is a projective variety using Segre embedding

${\displaystyle P^{n}\times P^{m}\to P^{(n+1)(m+1)-1},(x_{i},y_{j})\mapsto x_{i}y_{j}}$(in lexicographical order ).

• The fiber product of two projective varieties is a projective variety.
• Hypersurfaces are sets of roots of an irreducible homogeneous polynomial. Every irreducible closed subset of codimension 1 is a hypersurface.
• A smooth curve (i.e. curve with no singularities) is a projective variety if and only if it is complete . One example is elliptical curves that can be embedded in. (In general, any smooth complete curve can be embedded in.) Smooth complete curves of gender greater than 1 are called hyperelliptic curves if there is a finite morphism of degree 2 on the . ${\displaystyle P^{2}}$ ${\displaystyle P^{3}}$ ${\displaystyle P^{1}}$
• Abelian varieties have an ample bundle of lines and are therefore projective. Examples are elliptic curves, Jacobi varieties and K3 surfaces .
• Grassmann manifolds are projective varieties using Plücker embedding .
• Flag manifolds are projective varieties by means of embedding in a product of Graßmann manifolds.
• Compact Riemann surfaces (compact one-dimensional complex manifolds ) are projective varieties. According to Torelli's theorem , they are uniquely determined by their Jacobi variety.
• A compact two-dimensional complex manifold with two algebraically independent meromorphic functions is a projective variety. (Chow Kodaira)
• The Kodaira embedding theorem gives a criterion for when a Kähler manifold is a projective variety.

Invariants

• The Hilbert-Samuel polynomial of the homogeneous coordinate ring , if the projective variety is defined by the homogeneous prime ideal . From the Hilbert-Samuel polynomial, the dimension, the degree and the arithmetic gender of the variety result in particular . ${\displaystyle K\left[X_{0},\ldots ,X_{n}\right]/I}$ ${\displaystyle I}$
• The Picard group (the group of isomorphism classes of line bundles) and the Jacobi variety (the core of ). ${\displaystyle deg:Pic(X)\rightarrow \mathbb {Z} }$

Web links

• Bauer et al .: Geometry and topology through projective spaces