Veronese embedding

In algebraic geometry , a branch of mathematics , the Veronese embedding denotes an embedding of projective spaces in higher-dimensional projective spaces.

construction

Let and be natural numbers and . ${\displaystyle d}$ ${\displaystyle n}$ ${\displaystyle N={\tbinom {n+d}{d}}-1}$

The Veronese embedding

${\displaystyle \nu _{d}\colon P^{n}\to P^{N}}$

is defined by the fact that all monomials of degree are mapped in lexicographical order. ${\displaystyle \left[x_{0}:\ldots :x_{n}\right]}$ ${\displaystyle d}$

For example for : ${\displaystyle n=1}$

${\displaystyle \nu _{d}(\left[x_{0}:x_{1}\right])=\left[x_{0}^{d}:x_{0}^{d-1}x_{1}:x_{0}^{d-2}x_{1}^{2}:\ldots :x_{1}^{d}\right]}$

or for : ${\displaystyle d=2}$

${\displaystyle \nu _{2}(\left[x_{0}:\ldots :x_{n}\right])=\left[x_{0}^{2}:x_{0}x_{1}:\ldots :x_{0}x_{n}:x_{1}^{2}:\ldots :x_{1}x_{n}:\ldots :x_{n}^{2}\right]}$.

The Veronese mapping converts the polynomial equations originally existing between the variables into linear equations. This is often useful because linear equations are easier to deal with. One example is the application of the Lefschetz hypereplanes theorem on hypersurfaces in projective space: Hypersurfaces can be converted into hyperplanes by means of Veronese embedding, to which the hyperplane theorem can be applied.

Regularity

The image of the Veronese embedding is a projective variety . The Veronese embed is a regular mapping and has a regular inverse mapping .

If there is a projective variety, then is also a projective variety. ${\displaystyle Y\subset P^{n}}$ ${\displaystyle \nu _{d}(Y)\subset P^{N}}$

Rational normal curves

For , the images of the Veronese embedding are called rational normal curves . ${\displaystyle n=1}$

Examples

• ${\displaystyle d=1}$: The projective straight line is obtained . ${\displaystyle P^{1}}$
• ${\displaystyle d=2}$: One gets the parabola , in affine coordinates . ${\displaystyle \left[x^{2}:xy:y^{2}\right]}$ ${\displaystyle (y,y^{2})}$
• ${\displaystyle d=3}$: You get the twisted cube , in affine coordinates . ${\displaystyle \left[x^{3}:x^{2}y:xy^{2}:y^{3}\right]}$ ${\displaystyle (y,y^{2},y^{3})}$

Equivariance

The Veronese embedding is equivariant with respect to the irreducible representation . ${\displaystyle \nu _{d}\colon \mathbb {R} P^{1}\to \mathbb {R} P^{d}}$ ${\displaystyle SL(2,\mathbb {R} )\to SL(d+1,\mathbb {R} )}$

More generally there are for and for each Hitchin representation , i. H. any deformation of the composition of the irreducible representation with , an equivariate hyperconvex curve . In general, however, this is not given by polynomials, but only Holder continuous . ${\displaystyle \iota \colon \Gamma \subset SL(2,\mathbb {R} )}$ ${\displaystyle \iota }$ ${\displaystyle \mathbb {R} P^{1}\to \mathbb {R} P^{n}}$

Veronese area

The image of

${\displaystyle \nu _{2}\colon P^{2}\to P^{5}}$

is called the Veronese area .

The Veronese surface is the only 2-dimensional Severi variety .

literature

• Joe Harris: Algebraic Geometry, A First Course . Springer-Verlag, New York 1992. ISBN 0-387-97716-3

Web links

• Baldwin, Bérczi: Algebraic Geometry (Chapter 4)