Veronese surface

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In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space. It is the embedding of the projective plane according to the linear system of divisors given by the conic sections. It is named for Giuseppe Veronese (1854-1917). The embeddings into three dimensions of the Veronese surface over the reals are known as Steiner surfaces. Its generalization to an algebraic variety is known as the Veronese variety.

1 Definition
2 Veronese map
3 Subvarieties
4 See also
5 References

[edit] Definition

The Veronese surface is a mapping

$\nu:\mathbb{P}^2\to \mathbb{P}^5$

given by

$\nu: [x:y:z] \mapsto [x^2:y^2:z^2:yz:xz:xy]$

where $[x:\ldots]$ denotes homogeneous coordinates.

[edit] Veronese map

The Veronese map or Veronese variety generalizes this idea to mappings of general degree d. That is, the Veronese map of degree d is the map

$\nu_d:\mathbb{P}^n \to \mathbb{P}^{m}$

with m given by the binomial coefficient

$m={n+d \choose d} -1$

The map sends $[x_0:\ldots:x_n]$ to all possible monomials of total degree d, thus the appearance of the binomial coefficient is from consideration of the combinatorics.

One may define the Veronese map in a coordinate-free way, as

$\nu_d: \mathbb{P}V \to \mathbb{P}(\rm{Sym}^d V)$

where V is any vector space of finite dimension, and $Sym d V$ are its symmetric powers of degree d. This is homogeneous of degree d under scalar multiplication on V, and therefore passes to a mapping on the underlying projective spaces.

If the vector space V is defined over a field K which does not have characteristic zero, then the definition must be altered to be understood as a mapping to the dual space of polynomials on V. This is because for fields with finite characteristic p, the pth powers of elements of V are not rational normal curves, but are of course a line. (See, for example additive polynomial for a treatment of polynomials over a field of finite characteristic).

[edit] Subvarieties

The image of a variety under the Veronese map is again a variety; furthermore, these are isomorphic in the sense that the inverse map exists and is regular. More precisely, the images of open sets in the Zariski topology are again open. This may be used to show that any projective variety is the intersection of a Veronese variety and a linear space, and thus that any projective variety is isomorphic to an intersection of quadrics.