Editing Veronese surface

In [[mathematics]], the '''Veronese surface''' is an [[algebraic surface]] in five-dimensional [[projective space]], and is realized by the '''Veronese embedding''', the embedding of the [[projective plane]] given by the complete [[linear system of conics]]. It is named after [[Giuseppe Veronese]] (1854–1917). Its generalization to higher dimension is known as the '''Veronese variety'''.

The surface admits an embedding in the four-dimensional projective space defined by the projection from a general point in the five-dimensional space. Its general projection to three-dimensional projective space is called a [[Steiner surface]].

==Definition==
The Veronese surface is the image of the mapping
:<math>\nu:\mathbb{P}^2\to \mathbb{P}^5</math>
given by

:<math>\nu: [x:y:z] \mapsto [x^2:y^2:z^2:yz:xz:xy]</math>

where <math>[x:\cdots]</math> denotes [[homogeneous coordinates]]. The map <math>\nu</math> is known as the '''Veronese embedding.'''

==Motivation==
The Veronese surface arises naturally in the study of [[conic]]s. A conic is a degree 2 plane curve, thus defined by an equation:
:<math>Ax^2 + Bxy + Cy^2 +Dxz + Eyz + Fz^2 = 0.</math>
The pairing between coefficients <math>(A, B, C, D, E, F)</math> and variables <math>(x,y,z)</math> is linear in coefficients and quadratic in the variables; the Veronese map makes it linear in the coefficients and linear in the monomials. Thus for a fixed point <math>[x:y:z],</math> the condition that a conic contains the point is a [[linear equation]] in the coefficients, which formalizes the statement that "passing through a point imposes a linear condition on conics".

==Veronese map== 
{{main|Veronese map}}
The Veronese map or Veronese variety generalizes this idea to mappings of general degree ''d'' in ''n''+1 variables. That is, the Veronese map of degree ''d'' is the map

:<math>\nu_d\colon \mathbb{P}^n \to \mathbb{P}^m</math>

with ''m'' given by the [[multiset coefficient]], or more familiarly the [[binomial coefficient]], as:

:<math>m= \left(\!\!{n + 1 \choose d}\!\!\right) - 1 = {n+d \choose d} - 1.</math>

The map sends <math>[x_0:\ldots:x_n]</math> to all possible [[monomial]]s of [[total degree]] ''d'' (of which there are <math>m+1</math>); we have <math>n+1</math> since there are <math>n+1</math> variables <math>x_0, \ldots, x_n</math> to choose from; and we subtract <math>1</math> since the projective space <math>\mathbb{P}^m</math> has <math>m+1</math> coordinates. The second equality shows that for fixed source dimension ''n,'' the target dimension is a polynomial in ''d'' of degree ''n'' and leading coefficient <math>1/n!.</math>

For low degree, <math>d=0</math> is the trivial constant map to <math>\mathbf{P}^0,</math> and <math>d=1</math> is the identity map on <math>\mathbf{P}^n,</math> so ''d'' is generally taken to be 2 or more.

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

:<math>\nu_d: \mathbb{P}(V) \ni [v] \mapsto [v^d] \in \mathbb{P}(\rm{Sym}^d V)</math>

where ''V'' is any [[vector space]] of finite dimension, and <math>\rm{Sym}^d V</math> are its [[symmetric power]]s of degree ''d''.  This is homogeneous of degree ''d'' under scalar multiplication on ''V'', and therefore passes to a mapping on the underlying [[projective space]]s.

If the vector space ''V'' is defined over a [[field (mathematics)|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 ''p''th powers of elements of ''V'' are not [[rational normal curve]]s, but are of course a line. (See, for example [[additive polynomial]] for a treatment of polynomials over a field of finite characteristic).

=== Rational normal curve ===
{{see|Rational normal curve}}

For <math>n=1,</math> the Veronese variety is known as the [[rational normal curve]], of which the lower-degree examples are familiar.
* For <math>n=1, d=1</math> the Veronese map is simply the identity map on the projective line.
* For <math>n=1, d=2,</math> the Veronese variety is the standard [[parabola]] <math>[x^2:xy:y^2],</math> in affine coordinates <math>(x,x^2).</math>
* For <math>n=1, d=3,</math> the Veronese variety is the [[twisted cubic]], <math>[x^3:x^2y:xy^2:y^3],</math> in affine coordinates <math>(x,x^2,x^3).</math>

==Biregular==
The image of a variety under the Veronese map is again a variety, rather than simply a [[Constructible set (topology)|constructible set]]; furthermore, these are isomorphic in the sense that the inverse map exists and is [[regular function|regular]] – the Veronese map is [[biregular]]. More precisely, the images of [[open set]]s in the [[Zariski topology]] are again open.

==See also==
*The Veronese surface is the only [[Scorza variety|Severi variety]] of dimension 2

==References==
* Joe Harris, ''Algebraic Geometry, A First Course'', (1992) Springer-Verlag, New York. {{isbn|0-387-97716-3}}

[[Category:Algebraic varieties]]
[[Category:Algebraic surfaces]]
[[Category:Complex surfaces]]
[[Category:Tensors]]