The Veronese surface is the image of the mapping

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

given by

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

where ${\displaystyle [x:\cdots ]}$  denotes homogeneous coordinates. The map ${\displaystyle \nu }$  is known as the Veronese embedding.

The Veronese surface arises naturally in the study of conics. A conic is a degree 2 plane curve, thus defined by an equation:

${\displaystyle Ax^{2}+Bxy+Cy^{2}+Dxz+Eyz+Fz^{2}=0.}$ 

The pairing between coefficients ${\displaystyle (A,B,C,D,E,F)}$  and variables ${\displaystyle (x,y,z)}$  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 ${\displaystyle [x:y:z],}$  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".

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

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

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

${\displaystyle m=\left(\!\!{n+1 \choose d}\!\!\right)-1={n+d \choose d}-1.}$ 

The map sends ${\displaystyle [x_{0}:\ldots :x_{n}]}$  to all possible monomials of total degree d (of which there are ${\displaystyle m+1}$ ); we have ${\displaystyle n+1}$  since there are ${\displaystyle n+1}$  variables ${\displaystyle x_{0},\ldots ,x_{n}}$  to choose from; and we subtract ${\displaystyle 1}$  since the projective space ${\displaystyle \mathbb {P} ^{m}}$  has ${\displaystyle m+1}$  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 ${\displaystyle 1/n!.}$ 

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

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

${\displaystyle \nu _{d}:\mathbb {P} (V)\ni [v]\mapsto [v^{d}]\in \mathbb {P} ({\rm {{Sym}^{d}V)}}}$ 

where V is any vector space of finite dimension, and ${\displaystyle {\rm {{Sym}^{d}V}}}$  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).

Rational normal curve edit

For ${\displaystyle n=1,}$  the Veronese variety is known as the rational normal curve, of which the lower-degree examples are familiar.

• For ${\displaystyle n=1,d=1}$  the Veronese map is simply the identity map on the projective line.
• For ${\displaystyle n=1,d=2,}$  the Veronese variety is the standard parabola ${\displaystyle [x^{2}:xy:y^{2}],}$  in affine coordinates ${\displaystyle (x,x^{2}).}$ 
• For ${\displaystyle n=1,d=3,}$  the Veronese variety is the twisted cubic, ${\displaystyle [x^{3}:x^{2}y:xy^{2}:y^{3}],}$  in affine coordinates ${\displaystyle (x,x^{2},x^{3}).}$ 

The image of a variety under the Veronese map is again a variety, rather than simply a constructible set; furthermore, these are isomorphic in the sense that the inverse map exists and is regular – the Veronese map is biregular. More precisely, the images of open sets in the Zariski topology are again open.

• The Veronese surface is the only Severi variety of dimension 2

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