# 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

given by

where
denotes homogeneous coordinates. The map
is known as the **Veronese embedding.**

## Motivation

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

The pairing between coefficients and variables 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 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

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

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

The map sends
to all possible monomials of total degree *d*, thus the appearance of combinatorial functions; the
and
are due to projectivization. The last expression shows that for fixed source dimension *n,* the target dimension is a polynomial in *d* of degree *n* and leading coefficient

For low degree,
is the trivial constant map to
and
is the identity map on
so *d* is generally taken to be 2 or more.

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

where *V* is any vector space of finite dimension, and
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 *p*th 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

For the Veronese variety is known as the rational normal curve, of which the lower-degree examples are familiar.

- For the Veronese map is simply the identity map on the projective line.
- For the Veronese variety is the standard parabola in affine coordinates
- For the Veronese variety is the twisted cubic, in affine coordinates

## Biregular

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.

## See also

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

## References

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