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.
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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 a mapping
given by
where
Motivation
The Veronese surface arises naturally in the study of conics, specifically in formalizing the statement that five points determine a conic. A conic is a degree 2 plane curve, thus defined by an equation:
The pairing between coefficients
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
For low degree,
One may define the Veronese map in a coordinate-free way, as
where V is any vector space of finite dimension, and
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
For
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.
Biregularity has a number of important consequences. Most significant is that the image of points in general position under the Veronese map are again in general position, as if the image satisfies some special condition then this may be pulled back to the original point. This shows that "passing through k points in general position" imposes k independent linear conditions on a variety.
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.