Veronese Surface - 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, more familiarly the binomial coefficient, or more elegantly the rising factorial, 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 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).