Surface de Veronese

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. The Veronese surface is the image of the mapping given by where denotes homogeneous coordinates. The map 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: 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". 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 (of which there are ); we have since there are variables to choose from; and we subtract since the projective space has 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 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.