In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Veronese surface is an
algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...
in five-dimensional
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
, and is realized by the Veronese embedding, the embedding of the
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...
given by the complete
linear system of conics. It is named after
Giuseppe Veronese
Giuseppe Veronese (7 May 1854 – 17 July 1917) was an Italian mathematician. He was born in Chioggia, near Venice.
Education
Veronese earned his laurea in mathematics from the Istituto Tecnico di Venezia in 1872.
Work
Although Veronese's work w ...
(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
: