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.

The Veronese surface is a mapping

given by

where ${\displaystyle [x:\cdots ]}$ denotes homogeneous coordinates. The map ${\displaystyle \nu }$ is known as the Veronese embedding.

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:

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