Veronese mapping
A special regular mapping of a projective space; named after G. Veronese. Let be positive integers,
, and
,
projective spaces over an arbitrary field (or over the ring of integers), regarded as schemes; let
be projective coordinates in
, and let
,
, be projective coordinates in
. The Veronese mapping is the morphism
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given by the formulas ,
. The Veronese mapping may be defined in invariant terms as a regular mapping given by a complete linear system
, where
is a hyperplane section in
. The Veronese mapping is a closed imbedding; its image
is called a Veronese variety, and is defined by the equations
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where . For instance,
is the curve represented by the equation
in
. The degree of a Veronese variety is
. For any hypersurface
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in its image with respect to the Veronese mapping
is the intersection of the Veronese variety
with the hyperplane
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Owing to this fact, Veronese mappings may be used to reduce certain problems on hypersurfaces to the case of hyperplane sections.
References
[1] | I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 |
Comments
The image of in
under the Veronese imbedding (
,
) is called the Veronese surface.
References
[a1] | P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) pp. 178; 674; 179; 349; 525; 532; 535; 632; 743 MR0507725 Zbl 0408.14001 |
[a2] | R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 13; 170; 316; 381 MR0463157 Zbl 0367.14001 |
Veronese mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Veronese_mapping&oldid=13814