In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, ''trope'' is an archaic term for a singular (meaning special) tangent space of a
variety
Variety may refer to:
Arts and entertainment Entertainment formats
* Variety (radio)
* Variety show, in theater and television
Films
* ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont
* ''Variety'' (1935 film), ...
, often a
quartic surface
In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4.
More specifically there are two closely related types of quartic surface: affine and projective. An ''affine'' quartic surfac ...
. The term may have been introduced by , who defined it as "the reciprocal term to node". It is not easy to give a precise definition, because the term is used mainly in older books and papers on
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, whose definitions are vague and different, and use archaic terminology. The term ''trope'' is used in the theory of quartic surfaces in
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 ...
, where it is sometimes defined as a tangent space meeting the quartic surface in a conic; for example
Kummer's surface has 16 tropes.
, describes a trope as a tangent plane where the envelope of nearby tangent planes forms a conic, rather than a plane
pencil
A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage, and keeps it from marking the user's hand.
Pencils create marks by physical abrasion, leaving a trail ...
which we would expect for a generic point. The tangent plane would be tangent to the quartic along the conic, implying that the
Gauss map
In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere ''S''2. Namely, given a surface ''X'' lying in R3, the Gauss map is a continuous map ''N'': ''X'' → ''S''2 such that '' ...
would have a singular point.
See also
*
Glossary of classical algebraic geometry
The terminology of algebraic geometry changed drastically during the twentieth century, with the introduction of the general methods, initiated by David Hilbert and the Italian school of algebraic geometry in the beginning of the century, and later ...
References
* See page 202 for an early use of the term "trope".
*
*
* {{Citation , title=Classical Algebraic Geometry: A Modern View, first=Igor V. , last=Dolgachev, date=2012 , publisher=Cambridge University Press, isbn=978-1107017658
Algebraic geometry