In
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, the Zeuthen–Segre invariant ''I'' is an
invariant of a
projective surface found in a
complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
which was introduced by and rediscovered by .
The invariant ''I'' is defined to be ''d'' – 4''g'' – ''b'' if the surface has a
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 of ...
of
curves
A curve is a geometrical object in mathematics.
Curve(s) may also refer to:
Arts, entertainment, and media Music
* Curve (band), an English alternative rock music group
* Curve (album), ''Curve'' (album), a 2012 album by Our Lady Peace
* Curve ( ...
, non-singular of
genus
Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
''g'' except for ''d'' curves with 1 ordinary
node
In general, a node is a localized swelling (a "knot") or a point of intersection (a vertex).
Node may refer to:
In mathematics
* Vertex (graph theory), a vertex in a mathematical graph
*Vertex (geometry), a point where two or more curves, lines ...
, and with ''b'' base points where the curves are non-singular and transverse.
showed that the Zeuthen–Segre invariant ''I'' is χ–4, where χ is the topological
Euler–Poincaré characteristic introduced by , which is equal to the
Chern number ''c''
2 of the surface.
References
*
* Reprinted 2010
*
*
*
*
Algebraic surfaces
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