Geometric genus

Geometric genus

In algebraic geometry, the geometric genus is a basic birational invariant "p""g" of algebraic varieties, defined for non-singular complex projective varieties (and more generally for complex manifolds) as the Hodge number "h""n",0 (equal to "h"0,"n" by Serre duality). In other words for a variety "V" of complex dimension "n" it is the number of linearly independent holomorphic "n"-forms to be found on "V". This definition, as the dimension of

:"H"0("V",Ω"n")

then carries over to any base field, when Ω is taken to be the sheaf of Kähler differentials and the power is the (top) exterior power.

The definition of geometric genus is carried over classically to singular curves "C", by decreeing that

:"p""g"("C")

is the geometric genus of the normalization "C"′. That is, since the mapping

:"C"′ → "C"

is birational, the definition is extended by birational invariance.

The geometric genus is the first invariant "p""g" = "P1" of a sequence of invariants "Pn" called the plurigenera.

ee also

*Genus (mathematics)
*Arithmetic genus
*Invariants of surfaces

References

*


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Genus (disambiguation) — Genus, from the Latin genus, generis birth, race, stock, kind may refer to: *In Linnaean taxonomy, genus is the rank between family and species. * Genus (mathematics), a classifying property of a mathematical object ** Genus of a multiplicative… …   Wikipedia

  • Genus (mathematics) — In mathematics, genus has a few different, but closely related, meanings:TopologyOrientable surfaceThe genus of a connected, orientable surface is an integer representing the maximum number of cuttings along closed simple curves without rendering …   Wikipedia

  • Geometric invariant theory — In mathematics Geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper… …   Wikipedia

  • Geometric moray — Taxobox name = Geometric moray regnum = Animalia phylum = Chordata classis = Actinopterygii ordo = Anguilliformes familia = Muraenidae genus = Gymnothorax species = G. griseus binomial = Gymnothorax griseus binomial authority = (Lacepède,… …   Wikipedia

  • Geometric Tortoise — Taxobox name = Geometric Tortoise status = EN | status system = IUCN2.3 regnum = Animalia phylum = Chordata classis = Reptilia ordo = Testudines familia = Testudinidae genus = Psammobates species = P. geometricus binomial = Psammobates… …   Wikipedia

  • Arithmetic genus — In mathematics, the arithmetic genus of an algebraic variety is one of some possible generalizations of the genus of an algebraic curve or Riemann surface.The arithmetic genus of a complex manifold of dimension n can be defined as a combination… …   Wikipedia

  • List of geometric topology topics — This is a list of geometric topology topics, by Wikipedia page. See also: topology glossary List of topology topics List of general topology topics List of algebraic topology topics Publications in topology Contents 1 Low dimensional topology 1.1 …   Wikipedia

  • List of mathematics articles (G) — NOTOC G G₂ G delta space G networks Gδ set G structure G test G127 G2 manifold G2 structure Gabor atom Gabor filter Gabor transform Gabor Wigner transform Gabow s algorithm Gabriel graph Gabriel s Horn Gain graph Gain group Galerkin method… …   Wikipedia

  • David Mumford — in 1975 Born 11 June 1937 (1937 06 11) (age 74) …   Wikipedia

  • Enriques–Kodaira classification — In mathematics, the Enriques–Kodaira classification is a classification of compact complex surfaces into ten classes. For each of these classes, the surfaces in the class can be parametrized by a moduli space. For most of the classes the moduli… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”