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 geometric genus is a basic
birational invariant of
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
and
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...
s.
Definition
The geometric genus can be defined for
non-singular complex projective varieties and more generally for
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...
s as the
Hodge number (equal to by
Serre duality
In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Ale ...
), that is, the dimension of the
canonical linear system plus one.
In other words, for a variety of
complex dimension In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety. These are spaces in which the local neighborhoods of points (or of non-singular points in the case of a variety) are modeled on ...
it is the number of linearly independent holomorphic -
forms to be found on .
[Danilov & Shokurov (1998), p. 53/ref> This definition, as the dimension of
:
then carries over to any base field, when is taken to be the sheaf of Kähler differentials and the power is the (top) ]exterior power
In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...
, the canonical line bundle.
The geometric genus is the first invariant of a sequence of invariants called the plurigenera.
Case of curves
In the case of complex varieties, (the complex loci of) non-singular curves are Riemann surfaces. The algebraic definition of genus agrees with the topological notion. On a nonsingular curve, the canonical line bundle has degree .
The notion of genus features prominently in the statement of the Riemann–Roch theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It re ...
(see also Riemann–Roch theorem for algebraic curves) and of the Riemann–Hurwitz formula. By the Riemann-Roch theorem, an irreducible plane curve of degree ''d'' has geometric genus
:
where ''s'' is the number of singularities when properly counted.
If is an irreducible (and smooth) hypersurface in the projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...
cut out by a polynomial equation of degree , then its normal line bundle is the Serre twisting sheaf , so by the adjunction formula, the canonical line bundle of is given by
:
Genus of singular varieties
The definition of geometric genus is carried over classically to singular curves , by decreeing that
:
is the geometric genus of the normalization
Normalization or normalisation refers to a process that makes something more normal or regular. Science
* Normalization process theory, a sociological theory of the implementation of new technologies or innovations
* Normalization model, used in ...
. That is, since the mapping
:
is birational, the definition is extended by birational invariance.
See also
*Genus (mathematics)
In mathematics, genus (: genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1.
Topology
Orientable surfaces
The genus of a co ...
* Arithmetic genus
* Invariants of surfaces
Notes
References
*
* {{cite book , author1=V. I. Danilov , author2=Vyacheslav V. Shokurov , title=Algebraic curves, algebraic manifolds, and schemes , publisher=Springer , year=1998 , isbn=978-3-540-63705-9
Algebraic varieties