In algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, the geometric genus is a basic birational invariant

Birational invariant

In algebraic geometry, a birational invariant is a quantity or object that is well-defined on a birational equivalence class of algebraic varieties...

 p_g of algebraic varieties and complex manifold

Complex manifold

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....

s.

Definition

The geometric genus can be defined for non-singular complex projective varieties and more generally for complex manifold

Complex manifold

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....

s as the Hodge number h^n,0 (equal to h^0,n by Serre duality

Serre duality

In algebraic geometry, a branch of mathematics, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n . It shows that a cohomology group Hi is the dual space of another one, Hn−i...

), that is, the dimension of the canonical linear system.

In other words for a variety V of complex dimension

Complex dimension

In mathematics, complex dimension usually refers to the dimension of a complex manifold M, or complex algebraic variety V. If the complex dimension is d, the real dimension will be 2d...

 n it is the number of linearly independent holomorphic n-forms

Differential form

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...

 to be found on V. This definition, as the dimension of

H⁰(V,Ωⁿ)

then carries over to any base field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

, when Ω is taken to be the sheaf of Kähler differential

Kähler differential

In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes.-Presentation:The idea was introduced by Erich Kähler in the 1930s...

s and the power is the (top) exterior power, the canonical line bundle

Canonical bundle

In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n is the line bundle\,\!\Omega^n = \omegawhich is the nth exterior power of the cotangent bundle Ω on V. Over the complex numbers, it is the determinant bundle of holomorphic n-forms on V.This is the dualising...

.

The geometric genus is the first invariant p_g = P₁ of a sequence of invariants P_n called the plurigenera.

The 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 2g-2.

The notion of genus features prominently in the statement of the Riemann-Roch theorem (see also Riemann-Roch theorem for algebraic curves) and of the Riemann-Hurwitz formula

Riemann-Hurwitz formula

In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. It therefore connects ramification with algebraic topology, in this case...

.

If C is an irreducible (and smooth) hypersurface in the projective plane

Algebraic geometry of projective spaces

Projective spaces play a central role in algebraic geometry. The aim of this article is to define the notion in terms of abstract algebraic geometry and to describe some basic uses of those spaces.- Homogeneous polynomial ideals:...

 cut out by a polynomial equation of degree d, then its normal line bundle is the Serre twisting sheaf , so by the adjunction formula, the canonical line bundle of C is given by .

Genus of singular varieties

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.

See also

• Genus (mathematics)
  Genus (mathematics)
  In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
  
  
• Arithmetic genus
  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 projective complex manifold...
  
  
• Invariants of surfaces