Geometric genus

In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds.

Definition

The geometric genus can be defined for non-singular complex projective varieties and more generally for complex manifolds as the Hodge number (equal to by Serre duality), that is, the dimension of the canonical linear system plus one.

In other words for a variety of complex dimension 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, 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 (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

:$g=\frac-s,$

where ''s'' is the number of singularities when properly counted

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

: \mathcal K_C = \left \mathcal K_ + \mathcal O(d) \right = \mathcal O(d-3)_

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 . That is, since the mapping

:

is birational, the definition is extended by birational invariance.

See also

*Genus (mathematics)
* 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