Geometric genus

In algebraic geometry, the geometric genus is a basic birational invariant $pg$ 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 $hn,0$ (equal to $h0,n$ by Serre duality), that is, the dimension of the canonical linear system plus one.

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



then carries over to any base field, when $H0(V,&Omega;n)$ 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 $&Omega;$ of a sequence of invariants $pg = P1$ 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 $Pn$.

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


 * $$g=\frac{(d-1)(d-2)}{2}-s,$$

where s is the number of singularities.

If $C$ is an irreducible (and smooth) hypersurface in the projective plane cut out by a polynomial equation of degree $d$, then its normal line bundle is the Serre twisting sheaf $\mathcal O$$2g &minus; 2$, so by the adjunction formula, the canonical line bundle of $C$ is given by


 * $$ \mathcal K_C = \left[ \mathcal K_{\mathbb P^2} + \mathcal O(d) \right]_{\vert C} = \mathcal O(d-3)_{\vert C} $$

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



is the geometric genus of the normalization $(d)$. That is, since the mapping



is birational, the definition is extended by birational invariance.