Geometric genus
# Geometric genus

In algebraic geometry, the geometric genus is a basic birational invariant $p_g$ 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 $h^{n,0}$ (equal to $h^{0,n}$ 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

    $H^0(V,Ω^n)$

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 canonical line bundle.

The geometric genus is the first invariant $p_g$ = $P_1$ of a sequence of invariants $P_n$ 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 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. 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 ${\mathcal {O}}$(d), 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

    $p_g(C)$

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

    C′ → C

is birational, the definition is extended by birational invariance.

