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 of dimension n 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.^[1] This definition, as the dimension of

H⁰(V,Ωⁿ)

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 = P₁ 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

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

where s is the number of singularities when properly counted.

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 ⁠${\displaystyle {\mathcal {O}}}$⁠(d), so by the adjunction formula, the canonical line bundle of C is given by

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

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.

See also

• Genus (mathematics)
• Arithmetic genus
• Invariants of surfaces