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Concept
Geometric genus
Summary
In algebraic geometry, the geometric genus is a basic birational invariant p_g of algebraic varieties and complex manifolds.
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.
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
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
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.
Official source
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