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 of dimension n 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
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 pg = P1 of a sequence of invariants Pn 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 = ( d − 1 ) ( d − 2 ) 2 − s , {\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 O {\displaystyle {\mathcal {O}}} (d), so by the adjunction formula, the canonical line bundle of C is given by
K C = [ K P 2 + O ( d ) ] | C = O ( d − 3 ) | C {\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
pg(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
Notes
References P. Griffiths; J. Harris (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. p. 494. ISBN 0-471-05059-8. V. I. Danilov; Vyacheslav V. Shokurov (1998). Algebraic curves, algebraic manifolds, and schemes. Springer. ISBN 978-3-540-63705-9.
