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 $V$ .^[1] 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 $2 g - 2$ .

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 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 ⁠ ${\mathcal {O}}$ ⁠ $(d)$ , 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' \to C

is birational, the definition is extended by birational invariance.

Notes

^ Danilov & Shokurov (1998), p. 53

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.

[1] Danilov & Shokurov (1998), p. 53

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