The 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.
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, so by the adjunction formula, the canonical line bundle of C is given by .
Read more about this topic: Geometric Genus
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