Editing Canonical bundle (section)

====Low genus====

Suppose that ''C'' is a smooth algebraic curve of genus ''g''. If ''g'' is zero, then ''C'' is '''P'''<sup>1</sup>, and the canonical class is the class of &minus;2''P'', where ''P'' is any point of ''C''. This follows from the calculus formula ''d''(1/''t'') = &minus;''dt''/''t''<sup>2</sup>, for example, a meromorphic differential with double pole at the origin on the [[Riemann sphere]]. In particular, ''K''<sub>''C''</sub> and its multiples are not effective. If ''g'' is one, then ''C'' is an [[elliptic curve]], and ''K''<sub>''C''</sub> is the trivial bundle. The global sections of the trivial bundle form a one-dimensional vector space, so the ''n''-canonical map for any ''n'' is the map to a point.