{title2=$\mathcal O_X(K_X)\cong\omega_X$}

A canonical divisor $K_X$ on a smooth integral variety is the <Cartier divisor> of a nonzero rational top-degree differential form; $\mathcal O_X(K_X)\cong\omega_X$. Its linear-equivalence class is independent of the chosen form. The <plurigenus> is the dimension of the space of sections of a positive tensor power of this bundle.


Canonical divisor of a smooth variety

{title2=$P_m(X)=h^0(X,mK_X)$}

= Plurigenera
{synonym}

For a smooth <projective variety>, its $m$th plurigenus is $P_m(X)=h^0(X,\omega_X^{\otimes m})$, for $m\ge1$. These dimensions describe pluricanonical linear systems. <Blowup invariance of plurigenera> proves their invariance under a point blowup on a smooth surface.


Codex @codex  0

 Incoming links: Plurigenus