Multigraded Fujita Approximation

Abstract

The original Fujita approximation theorem states that the volume of a big divisor $D$ on a projective variety $X$ can always be approximated arbitrarily closely by the self-intersection number of an ample divisor on a birational modification of $X$. One can also formulate it in terms of graded linear series as follows: let $W_{\bullet} = \{W_k \}$ be the complete graded linear series associated to a big divisor $D$: \[ W_k = H^0\big(X,\mathcal{O}_X(kD)\big). \] For each fixed positive integer $p$, define $W^{(p)}_{\bullet}$ to be the graded linear subseries of $W_{\bullet}$ generated by $W_p$: \[ W^{(p)}_{m}={cases} 0, &\text{if $p\nmid m$;} \mathrm{Image} \big(S^k W_p \rightarrow W_{kp} \big), &\text{if $m=kp$.} {cases} \] Then the volume of $W^{(p)}_{\bullet}$ approaches the volume of $W_{\bullet}$ as $p\to\infty$. We will show that, under this formulation, the Fujita approximation theorem can be generalized to the case of multigraded linear series.