Multigraded Fujita Approximation

TL;DR

This paper generalizes the Fujita approximation theorem to multigraded linear series, showing that the volume of these series can be approximated by simpler subseries, extending the classical approximation to a broader setting.

Contribution

It introduces a multigraded version of Fujita approximation, expanding the theorem's applicability to multigraded linear series and their volume approximation.

Findings

The volume of multigraded linear series can be approximated by finitely generated subseries.

The classical Fujita approximation is extended to multigraded settings.

The approximation converges as the grading parameter increases.

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.