On the Simultaneously Generation of Jets of the Adjoint Bundles

TL;DR

This paper studies the conditions under which certain adjoint line bundles on algebraic varieties can generate jets of specified order simultaneously, providing bounds that are linear in the jet order and applicable in arbitrary characteristics.

Contribution

It establishes new bounds for the simultaneous generation of jets of adjoint bundles, extending results to singular varieties and arbitrary characteristics.

Findings

Bound of m is linear in r, specifically m ≥ n + r + 1.

Results apply to varieties over arbitrary characteristics.

Extends jet generation results to singular varieties with dualizing complexes.

Abstract

In this paper, we investigate the problem of simultaneously generations of $r$-jets of $\omega_X\otimes L^{\otimes m}$ when $L$ is ample and base point free. It turns out that in this case, the bound of $m$ is optimistic, i.e. $m\geq f(r)=n+r+1$, which is linear in $r$. Our results work over arbitrary characteristics. We also treat the same problem when $X$ is singular where $\omega_X$ is replaced by the pushforward of canonical sheaves or the cohomology sheaves of dualizing complexes.