The authors' Fujita-type jet-ampleness conjecture

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$, let $L$ be an ample divisor on $X$, and let $\ell\geq 0$ be an integer. A line bundle or linear system is $\ell$-jet ample when it separates jets of total order prescribed by the definition in the source.

Fujita-type jet-ampleness conjecture. There exist integer-valued polynomial functions $a_\ell(n)$ and $b_\ell(n)$ depending only on $n$ such that

$$\bigl\lvert K_X+a_\ell(n)K_X+b_\ell(n)L \bigr\rvert$$

is $\ell$-jet ample. Moreover, the functions can be chosen so that, if $\operatorname{char}(k)>0$, the canonical linear system

$$\biggl\lvert S^0\Bigl(X,\omega_X\otimes_{\mathcal{O}_X}\mathcal{O}_X\bigl(a_\ell(n)K_X+b_\ell(n)L\bigr)\Bigr) \biggr\rvert$$

is $\ell$-jet ample.

This is proposed as a characteristic-independent replacement for Fujita's conjectures after positive-characteristic counterexamples. The paper presents it as the central conjecture and proves the variant for smooth surfaces in arbitrary characteristic and for smooth complex projective varieties of arbitrary dimension.