Moving Seshadri constants and effective Fujita-type conjectures

Fujita's conjecture is known to be false in positive characteristic. We conjecture and give an approach to a new variant of Fujita's conjecture for the basepoint-freeness, very ampleness, and jet ampleness of linear systems of the form $\lvert K_X+aK_X+bL\rvert$. We extend the theory of moving Seshadri constants, previously established for smooth complex projective varieties by Ein, Lazarsfeld, Musta\c{t}\u{a}, Nakamaye, and Popa, to the more general setting of complete varieties over arbitrary fields. This theory is both an important component of our approach to this new conjecture and of independent interest. Using our approach, we prove our variant of Fujita's conjecture for smooth surfaces in arbitrary characteristic and for smooth complex projective varieties of arbitrary dimension.