Fujita’s conjecture states: If L is an ample divisor on a smooth projective variety X of dimension n, then K + (n+1)L is globally generated and K + (n+2)L is very ample. An affirmative answer to Fujita’s conjecture would give an effective way to embed smooth projective varieties in projective space given only the datum of an ample divisor. While recent work of Gu, Zhang, and Zhang shows that Fujita’s conjecture is false over fields of positive characteristic, it remains an open question whether there exist uniform constants a and b only depending on n such that aK + bL is globally generated or very ample. Such constants are known to exist over fields of characteristic zero.

In this talk, I will present my recent progress on this problem for surfaces over fields of arbitrary characteristic. I will also present my approach to this problem in arbitrary dimensions that reduces to finding lower bounds for Seshadri constants of divisors of the form cK + dL.