Fujita's freeness and very ampleness conjectures

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$, and let $L$ be an ample divisor on $X$.

Fujita's conjectures. The linear system

$$\bigl\lvert K_X+(n+1)L \bigr\rvert$$

is basepoint-free, and

$$\bigl\lvert K_X+(n+2)L \bigr\rvert$$

is very ample.

These are the classical freeness and very ampleness conjectures of Fujita. The paper notes that Fujita's conjectures are false in positive characteristic, motivating the variants studied there.