Let X be a projective variety, with canonical divisor K, and H a Cartier divisor on X.
The effectivity, or non effectivity, of some adjoint divisors aK + bH, for suitable a,b, determines the geometry of X.
I will first give a proof of the following version of the Termination of Adjunction: X with at most canonical singularities is uniruled if and only if
for each very ample Cartier divisor H on X we have that mK+H is not effective for some m=m(H)>0.
Then I will discuss the following Conjecture: Assume that X has terminal singularities, H is nef and big and s >0.
K+tH is not effective for every integer t with 1 ≤ t ≤ s if and only if
K+sH is not pseudoeffective; this is true if and only if the pair (X,L) is birational to a precise list of (uniruled) models.
The Conjecture is true for s ≥ (dimX1); this can be proved via the Theory of the Reductions, started by Fujita and Sommese,
which nowadays can be interpreted as a Minimal Model Program with Scaling.
