Advantage of discrepancy

Question

In the definition of Minimal model of projective variety, some authors use of discrepancy, and some others omit this condition. I am wondering to know the advantage of discrepancy In the definition of Minimal model.

Answer 1

Take $(X,B)$ and $(Y,B_Y)$, be the log canonical pairs and assume we have birational map $φ:X⟶Y$, we need to take $d(E,X,B)≤d(E,Y,B_Y)$ for any prime divisor $E$ on $X$. In fact discrepancy is measure of singularities, and this condition $d(E,X,B)≤d(E,Y,B_Y)$ saying that the singularities of $Y$ is at least as good as singularities of $X$

This is equivalent to say that there exists a common resolution $g:W→X$, and $h:W→Y$ s.t, $g^∗(K_X+B)≥h^∗(K_Y+B_Y)$ to preserve sections .

Heuristically we want always there exists a natural map from $$H^0(X,m(K_X+B))→H^0(Y,m(K_Y+B_Y))$$ and if that condition of discrepancy holds then we have such natural injective map