Pardon me if this question is trivial as I am just starting to learn the subject. Let $k$ be a number field. Given a smooth projective variety $X \subset \mathbb{P}_k^n $, a positive divisor $D$ in $\text{Div} _\bar{k} (X)$ and a prime divisor $Y$ in $\text{Div} _\bar{k} (X)$, can we say anything about $\text{deg}(D \cdot Y)$ besides being less than $\text{deg}(D)\cdot\text{deg}(Y)$? Any nontrivial lower bound? Under what condition (on $D$ and $Y$) do we have equality? (If $X = \mathbb{P}_k^n $, then we always have equality, so say we have a general $X$). Any good reference will be appreciated. Thank you.
