Definition 43.13.5. Let $X$ be a nonsingular variety.
Let $W,V \subset X$ be closed subvarieties with $\dim (W) = s$ and $\dim (V) = r$. We say that $W$ and $V$ intersect properly if $\dim (V \cap W) \leq r + s - \dim (X)$.
Let $\alpha = \sum n_ i [W_ i]$ be an $s$-cycle, and $\beta = \sum _ j m_ j [V_ j]$ be an $r$-cycle on $X$. We say that $\alpha $ and $\beta $ intersect properly if $W_ i$ and $V_ j$ intersect properly for all $i$ and $j$.