When we say, that, say, a surface contains ${\infty}^{k}$ lines, do we mean that it contains a k-parameter family of lines? Do we assume that this family is parametrized by a $P^{k}$, say, or we use this term more informally?
This is certainly a standard notation, but I didn't see its explanation in standard modern textbooks.



The meaning of ${\infty}^{k}$.
The terminology is fairly common in classical works on projective algebraic/differential geometry. I am not aware of its origins. Anyway it is used rather informally and only means
that you have a $k$-dimensional "continuous" family of objects. The parameter space usually is only local and should be thought as a small ball in $\mathbb C^k$. 


