Thank you! Actually I knew fact 1 you cite, I simple didn't think over it while writing the question since in my context I always work up to a general deformation of f. Moreover $∑a_i$ is independent of $f$ for every $[f]∈V$, which is consistent with example suggested by Jason Starr. I didn't know the second fact you cite, now I check it

Hi Damian, set-theoretically this is just the set of points $x \in X$ such that there exists a deformation of the curve $C \doteq f(\mathbb{P}^1)$ passing through $x$. I used the word "locus" cause I didn't try to prove that it is a (closed) subvariety in $X$, although this should be true (there a similar exercise in Debarre's book Higher dimensional algebraic geometry).