Let $f: X\to Y$ be a regular map of projective varieties that is closed (in the sense that it takes Zariski closed sets to Zariski closed sets). Let $V\subset X$ be a quasiprojective subvariety (i.e. locally closed and irreducible). Is $f(V)$ a quasiprojective subvariety of $Y$?
(I'm aware that under arbitrary regular maps, the image of quasiprojective need only be constructible, but I am assuming the map is closed).
I am also interested in the more general question in the topological category - see Difference of closed sets under closed map.
UPDATE: This has been resolved at https://mathoverflow.net/questions/335512/image-of-quasiprojective-variety-under-closed-map, where a counterexample of a blow-down is given.