Let $f:X\rightarrow Y$ be a regular map of smooth connected algebraic varieties (say over an algebraically closed field). I know that the image $f(X)$ is only a constructible set, in general, but I am interested in conditions that ensure $f(X)$ being an algebraic variety.

**A precise question:** suppose the differential $df$ has a constant rank. Is $f(X)$ an algebraic variety (or a locally closed subvariety of $Y$)?

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