# Name for the variety of preimages of a finite morphism

If $f:X\to Y$ is a finite morphism of degree $d$ between two varieties, you get a closed subset of the symmetric product $X^{(d)}$ (or perhaps rather the Hilbert scheme $X^{[d]}$), defined as the closure of the locus of points having the same image in $Y$. Is there a standard name for this locus?

• I guess there's no standard name (at least I've never seen one, and no one has answered yet). How about 'total ramification locus'? – Mattia Talpo Nov 28 '15 at 7:36