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?
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$\begingroup$ 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'? $\endgroup$– Mattia TalpoCommented Nov 28, 2015 at 7:36
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