Question

Hi, everybody.

Consider an ${\rm S}_{1}$- morphism $f:X\rightarrow S$ of reduced complex spaces. Assume that $f$ is open (universally open in Alg.geom), equidimensional with $n$-pure dimensional fiber, surjectiv. Let $U$ be the flat locus of $f$ (which is a dense open set).

Question: It is true that the codimension of $(X-U)\cap X_{s}$ is of codimension 2 in the fiber $X_{s}$ ?

Remark: We can refer to the Thm 15.2.2, p.226 and Prop 4.7.10 of [EGA].

Thank you very much...

Answer 1

Take $X=\mathbb{C}$, $S$= the cuspidal plane cubic $y^2=x^3$, and $f$= the normalization map $t\mapsto (t^2,t^3)$. This is a universal homeomorphism. The flat locus is $U=\mathbb{C}^*$, so $X\setminus U$ is the whole fiber at $(0,0)$.