0
$\begingroup$

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...

$\endgroup$
  • $\begingroup$ [EGA]? $\mbox{}$ $\endgroup$ – Joseph O'Rourke Sep 2 '10 at 0:04
  • 1
    $\begingroup$ "Éléments de géométrie algébrique" is the bible of this sect. $\endgroup$ – Donu Arapura Sep 3 '10 at 13:08
2
$\begingroup$

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 $(0,0)$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.