Let $f:X \to Y$ be a projective morphism between noetherian scheme. Is there any know condition under which there exists a functorial stratification of $Y$ i.e., there exists a filtration by locally closed subschemes $ \emptyset= S_0 \subset S_1 \subset S_2 \subset ... \subset S_r=Y$ such that $f^{-1}(S_i \backslash S_{i-1})$ is flat over $S_i\backslash S_{i-1}$ and for any morphism $g:T \to Y$, if the base change $X \times_Y T \to T$ is flat over $T$, then there exists an $i$ such that $g$ factor through $S_i\backslash S_{i-1}$? Any hint or reference will be most welcome.
$\begingroup$
$\endgroup$
3
-
$\begingroup$ Isn't this how it is constructed in Mumford's book, Curves on a surface? I do not have the book next to me, but please check if you have. $\endgroup$– MohanCommented May 11, 2018 at 16:28
-
$\begingroup$ This is definitely in Mumford's book, and in the end boils down to the fact that a projective morphism is flat if and only if the Hilbert polynomial is constant. The chapter is available on JSTOR. Let me know if you need a copy. $\endgroup$– Keerthi MadapusiCommented May 11, 2018 at 18:37
-
$\begingroup$ @Mohan I found the result on the stratification. Does this also guarentee the filtration i.e., does the strata glue in some sense? $\endgroup$– user43198Commented May 11, 2018 at 21:49
Add a comment
|