Timeline for Equi-dimensionality of special fibers in a flat family
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 3, 2016 at 15:22 | vote | accept | Qiao | ||
| Mar 3, 2016 at 12:58 | comment | added | Allen Knutson | I suppose I remembered the argument I gave because it also gets you that the special fiber is connected in codimension $1$ (which wouldn't be true without the properness assumption). | |
| Mar 2, 2016 at 22:00 | comment | added | nfdc23 | As t3suji says, it's easier without projectivity: can localize on $X$! For $f:X \rightarrow Y$ flat surjective of finite type with $Y$ noetherian and connected, if one fiber is equidimensional of dimension $d$, we claim all are. Link irreducible components of $Y$ and specialize from its generic points to reduce to base a dvr. Closures of generic fiber irred. components of $X$ are $Y$-flat, and they cover $X$ since $X$ is $Y$-flat, so enough to show for $X$ irred. with generic fiber of dimension $d$ that dim$(X_0)=d$ (so $X_0$ equidim'l by Zariski-localizing on $X$). Now EGA IV$_3$, 14.3.10. | |
| Mar 2, 2016 at 20:24 | comment | added | t3suji | I think Zariski's Main Theorem may be an overkill here, because the statement holds without the projectivity assumption. In fact, can't you just say that for every point $x\in X$, dimension of the fiber at x and dimension of X at x differ by 1 (Hartshorne, Proposition III.9.5). From this one easily sees that every component of X must dominate Y, therefore X has pure dimension d+1, and each fiber has pure dimension d. | |
| Mar 2, 2016 at 17:55 | history | answered | Allen Knutson | CC BY-SA 3.0 |