Timeline for reduced-ness of the fibres of stein factorisation
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jan 23, 2018 at 11:28 | comment | added | Sándor Kovács | @LaurentMoret-Bailly: I meant $Y$, the target of $f$, but I should really say "image" I suppose. (I had $Y=Z$ in mind as I said in the third sentence, so the target of both $f$ and $g$ were the same). | |
| Jan 22, 2018 at 8:09 | comment | added | Laurent Moret-Bailly | @SándorKovács: My target is normal. Do you mean the source? | |
| Jan 21, 2018 at 15:40 | comment | added | Sándor Kovács | @LaurentMoret-Bailly: right. I meant to say "such that the target is normal". | |
| Jan 20, 2018 at 8:59 | comment | added | Laurent Moret-Bailly | @SándorKovács: If, say, $Z=\mathrm{Spec}\,k$ and $X=\mathrm{Spec}\,(k[t]/t^2)$, then $Y=X$ anf $f$ is the identity. (I now realize that my phrase "is its own Stein factorization" is ambiguous). | |
| Jan 20, 2018 at 3:56 | comment | added | Sándor Kovács | How about just take any morphism with a non-reduced fiber. Then it is its own Stein factorisation, so no. You can even make $X,Y=Z$ smooth and $f$ flat. | |
| Jan 19, 2018 at 16:20 | comment | added | Laurent Moret-Bailly | As a matter of fact, you can even restrict the preceding example to $t=0$: then $Z=\mathrm{Spec}\,k$ and $X$ is the conic $x^2=0$ in $\mathbb{P}^2$, which is its own Stein factorization since $\mathrm{H}^0(X,\mathscr{O}_X)=k$. | |
| Jan 19, 2018 at 14:21 | comment | added | Laurent Moret-Bailly | Another (perhaps more elementary) example: take $Z=\mathrm{Spec}\,k[t]$, and let $X\subset \mathbb{P}^2_Z$ be the relative conic defined by $x^2=ty^2$. Then $Y=Z$ but the fiber at $t=0$ is not reduced. | |
| Jan 19, 2018 at 14:16 | comment | added | Jason Starr | Typo correction: "... at $0$ with exceptional divisor $E_1$." --> "... at $x_0$ with exceptional divisor $E_1$." | |
| Jan 19, 2018 at 12:36 | comment | added | Jason Starr | Some fibers may be nonreduced. Let $Z$ equal $\mathbb{A}^2_k$, and let $x_0$ be a smooth $k$-point. Form the blowing up, $\nu_1:Z_1\to Z,$ at $0$ with exceptional divisor $E_1$. For a smooth point $x_1\in E_1$, denote by $\nu_2:Z_2\to Z_1$ the blowing up at $x_1$ with exceptional locus $E_2$. For the strict transform $\widetilde{E}_1\subset Z_2$, there is a unique intersection point $x_2\in \widetilde{E}_1\cap E_2$. Denote the blowing up at $x_2$ by $\nu_3:X\to Z_2$. The composite $f=\nu_1\circ \nu_2\circ \nu_3$ has connected fibers, but the fiber over $x_0$ is nonreduced. | |
| Jan 19, 2018 at 12:26 | history | asked | user111251 | CC BY-SA 3.0 |