Timeline for Is $X\times \mathbb{P}^{n}$ connected for a smooth, proper and connected scheme $X$?
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
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| Aug 25, 2010 at 19:45 | comment | added | Jack Huizenga | Wouldn't noting that the map $\mathbb{P}^n_X \to X$ admits a section suffice? | |
| Aug 25, 2010 at 14:50 | comment | added | damiano | A natural place where this happens is for instance the case on the smooth stratification of the union $X$ of the two axes in the affine plane. Obviously the scheme $X$ is connected; on the other hand, we can look at the stratification of $X$ into smooth strata, and in this case we find the complement of the origin in $X$ and the origin. Clearly there is a surjective morphism from the disjoint union of the two strata to $X$, each fiber is connected (and consists of a single point), and the target is connected. Clearly again, the whole space is not connected! | |
| Aug 25, 2010 at 14:24 | comment | added | BCnrd | Not every continuous surjective map with connected fibers onto a connected base has connected source; must use that the map to the base is either open (since flat and locally of finite presentation; concretely in this case it is Zariski-locally an affine space, for which life is easy) or closed (because of properness). | |
| Aug 25, 2010 at 13:54 | history | answered | Charles Siegel | CC BY-SA 2.5 |