Timeline for Examples of varieties with every stable sheaf simple
Current License: CC BY-SA 3.0
8 events
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| Aug 26, 2017 at 15:39 | comment | added | user43198 | @R.vanDobbendeBruyn For me, simple means that $\mathrm{End}(\mathcal{F})=k$. | |
| Aug 26, 2017 at 12:52 | comment | added | R. van Dobben de Bruyn | Yeah. Not sure if it's standard terminology. And depending on which definition of simple you use, it may or may not be automatic (does simple mean that $\operatorname{End}(\mathscr F)$ is a division algebra, or that it equals $k$?). | |
| Aug 26, 2017 at 6:36 | comment | added | user43198 | @R.vanDobbendeBruyn Thanks. By "(geometrically) simple", you mean simple as well as its pull-back to algebraic closure is simple? | |
| Aug 26, 2017 at 2:33 | comment | added | R. van Dobben de Bruyn | Any geometrically stable sheaf $\mathscr F$ on a smooth projective $k$-variety $X$ is (geometrically) simple. Indeed, as you note, by flat base change it suffices to prove the result for $k = \bar k$. Then stability implies that a nonzero endomorphism $\phi \colon \mathscr F \to \mathscr F$ has to be surjective, since otherwise either $\operatorname{im} \phi$ or $\ker \phi$ has to be a proper subsheaf of larger slope. But a surjective endomorphism of a coherent sheaf is an isomorphism (see e.g. here). | |
| Aug 25, 2017 at 11:08 | history | edited | user43198 | CC BY-SA 3.0 |
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| Aug 25, 2017 at 10:58 | history | edited | Ben McKay | CC BY-SA 3.0 |
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| Aug 25, 2017 at 10:47 | history | edited | user43198 | CC BY-SA 3.0 |
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| Aug 25, 2017 at 8:16 | history | asked | user43198 | CC BY-SA 3.0 |