Timeline for Where is the representability of the moduli of curves with framed points proved?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 11, 2012 at 18:20 | comment | added | Jason Starr | @S. Carnahan -- I missed that you are asking about representability by a quasi-projective scheme. I think you already see one answer for yourself. I could ask Alexander Kirillov, Jr. what was his argument the next time that I see him (almost daily). | |
| Jan 10, 2012 at 22:30 | comment | added | S. Carnahan♦ | Thanks. I should have mentioned that freeness of the $PGL(5g+3n-5)$ action on the scheme of tricanonically embedded curves with framed points implies the stack is an algebraic space. In order to get a scheme, some more work is needed. Possibilities that come to mind include showing that all orbits lie in affine opens of $H_{g,n}$, giving an explicit Zariski cover, or writing down a compactification together with an ample line bundle. | |
| Jan 10, 2012 at 17:28 | comment | added | Moosbrugger | Right, and if the automorphism groups are trivial then it's an algebraic space. | |
| Jan 10, 2012 at 16:16 | comment | added | Jason Starr | @S. Carnahan -- The stack is always algebraic. So, as Moosbrugger suggests, you automatically get that your stack is algebraic. Once you know that your stack has finite, reduced stabilizer groups, then it follows that it is a Deligne-Mumford stack. Quite possibly your particular stack has never been proved to be DM in the literature. But this approach to proving a stack such as your is DM is developed, for instance, in Behrend's article "Gromov-Witten Invariants in Algebraic Geometry". | |
| Jan 10, 2012 at 15:11 | comment | added | S. Carnahan♦ | The moduli stack of pointed curves is not in general representable (i.e., for $n$ small compared to $g$), so this doesn't always help for representability of the total stack. | |
| Jan 10, 2012 at 14:56 | comment | added | Moosbrugger | The map to the moduli of pointed curves (without tangent vectors) seems to be trivially representable -- am I missing something? | |
| Jan 10, 2012 at 14:26 | history | asked | S. Carnahan♦ | CC BY-SA 3.0 |