Timeline for Rationality of Hilbert schemes?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 30, 2016 at 5:01 | vote | accept | Elle Najt | ||
| Dec 30, 2016 at 4:02 | comment | added | Elle Najt | @NoamD.Elkies Oh okay - I had it backwards, I thought you were trying to build a map from $M_g$. Thanks for your patience. | |
| Dec 30, 2016 at 3:42 | comment | added | Noam D. Elkies | Is all that really necessary? All I'm saying is that the image of the natural map $H_{d,g} \to {\cal M}_g$ is dense in ${\cal M}_g$; so I need only check pointwise that every genus-$g$ curve is in the image by finding some degree-$d$ embedding of it in ${\bf P}^3$. | |
| Dec 30, 2016 at 3:35 | answer | added | Noam D. Elkies | timeline score: 9 | |
| Dec 30, 2016 at 3:33 | comment | added | Elle Najt | @NoamD.Elkies You show that a fixed curve of genus $g$ can be embedded as a degree $d>>g$ curve in $P^3$. To get a flat family of space curves I think we can find a divisor in the universal curve over $M_g^0$ (curves with trivial automorphisms) which is degree d on each fiber, then use some form of relative Riemann-Roch to show it's very ample (maybe?), then embed this universal family in $M_g^0 \times P^N$, and argue that projecting to a particular$P^3$ still embeds the curves from an open set in $M_g^0$. I don't know how to find that divisor though ...maybe there is a better way? | |
| Dec 29, 2016 at 21:43 | comment | added | Noam D. Elkies | For $g>0$, if $d \gg g$ then the sections of any divisor of degree $d$ embed $C$ as a curve of degree $d$ in a projective space of dimension $\geq 3$. Now project to a generic ${\bf P}^3$ and you're done. | |
| Dec 29, 2016 at 21:36 | comment | added | Noam D. Elkies | The fact that ${\cal M}_g$ is of general type for $g \geq 24$ is "a famous result due to Harris, Mumford and Eisenbud" as G.Farkas puts it at mathematik.hu-berlin.de/~farkas/m22.pdf (which proves the same result for $g=22$). | |
| Dec 29, 2016 at 19:46 | history | edited | Elle Najt | CC BY-SA 3.0 |
added 43 characters in body
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| Dec 29, 2016 at 19:45 | comment | added | Elle Najt | @NoamD.Elkies Specifically I mean ... a reference for why $M_g$ is of general type, and for why when $d >> g$ $M_g$ dominates the corresponding Hilbert scheme. I'm also curious about the low degree cases as well, but I'm not sure where to study this material. | |
| Dec 29, 2016 at 19:39 | comment | added | Elle Najt | @NoamD.Elkies Thanks, I would be happy to accept that as an answer. Do you know a good reference for the higher genus case you describe? | |
| Dec 29, 2016 at 5:14 | comment | added | Noam D. Elkies | Surely it's rational for some $g$ and $d$, e.g. $(g,d)=(0,3)$ (rational normal cubics). Not for all $(g,d)$, though, because for $g$ large enough ${\cal M}_g$ is of general type, and dominates this Hilbert scheme once $d$ is large enough given $g$. | |
| Dec 29, 2016 at 3:08 | history | asked | Elle Najt | CC BY-SA 3.0 |