Timeline for Hilbert scheme of projectively normal elliptic curves
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 8, 2016 at 15:05 | comment | added | Abdelmalek Abdesselam | Even if you probably know that, I would mention the work of Tom Fisher at Cambridge dpmms.cam.ac.uk/~taf1000/research.html He studied these questions a lot, but he does not seem to have much for n>5. | |
| Mar 8, 2016 at 14:18 | comment | added | Jason Starr | Unfortunately, even the moduli space of genus 1 stable maps behaves badly. Of course there is some $\text{PGL}_n$-equivariant resolution of that moduli space. My colleague, Aleksey Zinger, has studied one particularly nice $\text{PGL}_n$-equivariant resolution of the moduli space of stable maps of genus 1 that has a good functorial property. | |
| Mar 8, 2016 at 14:11 | comment | added | Will Sawin | @JasonStarr No, I am not sure that I wrote it down correctly. Maybe any compactification, as long as it's $PGL_n$-equivariant, is fine for what Bhargava and Shankar want to do. In which case do I want to ask what the moduli space of stable maps is? Or maybe what the nicest point in the space of Bridgeland stability conditions is? | |
| Mar 8, 2016 at 14:05 | comment | added | Will Sawin | @Sasha Sure but such elliptic curves have a lot of extra structure. arxiv.org/pdf/1306.4424v1.pdf Theorem 2.1 describes this extra structure for the first case and Theorem 3.1 for the second case. | |
| Mar 8, 2016 at 10:02 | comment | added | Jason Starr | Just to answer my own question: it seems that the locus of pencils with a planar base locus contains the transform of several different loci in the Hilbert scheme. First, the union of a plane cubic and a line skew to the plane (intersecting the cubic) transforms to such. Next, plane quartics with two nodes and two embedded points at the nodes transform to pencils of quadrics whose base locus is a 2-plane with an embedded line. Also, plane quartics with a tacnode (or ramphoid cusp) transform to pencils of quadrics whose base locus is a union of a 2-plane and a skew line. | |
| Mar 8, 2016 at 9:31 | comment | added | Jason Starr | Certainly the closure of $X_n$ is not always smooth. This is not even true for the Hilbert scheme of genus $0$ curves, which is a big part of the reason that we use the moduli spaces of stable maps. By the way, in your description of $X_4$, what happens for a pencil of quadrics that contains a $2$-plane in its base locus? Are you certain you wrote down the Hilbert scheme, rather than one of its birational modifications (e.g., obtained by varying the Bridgeland stability condition)? | |
| Mar 8, 2016 at 8:33 | comment | added | Sasha | For $n = 6$, I guess, one can use the fact, that an elliptic curve of degree 6 can be represented as a codimension 2 linear section of $P^1 \times P^1 \times P^1$, or as a codimension 3 linear section of $P^2\times P^2$. | |
| Mar 8, 2016 at 4:40 | history | asked | Will Sawin | CC BY-SA 3.0 |