Timeline for Moduli of smooth curves
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 15, 2023 at 13:22 | comment | added | Mikhail Katz | @SamNead, at any rate, I think the hyperbolic approach is more elementary. You can construct hyperbolic surfaces by gluing together right-angled hyperbolic polygons. So constructing surfaces with desired properties just boils down to the existence of certain rather simple polygons in the hyperbolic plane, with no fancy moduli space theory. See e.g., Hyperbolic octagons and Teichmüller space in genus 2. Aline Aigon-Dupuy; Peter Buser; Michel Cibils; Alfred F. Künzle; Frank Steiner. J. Math. Phys. 46, 033513 (2005) doi.org/10.1063/1.1850177 based on earlier work by Buser and others. | |
| Oct 15, 2023 at 13:04 | comment | added | Mikhail Katz | So you are using "exits" in the sense of "tends to infinity in the moduli space", or "approaches the boundary at infinity". Is this a standard term in algebraic geometry? I am not sufficiently familiar with the field. If not, the term is not immediately understood. | |
| Oct 15, 2023 at 13:02 | comment | added | Sam Nead | It exists and it exits (moduli space). The latter is the hard part. | |
| Oct 15, 2023 at 13:00 | comment | added | Mikhail Katz | @SamNead, do you mean "exists"? | |
| Oct 15, 2023 at 12:56 | comment | added | Sam Nead | Yes, I know that we can metrically pinch a curve, and you know that too.... but it is not going to help the original poster. (And, if we are going to pinch a curve then I prefer the family I gave in my answer. You can prove, using the euclidean metric, that there is a large modulus annulus about the points $z_{2g + 1}$ and $z_{2g + 2}$. Since moduli of annuli are conformal invariants, my family exits). | |
| Oct 15, 2023 at 12:51 | comment | added | Mikhail Katz | One way of doing it is by looking at the particular case of the space of tori with one (geodesic) circle component, and showing that the boundary circle can be arbitarily short. Then one can double this torus with boundary to get a genus 2 surface with small systole. Any work with hyperbolic surfaces is going to involve more calculation than the case of elliptic curves, but it is easy enough to find papers dealing with the case of torus with boundary. @SamNead | |
| Oct 15, 2023 at 12:45 | comment | added | Sam Nead | The problem with this answer is that it requires two more pieces of work from the original asker: why is this "new" moduli space homeomorphic to the old one? And (as you mention): why are there hyperbolic surfaces with systoles tending to zero? | |
| Oct 15, 2023 at 12:17 | history | answered | Mikhail Katz | CC BY-SA 4.0 |