Timeline for Smoothness of ramification divisor
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 24, 2020 at 22:20 | answer | added | aglearner | timeline score: 2 | |
| Jul 14, 2018 at 9:18 | comment | added | Frank | Even that is too optimistic. It might help to take a look at this paper in combination with Jason's comment above arxiv.org/abs/0811.0467 | |
| Jul 13, 2018 at 19:57 | comment | added | Daniel Loughran | Possible way to save my question: Is the branch divisor always a simple normal crossings divisor? | |
| Jul 12, 2018 at 23:06 | comment | added | Dmitri Panov | Here is a classical example when the branching divisor is not smooth. Take $\mathbb C^2\to \mathbb C^2$ with $(x,y)\to (x^2,y^2)$. This extends to a map $\mathbb CP^2\to \mathbb CP^2$. | |
| Jul 12, 2018 at 21:55 | comment | added | Daniel Loughran | I see, I had no idea of this theory. I'm just familiar with the case of double covers, where I believe the branch curve is always smooth. | |
| Jul 12, 2018 at 21:41 | comment | added | Jason Starr | I do not understand this question. For a smooth surface $X$ in $\mathbb{P}^n$, for a general linear projection from $X$ to $Y=\mathbb{P}^2$, the branch divisor is almost never smooth. Classically, the double points, cusps, etc., of the branch divisor were the invariants that were studied in trying to understand the geometry of polarized surfaces. For instance, for a hypersurface in $\mathbb{P}^3$, the double points correspond to bitangent lines that contains a specified general point of $\mathbb{P}^3$. These numbers were stand-ins for Chern numbers prior to Chern. | |
| Jul 12, 2018 at 21:00 | comment | added | Daniel Loughran | In this paper they seem to consider normal surfaces, rather than just smooth surfaces. Can you be more specific? | |
| Jul 12, 2018 at 20:56 | comment | added | Ja ok | You might find useful results in Edixhoven-de Jong- Schepers article Covers of surfaces with fixed branch locus | |
| Jul 12, 2018 at 20:40 | history | asked | Daniel Loughran | CC BY-SA 4.0 |