Timeline for regularity of finite flat branched covers
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 16, 2016 at 9:04 | vote | accept | IMeasy | ||
| May 8, 2013 at 15:04 | comment | added | Francesco Polizzi | Alternatively, one can use Hurwitz formula to define the branch locus, at least when $C$ is Gorenstein. In fact, $K_C = f^*K_S + R$, where $R$ is the ramification divisor. Then $D=f_*R$ as a cycle, so it makes sense to speak of non-reduced branch divisor. For instance, in Joel example the ramification divisor is the intersection of the two lines. It is linearly equivalent to a general fibre of the covering, which is given by two distinct points sent by $f$ to the same point. So $f_*R=D$ is a point counted with multiplicity two. | |
| May 8, 2013 at 14:56 | comment | added | Francesco Polizzi | Ok, so let use the following definition that can be found in the paper by Iversen "Numerical invariants of multiple planes", p. 971. Let $x \in S$ and $y_1, \ldots, y_r$ be the points of $f^{-1}x$. Let $d_i$ be the discriminant of the extension $\widehat{\mathcal{O}_x} \to \widehat{\mathcal{O}_{y_i}}$. Then a local equation of $D$ in $x$ is given by $\prod d_i$. | |
| May 8, 2013 at 14:41 | comment | added | quim | @Francesco: We should be able to see that the schematic branch locus has multiplicity two by looking at the ring map $\mathbb{C}[x]\rightarrow\mathbb{C}[x,y]/(x^2-y^2)$. So Joël's question is pertinent: what is the definition of branch locus? If this is not a counterexample for the OP's definition of branched along $D$ (because it is actually branched along $2D$, which is not regular) then maybe your deleted (why?) answer was the right one. | |
| May 8, 2013 at 14:41 | comment | added | Joël | Ah okay. So what is the definition of the schematic branch locus? | |
| May 8, 2013 at 13:41 | comment | added | Francesco Polizzi | Joel: right. Anyway, in your example the schematic branch locus is not reduced (is one point counted with multiplicity two). Indeed, your cover is the limit of a family of flat double covers $f_t \colon \mathbb{P}^1 \to \mathbb{P}^1$, all branched in two points, when the two points come together. | |
| May 8, 2013 at 13:37 | answer | added | Francesco Polizzi | timeline score: 4 | |
| May 8, 2013 at 13:10 | comment | added | Joël | I am not sure what branched means. But if $D$ is an affine line over $\spec \mathbb C$, $S$ a closed point of $D$ (so $D$ and $S$ are clearly regular), and $C$ the union of two copies of $D$, intersecting at the point $S$, then the natural morphism $C \rightarrow D$ (the one sending each irreducible components of $D$ to $C$ by the identity map) is finite flat, and étale outside $S$, while $C$ is not regular. | |
| May 8, 2013 at 9:27 | history | asked | IMeasy | CC BY-SA 3.0 |