Timeline for Weierstrass Models and Canonical Models
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 21, 2013 at 10:13 | comment | added | TomPGR | Thank you both, this is very helpful. I have some reading to do! | |
| Feb 15, 2013 at 17:14 | comment | added | Qing Liu | continued: does not change when contracting $X$ to the canonical model $W$ (the latter has only rational singularities), this can be found in "Néron models", Theorem 9.7/1. | |
| Feb 15, 2013 at 17:10 | comment | added | Qing Liu | @nosr: the isomorphism is explained in S. Bloch: "de Rham cohomology and conductors of curves", Duke Math. J. 54 (1987), Lemma 1.2(i) when the model $X$ is regular. If we contract $X$ to the canonical model, this does not change the generic fiber, and the $H^1$ of the closed fiber doesn't change either (by explicit computations, e.g. Dolgacev: "On the purity of the degeneration loci of families of curves". Invent. Math. 8 (1969), Prop. 2.4). This can also be explained (at least when the curve has a rational point) by the fact that the neutral component of the Néron model of $Jac(X_{K})$... | |
| Feb 15, 2013 at 17:04 | comment | added | Qing Liu | @TomPGR:The canonical model in higher genus is obtained by contracting some $(-2)$-rational curves, and all such configurations of rational curves is classified by M. Artin (see my book, 10.1.53). They all appear in Kodaira-Néron's classification for elliptic curves. So the computation of zeta function is the same as for elliptic curves. | |
| Feb 13, 2013 at 5:46 | comment | added | user28172 | Let $R$ be a strictly henselian dvr with fraction field $K$ and residue field $k$, and $X$ a proper flat $R$-scheme with generic fiber $X_K$ geometrically connected and smooth of dimension 1 and positive genus. Assume $X$ is the minimal regular proper model of $X_K$. Choose a prime $\ell \in R^{\times}$. For $G_K := {\rm{Gal}}(K_s/K)$, you want $H^1(X_k, \mathbf{Q}_{\ell}) \rightarrow H^1(X_{K_s},\mathbf{Q}_{\ell})^{G_K}$ to be an isomorphism. This problem can be "localized" via vanishing cycles, and in general is subtle (but is tractable when $X_k$ is semistable). Let's see what Q. Liu says. | |
| Feb 12, 2013 at 13:46 | history | asked | TomPGR | CC BY-SA 3.0 |