Timeline for Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces
Current License: CC BY-SA 3.0
12 events
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| Apr 21, 2013 at 7:17 | comment | added | Laurent Moret-Bailly | @Omprokash Das: it is not true in general that you can embed $k(x)$ into $\mathcal{O}_{X,x}$ or even into the function field of $X$. Example: assume $k$ is not algebraically closed, take $X=\mathbb{P}^n_k$ and take for $x$ a closed point with residue field $\neq k$. Variant: same $X$ ($n\geq2$, $k$ arbitrary), $x$= generic point of a non-unirational subvariety, e.g. a curve of positive genus. | |
| Apr 20, 2013 at 19:40 | comment | added | Angelo | It follows easily from the fact that a finitely dimensional algebra is Jacobson; in particular, the closed points are dense. So, if there is only one closed point, the spectrum consists of that point, so the algebra is 0-dimensional, hence artinian. | |
| Apr 20, 2013 at 19:32 | comment | added | Omprokash | Thanks Angelo, could you please give a proof of this fact or probably suggest a reference? | |
| Apr 20, 2013 at 19:13 | comment | added | Angelo | No, a localization is practically never finitely generated. The local rings that are of finite type over a field are artinian. | |
| Apr 20, 2013 at 18:05 | comment | added | Omprokash | Hi Laurent, You are right, $S$ does bot have any ''natural'' $k(x)$-scheme structure but there is a obvious ''non natural'' $k(x)$-scheme structure, since I started with a Variety $X$, so $k(x)\hookrightarrow \mathcal{O}_{X,\ x}$ and the only reason for doing this is to get a finite type morphism so that I can say $S$ is a variety. Now that you asked about it, I am little worried, is $\mathcal{O}_{X,\ x}$ a finitely generated $k(x)$ algebra ? I always thought it is, but now I am not so sure! | |
| Apr 20, 2013 at 16:01 | comment | added | Omprokash | Hi Karl and Angelo, thanks for taking time to read my problem. Here is exactly what I have and what I need. Let $X$ be a normal variety over an algebraically closed filed $k$ of char $p>0$. Let $x$ be codim $2$ point of $X$. Assume $S=\text{Spec }\mathcal{O}_{X,\ x}$. Then $S$ is a surface over the field $k(x)$. Now by Lipman, $S$ has a resolution of singularities, call $f:Y\to S$. Let $B$ be a $\mathbb{Q}$-Cartier $\mathbb{Q}$ divisor such that $B$ is $f$-nef and $f$-big and the fractional part of $B$ has SNC support. Then $R^if_*\mathcal{O}_X(K_Y+\lceil B\rceil)=0$ for all $i>0$. | |
| Apr 20, 2013 at 14:42 | comment | added | Laurent Moret-Bailly | I don't understand "over $k(x)$". Clearly $S$ is a two-dimensional local scheme with closed point $\mathrm{Spec}\,k(x)$, but in general it is not a $k(x)$-scheme in any natural way. | |
| Apr 20, 2013 at 12:03 | comment | added | Angelo | Since the usual version of the Kawamata-Viehweg vanishing theorem generalized Kodaira's, you'd better write down the exact statement that you need. | |
| Apr 20, 2013 at 5:18 | comment | added | Karl Schwede | Presumably for surfaces non-algebraically closed fields, depending on exactly what you want, perhaps things are ok by some base change (perhaps as long as your exceptional set is geometrically whatever)? If you are looking for references, I know that some Grauert-Riemenschneider-type statements are even true for excellent two dimensional rings (see for example the work of Lipman on resolution of 2 dimensional singularities and also his work on rational singularities in any characteristic). Can you be more precise in terms of what exactly you need? | |
| Apr 19, 2013 at 19:53 | comment | added | Omprokash | Yes Angelo you are right, Kadaira vanishing does fail in char positive even on a surface, but some version of Kawamata-Viehweg vanishing theorem which is sufficient to the run the Minimal Model Program continues to hold even in positive characteristic for surfaces. You can look for a reference in the paper I mentioned above or in a preprint by Kollar and Kovacs, "Birational Geometry of Log Surfaces" | |
| Apr 19, 2013 at 19:30 | comment | added | Angelo | Doesn't Kodaira vanishing fail for surfaces in positive characteristic? Raynaud gave counterexamples. | |
| Apr 19, 2013 at 19:13 | history | asked | Omprokash | CC BY-SA 3.0 |