Omprokash Das
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Registered User
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I am a Graduate Student at the University of Utah studying Birational Geometry.
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Apr 21 |
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Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces Thanks Laurent! |
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Apr 20 |
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Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces Got it! Thanks again. |
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Apr 20 |
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Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces Thanks Angelo, could you please give a proof of this fact or probably suggest a reference? |
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Apr 20 |
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Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces 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! |
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Apr 20 |
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Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces 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 $i>0$. |
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Apr 19 |
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Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces 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" |
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Apr 19 |
asked | Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces |
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Apr 17 |
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Adjunction Formula for Weil Divisors on a Normal Variety X Corrected spelling |
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Apr 17 |
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Adjunction Formula for Weil Divisors on a Normal Variety X That is just a typo! I mean Divisor, sorry about that! |
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Apr 17 |
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Adjunction Formula for Weil Divisors on a Normal Variety X deleted 157 characters in body |
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Apr 17 |
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Adjunction Formula for Weil Divisors on a Normal Variety X Improved the description of the problem.; edited body |
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Apr 17 |
asked | Adjunction Formula for Weil Divisors on a Normal Variety X |
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Mar 27 |
awarded | ● Commentator |
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Mar 27 |
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How to Construct a ‘'Nice’' Birational Model in Characteristic p>0 Hi Will, I realized that I have asked some stupid questions above! Also the map $\phi$ I defined above doesn't make sense! Sorry about that. But could you please tell how did you say '' We clearly have a natural immersion from $\Spec\ B[y/x]$ to this blow-up'' ? I need this very badly for my work. Thanks! I am waiting for your reply. |
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Mar 23 |
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How to Construct a ‘'Nice’' Birational Model in Characteristic p>0 Since the normalization morphism is finite so I am hopping that the inverse image of $S'$ will be normal too! I am could be wrong, I haven't checked it yet. |
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Mar 23 |
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How to Construct a ‘'Nice’' Birational Model in Characteristic p>0 there is a closed immersion from $Spec\ B[y/x]$ to $\tilde{X}$. This how I see it: Let $\phi: \oplus_{d\geq 0} (x',y')^d \to B[y/x][t]$ be graded morphism defined naturally, then $\phi (C_+)= (B[y/x][t])_+$, where $C_+=\oplus_{d> 0} (x',y')^d$. So there is a closed immersion from $Spec\ B[y/x]$ to $\tilde{X}$. Now for your question that once we have the strict transform $S'$ of $S$ as normal in $Y$, then how do we know whether $Y$ is normal or not. What I am thinking about it is that take the normalization $\tilde{Y}$ of $Y$ and then the inverse image of $S'$ inside it |
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Mar 23 |
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How to Construct a ‘'Nice’' Birational Model in Characteristic p>0 Hi Will, thanks for your reply. I did something very similar as you. I took directly the the normalization of $B$ and then similar calculation as you but without the language of blow up, however I liked your arguments over mine. I have some questions though, you said there is a natural immersion from $Spec\ B[y/x]$ to the blow up, say $\tilde{X}$ and then you argued that since $Spec\ B[y/x]$ is proper over $A$, so it's a closed immersion. I have two questions here: (1) How do you say $Spec\ B[y/x]$ is proper over $A$ ? (2) I find it more natural to say directly that |
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Mar 23 |
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How to Construct a ‘'Nice’' Birational Model in Characteristic p>0 Then $S'$ is normal |
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Mar 23 |
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How to Construct a ‘'Nice’' Birational Model in Characteristic p>0 Hi Will, by $S'$ normal I mean every local ring of $S'$ is integrally closed domain, I don't need $S'$ to be irreducible, even though in the situation stated above it will be irreducible since $S$ is. Your example doesn't quite fit to my problem, your $S$ is union of two lines but mine is an irreducible closed subset of co-dimesion $1$. However even in your example it is possible to make $S'$ normal, since resolution of singularity exists in dimension $2$, therefore we can choose a log resolution $g:Y\to X$ such that support $S'=f^{-1}_*S$ is disjoint union of smooth irreducible components. |
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Mar 22 |
awarded | ● Autobiographer |
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Mar 22 |
asked | How to Construct a ‘'Nice’' Birational Model in Characteristic p>0 |
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Mar 1 |
awarded | ● Supporter |
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Mar 1 |
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On a Strongly F-regular Pair (X, \Delta) Thank you very much Karl for your reply! Now I know that it's not gonna work the way I was trying to make it work. To be honest I was expecting that you would answer something when I asked this problem yesterday, :-) Anyway, I looked at your paper and it seems $P^0$ might be a good candidate for me! I have the appropriate assumption on $M-K_X-\Delta$ that it's ample. So, under this assumption is the relation true replacing $S^0$ by $P^0$ on both sides? I will read the paper carefully. Thanks again. |
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Feb 28 |
awarded | ● Editor |
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Feb 28 |
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On a Strongly F-regular Pair (X, \Delta) deleted 4 characters in body |
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Feb 28 |
awarded | ● Student |
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Feb 28 |
asked | On a Strongly F-regular Pair (X, \Delta) |
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Nov 20 |
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About the Definition of Flat Morphism (Flat Sheaf) Thanks to all of you guys for your valuable comments! |
