Matt
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Registered User
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May 2 |
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morphism between two elliptic curves over a local field Thank you!! That was exactly the point I was missing. |
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May 1 |
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morphism between two elliptic curves over a local field Is there some obvious Galois cohomology way to see this? Since twists are parametrized by $H^1(G_{\overline{K}/K}, Aut(X))$ doesn't this say that this group is trivial? |
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Apr 6 |
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trivial deformation of a smooth affine scheme over complete DVR Ah, thanks! I've clearly been indoctrinated by only thinking about liftability of smooth varieties. |
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Apr 6 |
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trivial deformation of a smooth affine scheme over complete DVR I'm confused Angelo. The question says $X$ is affine and the obstruction to such a lifting lies in $H^2$ which vanishes, so doesn't such a lift always exist? |
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Mar 25 |
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Existence of non-split vector bundles on smooth projective varieties Even if it is well-known, isn't it still fairly standard to include these types of things as corollaries? It shows that your work has applications to things that people care about and gives alternate proofs of known results. Both these things seem worth it. |
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Mar 20 |
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About equivalent statements of the Birch and Swinnerton-Dyer Conjecture I actually have no idea what your question is from this. Saying what your question is about is quite a bit different than asking a question directly. |
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Mar 15 |
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Family with a fixed special fiber over finite fields Alright, then I think you mean that $X'$ should be over $\mathbb{Z}_p$ otherwise the dimensions won't make any sense. Every smooth projective curve always lifts to characteristic $0$ because the obstructions to deforming both the curve and an ample line bundle lie in $H^2$ which vanishes since it is a curve. So by Grothendieck's Existence Theorem the formal lift algebraizes. Surfaces are more delicate. There are known results like every K3 surface lifts to characteristic $0$, but there are also known surfaces that do not lift. |
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Mar 15 |
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Family with a fixed special fiber over finite fields This might be standard, but a variety over $\mathbb{Q}_p$ with "special fiber ..." just means a scheme $\mathfrak{X}/\mathbb{Z}_p$ whose generic fiber is $X'$ and special fiber is $X$ right? There are lots of examples of when this can't happen. Sometimes they formally lift, but aren't projective. Sometimes the deformation theory is highly obstructed. I'm not sure there can ever be some general conditions on $X$ that would guarantee a lift. Are you interested in a more specific type of variety or at least of a particular dimension. |
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Mar 15 |
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Definition of relative Picard functor If the map is $f:X\to S$, then you could define $Pic_{X/S}(T)=H^0(T, R^1f_{T*}\mathbb{G}_m)$. The notation is really that we're defining the functor of points of something. When doing that it is pretty standard to say the $T$-points are blah up to equivalence where equivalence means blah. So that notation is just meant to say the points are elements of $Pic(X\times T)$ where two line bundles are equivalent if they differ by the pullback of something from $T$. The quotient is by an equivalence relation and not necessarily a subgroup. Thanks Ryan, that was the obvious thing to try. |
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Mar 14 |
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Definition of relative Picard functor Oops, I guess no one can read that equation. There really needs to either be a sample feature or an edit comment button. Having neither is absurd. |
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Mar 12 |
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Tate conjecture for abelian varieties over a finitely generated extension of an algebraically closed field This might help, mathoverflow.net/questions/116900/… |
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Feb 28 |
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Essential geometric morphisms on the étale site. So $Sh(X)$ means sheaves on the small étale site? I only ask because it is stated that $f$ is étale in the question, but the site is only mentioned in the subject line. |
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Feb 16 |
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Geometric fibers of schemes. "Why does exactly this the right job?" For what? I'd argue that this doesn't do the right thing if what you are interested in is arithmetic properties of the fiber. This gives you "geometric" information about the fiber and hence is the "geometric fiber." Where did you see that this did the "right job?" We need this information if we want to answer why it did it. |
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Feb 12 |
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Open subset in the flat topology on Spec(R) Also, for the record, that second comment was a reply to a now deleted comment. I wasn't repeating myself unprovoked. |
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Feb 12 |
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Open subset in the flat topology on Spec(R) I should probably stop before I upset some people, but seriously, the term is "generization." Look in Hartshorne or any textbook that talks about Zariski spaces or sober spaces or even PlanetMath: planetmath.org/encyclopedia/Generization.html |
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Feb 12 |
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Open subset in the flat topology on Spec(R) Two people have said this now, so it is making me nervous, but isn't the term "stable under generization"? |
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Jan 28 |
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A question from GTM 52 of Hartshone If you combine MBeasy and Matthieu's comments you get a "section 3" solution. The ideas needed are in the proof of Theorem 3.2 plus one tiny extra piece of commutative algebra about normal Noetherian domains. |
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Jan 18 |
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Retracted Mathematics Papers Adam, what legal reasons? If you refer to the recent "Science Fraud" issues, then you have nothing to fear unless you plan on actually accusing mathematicians of fraud. Pointing out that a paper has errors is not the same thing, and should be encouraged. Mistakes happen and they are honest mistakes with no misconduct behind them. |
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Dec 31 |
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Lifting to Characteristic 0 not over W Oh. I see. Thanks so much! This theorem kind of blows my mind now that I see how it works better. |
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Dec 31 |
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Lifting to Characteristic 0 not over W Ah, thanks. That's pretty good I guess. |
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Dec 30 |
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Lifting to Characteristic 0 not over W Sorry. I think I'm being dumb. This paper has always confused me (I looked at it when first trying to find this example to find examples that lift to $W_2$ but not $W_3$). Are you saying the example is some component of the Hilbert scheme of nonsingular curves? Or are you saying that we create it from starting over $\text{Spec}(\mathbb{Z})$? Anything that starts over $\text{Spec}(\mathbb{Z})$ shouldn't work (but I think I'm not understanding) because base change to $W$ should give you the lift to $W$. Four people upvoted this immediately, so it is probably me missing the obvious. |
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Dec 30 |
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Lifting to Characteristic 0 not over W Is this "answer" supposed to be a comment? Am I missing some joke (based on Jacob's comment)...? |
