User eugene eisenstein - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T22:40:35Z http://mathoverflow.net/feeds/user/7756 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/33931/possible-singularities-of-the-base-of-a-mori-fiber-space Possible singularities of the base of a Mori fiber space Eugene Eisenstein 2010-07-30T19:04:58Z 2010-10-16T07:15:40Z <p>Suppose X is a normal projective complex variety, (X, $\Delta$) is a klt pair and f : X $\to$ Z is a Mori fiber space given by a contraction of an extremal ray for this pair. Here I mean that the relative dimension is at least 1. Do we know anything about the singularities of Z? Z is normal more or less by construction and we do know that if X was Q-factorial then so is Z. Can we say anything more? Eg: is there a $\Delta'$ on Z with (Z, $\Delta'$) klt? Does Z have rational singularities? What if $\Delta$ = 0, is Z terminal? Canonical?</p> http://mathoverflow.net/questions/34016/what-are-the-k-rational-points-of-kt/34084#34084 Answer by Eugene Eisenstein for What are the k-rational points of k[[t]]? Eugene Eisenstein 2010-08-01T04:11:55Z 2010-08-01T04:11:55Z <p>Perhaps one answer to your question about deformations is something like the following. A deformation over a complete local ring A (such as k[[t]]) is just a family X $\to$ Spec(A). Suppose that the fibers belong to some sort of moduli space M, such as the moduli space of curves. In the functorial point of view of moduli spaces, the family X $\to$ Spec(A) corresponds to a morphism Spec(A) $\to$ M that assigns to a point of Spec(A) the moduli of the fiber over this point. So, one parameter formal deformations (by this I just mean that A = k[[t]]) correspond precisely to the morphisms Spec(k[[t]]) $\to$ M. The scheme Hom(k[[t]], M) is called the space of arcs in M. If we fix the central fiber of the deformation then we get the space of arcs in M at the point corresponding to the central fiber. The space of arcs is a subtle and important invariant of a singularity. One can think of an arc (that is, a morphism Spec(k[[t]]) $\to$ M) as follows: if we had a curve in M then the arc would be the collection of <a href="http://en.wikipedia.org/wiki/Jet_(mathematics)" rel="nofollow">jets</a> this curve determines, ie all the derivatives of all orders of the curve (think of the way morphisms Spec$(k[t]/(t^2)) \to M$ determine the tangent vectors at the image of the closed point). So these deformations are telling us something significant about the local structure of the moduli space.</p> <p>The construction of these one parameter formal deformations works regardless of the existence of any moduli space. It tells us what the space of arcs on the moduli space of whatever it is you are deforming should be.</p> http://mathoverflow.net/questions/28125/what-is-a-flop-and-when-are-they-conjectured-to-give-derived-equivalences/33244#33244 Answer by Eugene Eisenstein for What is a flop (and when are they conjectured to give derived equivalences)? Eugene Eisenstein 2010-07-25T01:47:22Z 2010-07-25T01:47:22Z <p>I'm not any kind of expert on this stuff and I'm not sure what the current state of this conjecture is, but Kawamata has conjectures in <a href="http://arxiv.org/PS_cache/math/pdf/0205/0205287v3.pdf" rel="nofollow">this paper</a> and <a href="http://arxiv.org/PS_cache/math/pdf/0311/0311139v2.pdf" rel="nofollow">this paper</a> regarding when two birational varieties have equivalent derived categories. He also discusses flops in the first paper.</p> <p>He has partial results, including: if $X$ is general type and $\mathcal{D}^b(X) \cong \mathcal{D}^b(Y)$ as triangulated categories then $X$ and $Y$ are K-equivalent. This generalizes the famous theorem of Bondal-Orlov that the bounded derived category of a Fano variety determines the variety. IIRC, in the proof of his theorem he takes the kernel of the Fourier-Mukai transform that gives the equivalence, shows that the support of the kernel (meaning the union of the supports of the cohomology sheaves of the kernel) has a component $Z$ dominating both varieties and uses $Z$ for the "roof" of the K-equivalence. The assumption that $X$ is general type is used to show that the projections from $Z$ are birational.</p> http://mathoverflow.net/questions/26606/the-fibers-of-m-g-n-to-m-g-and-the-fulton-macpherson-compactification/33229#33229 Answer by Eugene Eisenstein for The fibers of M_{g,n} \to M_g and the Fulton-MacPherson compactification Eugene Eisenstein 2010-07-24T21:16:27Z 2010-07-24T21:16:27Z <p>It may be worth mentioning briefly that the special case of $\overline{M_{0,n}}$ can be constructed fairly explicitly by blow-ups. The Fulton-MacPherson construction works. There are at least two additional constructions: an inductive one by Sean Keel in "Intersection theory on moduli space of stable N-pointed curves of genus zero" and the construction of Kapranov using what he calls Veronese curves towards the end of <a href="http://arxiv.org/PS_cache/alg-geom/pdf/9210/9210002v1.pdf" rel="nofollow">this paper</a> and in an earlier paper not on the arXiv that he sites. Kapranov's construction expresses $\overline{M_{0,n}}$ as a composition of blow-ups of $\mathbb{P}^{n-3}$ at an explicit sequence of (strict transforms of) linear subspaces.</p> http://mathoverflow.net/questions/32841/replacing-spectrum-with-valuations-of-a-field-an-alternative-to-schemes/32849#32849 Answer by Eugene Eisenstein for Replacing Spectrum with Valuations of a Field - An Alternative to Schemes? Eugene Eisenstein 2010-07-21T21:14:20Z 2010-07-21T21:14:20Z <p>This "space of valuations" to me sounds like the Riemann-Zariski space. Generally this should be pretty nasty, for surfaces eg it is some kind of limit of the system of all possible blow-ups. It is an old idea.</p> <p>For a different direction towards compactifying $Spec(\mathbb{Z})$, you can read about Arakelov theory and look at the recent papers of Connes and Consani on the arXiv about geometry over the "field" with one element.</p> http://mathoverflow.net/questions/21205/nef-divisors-with-few-global-sections/32543#32543 Answer by Eugene Eisenstein for Nef divisors with few global sections Eugene Eisenstein 2010-07-19T21:46:50Z 2010-07-19T21:46:50Z <p>Here is an example in a slightly different vein. There is a surface with a nef line bundle on it so that every power of the line bundle has global sections of dimension one. This example is possibly due to Cutkosky? I got it out of Rob Lazarsfeld's book PAGI.</p> <p>Let $C$ be a curve of genus $g \geq 1$ and let $P$ be a line bundle of degree zero that is not torsion, that is, $P^{\otimes m} \neq O_C$ for any $m \geq 1$. There are plenty of them since $Pic^0(C)$ is a non-trivial abelian variety and therefore torsion is countable. Consider the vector bundle $V = O_C \oplus P$ and let $X = \mathbb{P}(V)$ be the associated projective bundle. This is a ruled surface. Consider the line bundle $O_X(1)$.</p> <p>Let us compute the sections of $O_X(m)$. Recall that $H^0(X, O_X(m)) = H^0(C, Sym^m V)$. Since $P$ is degree zero and not torsion, $H^0(C, O_C(kP)) = 0$ for all $k \geq 1$. But $Sym^m V = O_C \oplus V'$ where $V'$ is a direct sum of line bundles of the form $O_C(kP)$ for $k \geq 1$. It follows that $h^0(X, O_X(m)) = 1$ for all $m \geq 1$.</p> <p>But I claim that $O_X(1)$ is nef. Indeed, $X$ is a surface and if $0 \neq D \in |O_X(1)|$ (we just showed it exists) then recall that, since $V = O_C \oplus P$ with $P$ degree zero, $(D^2) = deg(V) = deg(P) = 0$. So $D$ is nef.</p> <p>There is a similar but scarier example, due to Mumford, of a ruled surface so that $h^0(X, O_X(m)) = 0$ for all $m \geq 1$ and yet $O_X(m)$ is nef.</p> http://mathoverflow.net/questions/26546/about-b-divisors/32537#32537 Answer by Eugene Eisenstein for About b-divisors Eugene Eisenstein 2010-07-19T21:10:02Z 2010-07-19T21:10:02Z <p>It can be convenient sometimes, eg: <a href="http://arxiv.org/abs/math/0608260" rel="nofollow">http://arxiv.org/abs/math/0608260</a>. They use this language to define their positive intersection product, which is defined in terms of supremums over all choices of certain nef classes on all birational modifications. Note that they want classes, not numbers, for their statements about the derivative of the volume.</p> http://mathoverflow.net/questions/31605/does-negative-kodaira-dimension-imply-uniruled/32534#32534 Answer by Eugene Eisenstein for Does negative Kodaira dimension imply uniruled? Eugene Eisenstein 2010-07-19T21:02:07Z 2010-07-19T21:02:07Z <p>The conjecture you attribute to Mumford is also sometimes called weak non-vanishing, or just non-vanishing. As already mentioned, by BDPP one is reduced to proving that if $h^0(mK_X) = 0$ for all $m \geq 0$ then $K_X$ is not even pseudo-effective, that is, not numerically a limit of effective divisors. There is an alternative formulation of the abundance conjecture which is arguably easier to understand: if $K_X + \Delta$ is nef and $(X, \Delta)$ is klt then $K_X + \Delta$ is semi-ample. The point is that this says that running the MMP turns $K_X$ into a semi-ample divisor.</p> <p>My understanding is that the answer to question to 2 is something like the following. Siu explains and gives references in his introduction to Part II of his paper that there is something called the numerically trivial fibration that is similar to the Kodaira-Iitaka fibration but works for the numerical dimension. Its construction is analytic in nature. Siu claims that, if $\pi : X' \to Y$ is a realization of numerically trivial fibration for $K_X$ with $X'$ a birational model of $X$, then $\pi_* O_{X'}(mK_{X'/Y})$ carries some kind of metric with strictly positive curvature on a general fiber of $\pi$. This, in particular, shows that $\pi$ is not the identity map, which I think is now known to imply the abundance conjecture, though I am not sure if this is how Siu proceeds. I am not familiar with the details of the argument.</p> <p>As for question 3, it is hard to say. The famous example is Siu's proof of the deformation invariance of plurigenera for all Kodaira dimensions, this is currently a purely analytic argument that involves taking limits of pluri-subharmonic functions to obtain a semipositive singular metric on $O_X(mK_X)$ that may not have analytic singularities but has the right multiplier ideal: see Paun's "Siu's invariance of plurigenera: a one-tower proof." This has resisted attempts to prove it algebraically so far. On the other hand, the algebraic approach allows reduction to positive characteristic. It may be that, ultimately, anything you can do with one you can do with the other. Of course, "algebraic" here allows the Kodaira vanishing theorem, which can be deduced from the homological statement that the map $H^i(X, \mathbb{C}) \to H^i(X, O_X)$ induced by the natural inclusion of sheaves is surjective.</p> http://mathoverflow.net/questions/34777/stability-of-vector-bundle Comment by Eugene Eisenstein Eugene Eisenstein 2010-08-06T16:43:56Z 2010-08-06T16:43:56Z Are X and Y irreducible curves? If not, is f finite? If you don't have control over $R^1 f_* F$ for subsheaves $F$ of $f^* E$ there isn't much hope I don't think. http://mathoverflow.net/questions/34717/analytic-tools-in-algebraic-geometry Comment by Eugene Eisenstein Eugene Eisenstein 2010-08-06T05:48:06Z 2010-08-06T05:48:06Z I really hesitate to try to answer any of this. All I'll say is, in birational geometry, the analog of Sard's theorem and moving lemmas works well in the vanilla Zariski topology. The notion of &quot;general&quot; and &quot;very general&quot; seems adequate so far. For uncountable fields, the complement of a countable union of hypersurfaces is dense is an algebraic Baire category theorem. Of course, over countable fields like $\overline{\mathbb{Q}}$ the matter is very different. But then some of the things that follow from arguments about very general points are expected to fail in that setup, eg cycles. http://mathoverflow.net/questions/32841/replacing-spectrum-with-valuations-of-a-field-an-alternative-to-schemes/32849#32849 Comment by Eugene Eisenstein Eugene Eisenstein 2010-07-25T23:22:30Z 2010-07-25T23:22:30Z Thank you Brian, you are absolutely right. I should have said &quot;the topology can be different from what we are used to on schemes&quot; instead of &quot;nasty.&quot; Also thank you for mentioning Borger.