User ru - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T14:42:33Z http://mathoverflow.net/feeds/user/13350 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115904/existence-of-open-dense-subset-in-a-lie-group Existence of open dense subset in a Lie group Ru 2012-12-09T18:22:49Z 2012-12-09T18:22:49Z <p>Let $G=S\cdot R$ be a Levi-Malcev decomposition of a connected complex Lie group and $\Gamma$ a discrete subgroup of $G$ such that the subgroups<br> $\Gamma_g := \Gamma\cap (gSg^{-1})$ are finite for all $g\in G$. Is it true that there exist a dense open subset W of G such that the<br> $\Gamma_g$ are all isomorphic for all $g\in W$? </p> http://mathoverflow.net/questions/112618/leray-spectral-sequence Leray Spectral Sequence Ru 2012-11-16T22:48:07Z 2012-11-17T02:27:44Z <p>Let $f:X\to Y$ be a smooth map between paracompact differential manifolds $X$ and $Y$. Let $U$ be an open and dense subset of $Y$. For any $y\in U$, let $f^{-1}(y)=F$ be a generic fiber that is a submanifold of $F$.<br> Assume the singular fibers are $F/\Gamma_t$, where for each $t\in Y\setminus U$, $\Gamma_t$ is a finite subgroup (depending on $t$) of the automorphism group of $F$ that is acting properly discontinuously on $F/\Gamma_t$, i.e., the latter is also a smooth manifold.<br> If $\Gamma_t$ is the identity for all $t$, and $f$ is a fibration, then there is a Leray spectral sequence relating the homology of $X$ to that of $F$ and $Y$. Is there some spectral sequence for the case when $\Gamma_t$ is not always the identity, and if so what? A reference for this would be appreciated too.</p> http://mathoverflow.net/questions/87255/linear-equivalence-of-divisors-in-smooth-algebraic-surface Linear equivalence of divisors in smooth algebraic surface Ru 2012-02-01T17:08:30Z 2012-02-01T17:08:30Z <p>Let's assume $X$ is a smooth algebraic surface and $C$ a curve containing a smooth point $p_0\in X$, then there exist divisors $H_1$ and $H_2$ non of which contain $p_0$ such that $C+H_1$ is linearly equivalent to $H_2$.</p> http://mathoverflow.net/questions/86888/orbits-of-the-action-of-a-6-on-mathbbp-2 Orbits of the action of $A_6$ on $\mathbb{P}_2$ Ru 2012-01-28T08:59:50Z 2012-01-28T08:59:50Z <p>By a paper of Scott Crass <a href="http://xxx.lanl.gov/pdf/math/9903111v1.pdf" rel="nofollow">http://xxx.lanl.gov/pdf/math/9903111v1.pdf</a> we know that $A_6$ (Permutation on 6 elements) is an automorphism group of $\mathbb{P}_2$ which fix a sextic. What is the geometry of this action i.e., what are the orbits of this action explicitly.</p> http://mathoverflow.net/questions/75372/automorphism-of-enrique-surface automorphism of Enrique surface Ru 2011-09-14T03:11:06Z 2011-09-21T20:20:33Z <p>What is the fixed point set of an order two automorhism group of an Enriques surface.</p> http://mathoverflow.net/questions/75158/selfintersection-of-curves-inside selfintersection of curves inside Ru 2011-09-11T19:20:57Z 2011-09-11T19:20:57Z <p>What are all the possibilities of the self intersection number of a smooth curve inside an Enriques surface?</p> http://mathoverflow.net/questions/75021/signature-of-picx signature of $Pic(X)$ Ru 2011-09-09T17:49:42Z 2011-09-09T18:06:35Z <p>Let $X$ be a $K3$ surface and $\sigma$ an antisymplectic involution on $X$ and so $X$ is algebraic. 1.Why the signature of $Pic(X)$ is $(1,\rho -1)$? (This is well known but I cant find any direct proof) 2. Lets assume the set of fixed point set of $\sigma$ which we denote by $Fix(\sigma)$ contains a smooth curve say $D_g$ of genus $g\geq 2$, let's assume we have other curve in $Fix(\sigma)$ which is elliptic say $D$, i.e., we should have $D^2=0$ and $D.D_g=0$ then why this is a contrary to the fact that the signature of $Pic(X)$ is $(1,\rho -1)$. 3. How do we see curves inside $Pic(X)$? I mean curves are inside $X$ and $Pic(X)$ is a lattice!</p> http://mathoverflow.net/questions/74113/every-involution-of-an-enriques-surface-is every involution of an Enriques surface is Ru 2011-08-31T00:52:33Z 2011-09-01T08:04:13Z <p>Is it true that every involution $\sigma$ (i.e., $\sigma^2=identity$) of an Enriques surface $X$ acts trivially on $K_X^{\otimes 2}$ i.e., for any $\omega\in K_X^{\otimes 2}$ we have $\sigma^* \omega=\omega$, where by $K_X^{\otimes 2}$ we mean the tensor 2 of the conanical bundle of $X$.</p> http://mathoverflow.net/questions/112618/leray-spectral-sequence/112621#112621 Comment by Ru Ru 2012-11-23T00:51:35Z 2012-11-23T00:51:35Z Thanks Algori! Have a good day. http://mathoverflow.net/questions/112618/leray-spectral-sequence/112621#112621 Comment by Ru Ru 2012-11-19T18:34:37Z 2012-11-19T18:34:37Z Thanks again. Do you know of any reference about homological case? http://mathoverflow.net/questions/112618/leray-spectral-sequence/112621#112621 Comment by Ru Ru 2012-11-18T20:48:36Z 2012-11-18T20:48:36Z Thanks Algori, As you mentioned spectral sequence argument seems to work for cohomology - how about homology? http://mathoverflow.net/questions/76348/normal-subgroups-of-borel-group Comment by Ru Ru 2011-09-25T20:56:53Z 2011-09-25T20:56:53Z PS: everything is complex. http://mathoverflow.net/questions/76348/normal-subgroups-of-borel-group Comment by Ru Ru 2011-09-25T20:56:09Z 2011-09-25T20:56:09Z Thanks, Normal Lie Subgroups. If it was just calculation (huge) I would call it homework, but if one wanna do it by considering the Lie Algebra which I dont know yet it is not a homework. http://mathoverflow.net/questions/75372/automorphism-of-enrique-surface/75488#75488 Comment by Ru Ru 2011-09-21T18:53:13Z 2011-09-21T18:53:13Z Dear Rita,Thanks. I will be pleased if you could tell me where I can find the proof of the above formulas and that the fixed point locus are isolated curves and points? http://mathoverflow.net/questions/75372/automorphism-of-enrique-surface Comment by Ru Ru 2011-09-15T01:40:46Z 2011-09-15T01:40:46Z That what it is? Curves?points? how many? http://mathoverflow.net/questions/75372/automorphism-of-enrique-surface Comment by Ru Ru 2011-09-14T05:05:36Z 2011-09-14T05:05:36Z Ill edit it, if that bothers you. I was wondering if someone could analyze this if he/she doesnt have an answer. http://mathoverflow.net/questions/75158/selfintersection-of-curves-inside Comment by Ru Ru 2011-09-11T20:22:54Z 2011-09-11T20:22:54Z Dear Elkies.Thanks. http://mathoverflow.net/questions/75158/selfintersection-of-curves-inside Comment by Ru Ru 2011-09-11T19:54:25Z 2011-09-11T19:54:25Z @Ottem: Thanks. Please correct me if I am wrong. Adjunction formula says $K_D=K_X.D \|D + D.D$, if $K_X$ is trivial then your formula is correct. But for Enrique surface $K_X$ is not trivial. http://mathoverflow.net/questions/75021/signature-of-picx Comment by Ru Ru 2011-09-09T18:45:54Z 2011-09-09T18:45:54Z Thanks Artie, Ok, if I have a curve then I can associate an element of the picard group. I understand this now. Suppose I give a you an element of the picard group of $X$. How can I say if it is a curve of genus $g$ or it is an elliptic curve inside $X$? http://mathoverflow.net/questions/74113/every-involution-of-an-enriques-surface-is/74127#74127 Comment by Ru Ru 2011-09-06T02:37:48Z 2011-09-06T02:37:48Z @Christian: Sorry to bother you again. Why $\tilde{\sigma}^2\in &lt;\tau&gt;$. I know that $&lt;\tilde{\sigma}&gt;=&lt;\sigma&gt;\times&lt;\tau&gt;$ where by $&lt;&gt;$ I mean the group generated by that automorphism. Thanks http://mathoverflow.net/questions/74113/every-involution-of-an-enriques-surface-is/74225#74225 Comment by Ru Ru 2011-09-05T17:23:47Z 2011-09-05T17:23:47Z Thanks and what is $\chi(\mathcal{O}_X)$. You please just tell the name and I will search. Thanks http://mathoverflow.net/questions/74113/every-involution-of-an-enriques-surface-is/74225#74225 Comment by Ru Ru 2011-09-05T16:33:41Z 2011-09-05T16:33:41Z Edit my last commment: I was meaning $&lt;\tilde{\sigma}&gt;$ instead of $&lt;\tilde{\sigma}^2&gt;$ http://mathoverflow.net/questions/74113/every-involution-of-an-enriques-surface-is/74225#74225 Comment by Ru Ru 2011-09-05T16:28:47Z 2011-09-05T16:28:47Z @Torsten and Chritian: Thanks for you answer. I am only a student so let me ask some dum questions: 1.I was wondering if what do you mean by $\chi(\mathcal{O}_X)$? 2.Torsten: Do you mean $\tilde{\sigma}$ in Christians notation by $\sigma$? 3. Why $\tilde{\sigma}^2 \in &lt;\tau&gt;$? I know that $&lt;\tilde{\sigma}^2&gt;=&lt;\tau&gt;\times&lt;\sigma&gt;$ Torsten: Can you please explain your answer specifically? Thanks