User moon - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T01:34:28Z http://mathoverflow.net/feeds/user/4643 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/121536/efficient-algorithm-finding-a-solution-of-system-of-linear-inequalities Efficient algorithm finding 'a' solution of system of linear inequalities Moon 2013-02-11T22:51:42Z 2013-02-11T23:38:29Z <p>I'm working on rational number field $\mathbb{Q}$.</p> <p><strong>Is there an efficient algorithm finding a solution of system of linear inequalities?</strong></p> <p>In many computer algebra systems like Sage or Maple, there are functions finding the whole solution set, but in my problem (approximately 40 dimensional vector space with 600 inequalities) it seems that the computation is too heavy. Also, in my situation I don't need the whole set - just a single solution is sufficient. What is a good method to find a solution?</p> http://mathoverflow.net/questions/120025/blow-up-of-segre-primal-and-mathcalm-0-6/120078#120078 Answer by Moon for blow up of segre primal and $\mathcal{M}_{0,6}$ Moon 2013-01-28T04:35:26Z 2013-01-28T04:35:26Z <p>As Steven said, there is a morphism $\overline M_{0,6} \to \tilde{X}$ because the inverse image sheaf of the ideal of double points generates a Cartier divisor. Now $\tilde{X}$ is nonsingular, so to check the fact $\overline{M}_{0,6} \to \tilde{X}$ is an isomorphism it suffices to show injectivity of the map, or show the equality of Picard numbers. </p> <p>In general, the birational morphism $\overline M_{0,n} \to (\mathbb{P}^1)^n//SL_2$ is a composition of smooth blow-ups and Kirwan's desingularization. Consult <a href="http://arxiv.org/abs/1002.2461" rel="nofollow">http://arxiv.org/abs/1002.2461</a>.</p> http://mathoverflow.net/questions/84973/example-of-rational-projective-variety-of-picard-number-1 Example of rational projective variety of Picard number 1 Moon 2012-01-05T16:44:29Z 2012-01-05T17:12:42Z <p>Is there any example (or more ambitiously, classification) of $X$ with following properties? </p> <ul> <li>$X$ is a variety over $\mathbb{C}$;</li> <li>$X$ is projective and normal;</li> <li>$\rho(X) = 1$;</li> <li>$X$ is birational to $\mathbb{P}^n$.</li> </ul> <p>Also, I want to hear a result after adding a singularity condition: How about when $X$ is $\mathbb{Q}$-factorial? How about $X$ is non-singular?</p> <p>I can't find an example which is not isomorphic to $\mathbb{P}^n$. The only result I know in this direction is Mori's theorem: If a nonsingular variety $X$ has ample tangent bundle, then $X$ is isomorphic to $\mathbb{P}^n$.</p> http://mathoverflow.net/questions/60689/checking-normality-of-variety Checking normality of variety Moon 2011-04-05T14:00:15Z 2011-04-05T18:50:54Z <p>All varieties here is defined over complex number. Let $X$ be a normal projective irreducible variety and let $Y$ be a projective irreducible variety. Suppose that there is a bijective morphism $f : X \to Y$. This does not imply that $Y$ is normal. One can easily construct a counterexample for curve case. </p> <p>My question is:</p> <p>1) Is it true if $Y$ is regular in codimension 1?</p> <p>2) Is there any extra condition guarantee normality of $Y$?</p> <p>I searched some related questions and answers such as</p> <p><a href="http://mathoverflow.net/questions/12688/nonsingular-normal-schemes/12689#12689" rel="nofollow">http://mathoverflow.net/questions/12688/nonsingular-normal-schemes/12689#12689</a> or</p> <p><a href="http://mathoverflow.net/questions/60097/checking-whether-a-variety-is-normal" rel="nofollow">http://mathoverflow.net/questions/60097/checking-whether-a-variety-is-normal</a> ,</p> <p>but I cannot find or prove a method working on my situation. In my problem, finding local defining equations is almost impossible.</p> <p>Edit: One of motivation is a kind of inverse problem of Zariski's main theorem. Suppose that $g : X \to Y$ is a morphism between projective varieties with connected fiber from normal variety $X$. By taking Stein factorization, we can reduce to the situation of question.</p> http://mathoverflow.net/questions/37570/technique-to-prove-basepoint-freeness Technique to prove basepoint-freeness Moon 2010-09-03T02:16:21Z 2010-10-02T17:22:15Z <p>Let $X$ be a smooth projective variety over $\mathbb{C}$. And let $L$ be a big and nef line bundle on $X$. I want to prove $L$ is semi-ample($L^m$ is basepoint-free for some $m > 0$).</p> <p>The only way I know is using Kawamata basepoint-free theorem:</p> <p>Theorem. Let $(X, \Delta)$ be a proper klt pair with $\Delta$ effective. Let $D$ be a nef Cartier divisor such that $aD-K_X-\Delta$ is nef and big for some $a > 0$. Then $|bD|$ has no basepoints for all $b >> 0$.</p> <p><strong>Question. What other kinds of techniques to prove semi-ampleness or basepoint-freeness of given line bundle are?</strong></p> <p>Maybe I miss some obvious method. Please don't hesitate adding answer although you think your idea on the top of your head is elementary.</p> <p>Addition : In my situation, $X$ is a moduli space $\overline{M}_{0,n}$. In this case, Kodaira dimension is $-\infty$. More generally, I want to think genus 0 Kontsevich moduli space of stable maps to projective space, too. $L$ is given by a linear combination of boundary divisors. It is well-known that boundary divisors are normal crossing, and we know many curves on the space such that we can calculate intersection numbers with boundary divisors explicitely.</p> http://mathoverflow.net/questions/32766/torsion-freeness-of-picard-group Torsion-freeness of Picard group Moon 2010-07-21T09:04:22Z 2010-07-21T22:43:19Z <p>Let $X$ be a complex normal projective variety.</p> <p><strong>Is there any sufficient condition to guarantee the torsion-freeness of Picard group of $X$?</strong></p> <p>One technique I sometimes use is following: If $X$ can be represented by GIT quotient $Y//G$ for some projective variety $Y$ with well-known Picard group, then by using Kempf's descent lemma, we can attack the computation of integral Picard group. </p> <p>Of course, if we can make a sequence of smooth blow-ups/downs between $X$ and $X'$ with well known Picard group, then we can get the information of $\mathrm{Pic}(X)$ from $\mathrm{Pic}(X')$.</p> <p>Is there any way to attack this problem?</p> http://mathoverflow.net/questions/18460/dualizing-sheaf-of-reducible-variety Dualizing sheaf of reducible variety? Moon 2010-03-17T03:07:16Z 2010-04-10T20:35:22Z <p>Sorry for my poor English.</p> <p>Let $X$ be a reducible projective variety. </p> <p><strong>My question is:</strong></p> <ol> <li>How can I compute the dualizing sheaf of $X$ and express it in an explicit way?</li> <li>Is there a method to get dualizing sheaf of whole reducible variety $X$ from the information of dualizing sheaves of its irreducible components?</li> </ol> <p>Currently I'm not concern the general case, but I want a few accessible concrete examples such as:</p> <ol> <li>reducible hypersurfaces,</li> <li>union of toric varieties glued at isomorphic orbits(Alexeev calls it stable toric variety).</li> </ol> <p>The reason why I concern is to understand the limit in moduli spaces of stable pairs. </p> http://mathoverflow.net/questions/18677/cohomology-rings-of-gl-nc-sl-nc/18714#18714 Answer by Moon for Cohomology rings of GL_n(C), SL_n(C) Moon 2010-03-19T03:34:11Z 2010-03-19T05:31:20Z <p>For classical groups such as $GL_n(F)$, $SL_n(F)$ for $F = \mathbb{R}, \mathbb{C}$, $SU(n)$, $U(n)$ and $O(n)$, you may find the cohomology ring structure and its proof in M. Mimura and H. Toda, Topology of Lie groups I, translations of mathematical monographs, vol 91.</p> http://mathoverflow.net/questions/84973/example-of-rational-projective-variety-of-picard-number-1 Comment by Moon Moon 2012-01-05T21:45:03Z 2012-01-05T21:45:03Z Oh, I thought it too difficult way... Thank you for comments! http://mathoverflow.net/questions/60689/checking-normality-of-variety/60693#60693 Comment by Moon Moon 2011-04-08T06:00:35Z 2011-04-08T06:00:35Z Thank you. I'll try it. http://mathoverflow.net/questions/60689/checking-normality-of-variety/60693#60693 Comment by Moon Moon 2011-04-07T02:16:44Z 2011-04-07T02:16:44Z Dear Karl, thank you for great answer. I have another question. To show the normality of given variety, is it the best way checking R1 and S2? In my situation, given variety $X$ is an incident variety of points and curves in projective space. By dimension estimate, R1 is not too difficult, but I can't figure out how can I prove S2 (or CM) condition. http://mathoverflow.net/questions/27249/what-does-the-ample-cone-look-like/27339#27339 Comment by Moon Moon 2010-11-05T01:43:18Z 2010-11-05T01:43:18Z To Michael Thaddeus, Thank you for a nice answer and comment, but I can't imagine that cone with finitely many extremal rays may not be polyhedral. Is there a counterexample or am I missing some obvious picture? http://mathoverflow.net/questions/37570/technique-to-prove-basepoint-freeness/37612#37612 Comment by Moon Moon 2010-09-04T10:49:34Z 2010-09-04T10:49:34Z Anyway, thank you for answer. By the way, what is the result of Mourougane and Russo? Do you mean the theorem 2.3.9 in PAG? http://mathoverflow.net/questions/37570/technique-to-prove-basepoint-freeness/37612#37612 Comment by Moon Moon 2010-09-04T10:46:02Z 2010-09-04T10:46:02Z You are right, nefness + bigness does not guarantee semi-ampleness. My question is following: Is there any sufficient condition to get semi-ampleness? With which conditions we get semi-ample property? http://mathoverflow.net/questions/37570/technique-to-prove-basepoint-freeness Comment by Moon Moon 2010-09-04T06:43:59Z 2010-09-04T06:43:59Z I edited the question, thanks. http://mathoverflow.net/questions/37172/what-are-some-open-problems-in-algebraic-geometry/37173#37173 Comment by Moon Moon 2010-08-31T07:01:26Z 2010-08-31T07:01:26Z Farkas proved that $\overline{M}_g$ is of general type for $g = 22$. http://mathoverflow.net/questions/18460/dualizing-sheaf-of-reducible-variety/18504#18504 Comment by Moon Moon 2010-04-12T01:58:20Z 2010-04-12T01:58:20Z Thank you! I'll check the paper. http://mathoverflow.net/questions/20138/why-is-proj-of-any-graded-ring-isomorphic-to-proj-of-a-graded-ring-generated-in-d Comment by Moon Moon 2010-04-02T10:15:59Z 2010-04-02T10:15:59Z Red book(old print), 403p.