User blade - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T17:42:30Z http://mathoverflow.net/feeds/user/14105 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109164/synthetic-approach-to-hyperbolic-geometry Synthetic approach to hyperbolic geometry? Blade 2012-10-08T18:22:25Z 2012-10-09T17:40:38Z <p>Hello, I am looking for a source that discusses and teaches hyperbolic geometry from a synthetic approach (As opposed to the common analytinc approach in the poincare disk). I am looking for something more in spirit with eucld's elements or hilbert's geometry book.</p> <p>Thank you </p> http://mathoverflow.net/questions/93123/hypersurfaces-in-toric-varieties-help-understand-a-proof-from-mikhalkins-paper Hypersurfaces in Toric Varieties, Help understand a proof from Mikhalkin's paper Blade 2012-04-04T14:18:20Z 2012-04-05T20:23:14Z <p>Hello, in G. Mikhalkin's Papaer "DECOMPOSITION INTO PAIRS-OF-PANTS FOR COMPLEX ALGEBRAIC HYPERSURFACES": <a href="http://arxiv.org/pdf/math/0205011.pdf" rel="nofollow">http://arxiv.org/pdf/math/0205011.pdf</a> There is a lemma about the relation between intersection of a hypersurface with the boundary divisors in a toric variety and the truncation of the defining polynomial of a hypersurface to some face of the newton polytope. (Lemma 2.20. Page 15).</p> <p>The lemma basically says that If we take a hypersurface with newton polytope P, and consider it's closure in the projective toric variery corresponding the the lattice polytope P, then to find it's intersection with the boundary divisors, it is enough to truncate the polynomial the the corresponding face of the newton polytope and take the zeroes of the truncated polynomial.</p> <p>The proof given there is a one-liner about the order of vanishing of some monomials and I don't understand why it proves the claim.</p> <p>I would appreciate if someone could explain this to me.</p> <p>Thank you</p> http://mathoverflow.net/questions/86271/what-is-the-advantage-of-the-approach-of-valuations-to-the-riemann-roch-theorem-f What is the advantage of the approach of valuations to the Riemann-Roch Theorem for curves (a la Chevalley)? AKA theory of algebraic functions in one variable Blade 2012-01-21T01:01:52Z 2012-01-21T14:03:48Z <p>Hello, Some books and courses take the approach to Riemann-Roch for curves considering only the algebraic viewpoint of algebraic extensions of a field k(t) without going into any algebraic geometry (no varieties, no topology, no dimension, no sheafs etc') divisors are defined using equivalence classes of valuations of the field, differentials have some 'wierd' definition as well. and riemann roch is proved only in that language, with no regard to concepts such as bundles. This can be seen in books such as Chevalley, Introduction to the Theory of Algebraic Functions of One Variable. Or even in Neukirch Algebraic number theory.</p> <p>Since I know of treatments of Riemann-Roch in the general setting for bundles over algebraic curves in books such as Kempf, Algebraic Varieties, or even in Vakil's notes: <a href="http://math.stanford.edu/~vakil/725/bagsrr.pdf" rel="nofollow">http://math.stanford.edu/~vakil/725/bagsrr.pdf</a></p> <p>My question is, what is the advantage of the approach using only algebra and the language of valuations instead of using concepts from algebraic geometry (such as sheaf)</p> <p>Thanks</p> http://mathoverflow.net/questions/82661/book-recommendation-for-ergodic-theory-and-or-topological-dynamics Book recommendation for ergodic theory and/or topological dynamics? Blade 2011-12-05T01:00:59Z 2011-12-09T03:50:54Z <p>Hello,</p> <p>I'd like to hear your opinion for ergodic theory books which would suit a beginner (with background in measure theory, real analysis and topological groups). I am looking for something well structured, well motivated, and perhaps with application to other fields.</p> <p>any such book exists?</p> <p>I tried a book by nadkarni, and could not read through it, seemed to concise to me, and tried the book by Petersen which I felt was accessible but didn't follow a clear path, jumping from subject to subject with lots of different object or properties.</p> <p>What are your recommendations on the subject?</p> http://mathoverflow.net/questions/72764/euler-characteristic-in-a-neighborhood-of-a-singularity-of-complex-curve-and-def Euler Characteristic in a neighborhood of a Singularity of Complex Curve, and Deformations Blade 2011-08-12T12:22:22Z 2011-08-16T16:37:15Z <p>Hello all. Let's say we have a one parameter (complex) family of complex curves on a family of corresponding surfaces. i.e, think of the whole thing living inside some threefold, fibered along corresponding surfaces.</p> <p>Let's look on the central fiber, we have a curve $C_0$ on a surface, and assume we have some isolated singularity there at point $z$ . pick a small closed ball around the singularity. </p> <p>Topologically $(C_0 \cap B_z )$ is a bouquet of discs. now, lets vary the parameter and look what happens to this fragment of the curve. Look at the Normalization of $(C_t \cap B_z )$. it's a collection of surfaces with some holes and handles. I want to compare the Euler Characteristic of this normalization with the Euler Characteristic of the normalization of the corresponding fragment of the central fiber, $C_0$.</p> <p>Let's say I know that the Euler Characteristic (Topological) has changed in the deformation by at most 2.</p> <p>Now what can it mean? Either some branches joined by a handle. or we could have added holes. EDIT: (by a handle I mean a tube, topologically $S^1\times[0,1]$ joining the two branches, like in the deformation $x^2 + y^2 = t$ : for $t=0$ we have bouquet of two discs and the normalization separates them to two disjoint discs, and for $t\ne0$ they join by a handle). It still decreases Euler Characteristic by 2 because it cuts out two small discs)</p> <p>I find it hard to understand what does adding holes mean, or even whether it is entirely possible that holes add up in the deformation. can the Euler characteristic jump at an increment of 1 and not 2 (adding 1 hole)? Can I get two holes instead of a handle? and at what conditions of the arrangement of branches of the central fiber?</p> <p>I hope I made my question clear, If not, please tell me and I'll try to explain better. for now, that's the best I could do, well... since this whole thing is not very clear to me, It's also hard to be fluent and precise when asking the question.</p> <p>Thank you all in advance </p> http://mathoverflow.net/questions/66401/singular-homology-cohomology-as-a-derived-functor Singular Homology/Cohomology as a derived functor? Blade 2011-05-29T22:32:52Z 2011-05-30T15:49:49Z <p>Hello, Learning some Alg.geometry and Sheaf theory, I got used to the notion that cohomology arises naturally as a derived functor of some sort.</p> <p>This has led me thinking, singular cohomology, from algebraic topology, was never defined (In all books i've checked) as a derived functor, but just by giving cycles and boundaries. I could not figure out by myself any reasonable functor whose derived functors yield singular cohomology, So I pose this question out here.</p> <p>I hope this might shed some more insight on what singular cohomology actually measures.</p> <p>Thanks</p> http://mathoverflow.net/questions/62627/intersection-of-curves-on-projective-toric-surface-and-some-enumerative-questions Intersection of curves on projective toric surface and some enumerative questions Blade 2011-04-22T15:33:21Z 2011-04-23T05:49:07Z <p>Reading on the tropical approach to enumerative geometry I have come across the claim: given a projective toric surface from a polygon P, we can consider a tautological bundle of algebraic / holomorphic functions with newton polygon contained in P (i.e. take functions that in the dense torus can be written as a combination of monomial corresponding only to points in P).</p> <p>The claim I've come across (in Mikhalkin's paper [1] (section on severi varieties) or in Shustin's book [2] (proof of lemma 2.42 page 59-61) is that if we take the zeroes of such a function from our bundle, then in the general case it is a curve that is transversal to the boundary divisors of the toric surface. Moreover, "Any curve (which is not a toric divisor) crosses at least two toric divisors; take the Newton polygon of a curve equation, then its intersections with the toric divisors are as follows: for each side of that Newton polygon equipped with the outer normal, take the parallel supporting line for the polygon P defining the toric surface, if the intersection is a vertex, then the curve passes through the corresponding intersection point of toric divisors, if the intersection is a side, then the curve crosses the corresponding toric divisor; for a Newton segment (binomial curve), we take it twice - with both normals." [I quote an explanation given to me on what are 'rules' of intersection]</p> <p>A) I could not find any proof or explanation why it is true. I am currently less interested in a fully rigorous proof but more in an explanation why this is true.</p> <p>B) Another question come to my mind, does any other curve on the toric surface must come from zeroes of a function from the tautological bundle? if not, why do authors focus so much on curves that are zeroes of this bundle in the context of counting curves via n points? (i.e. GW invariants)?</p> <p>Thanks!</p> <p>And i apologize if this was a bit cumbersome way of asking my question, when things are not clear enough, even asking properly is somewhat difficult :)</p> <p>References</p> <p>[1] <a href="http://arxiv.org/abs/math/0312530" rel="nofollow">Mikhalkin's papaer </a><br/> [2] <a href="http://books.google.com/books?id=4JP7ofjHVh8C&amp;printsec=frontcover&amp;dq=Tropical+algebraic+geometry&amp;hl=en&amp;ei=NJ-xTfGdCcuu8QOr0KmWDA&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=1&amp;ved=0CCgQ6AEwAA#v=onepage&amp;q&amp;f=false" rel="nofollow">Shustin's (et al) book</a></p> http://mathoverflow.net/questions/60430/what-information-hilbert-polynomial-encodes-other-than-dimension-degree-and-arit What information Hilbert Polynomial encodes other than Dimension, Degree and Arithmetic Genus? Blade 2011-04-03T12:16:51Z 2011-04-03T16:31:33Z <p>Hello, Consider the hilbert polynomial for a projective scheme. The degree, dimension and arithmetic genus extract information from the lowest term and the highest term in the polynomial. What about all other terms? It would seem they encode some more info about our scheme. I could not find any reference to these coefficients, though. So my question is what else can we learn about our scheme from the hilbert polynomial?</p> <p>Thanks</p> http://mathoverflow.net/questions/109164/synthetic-approach-to-hyperbolic-geometry/109172#109172 Comment by Blade Blade 2012-10-10T00:42:19Z 2012-10-10T00:42:19Z I'll look into them all, but of those three, which would you recommend as the 'best'? http://mathoverflow.net/questions/109164/synthetic-approach-to-hyperbolic-geometry Comment by Blade Blade 2012-10-09T11:25:08Z 2012-10-09T11:25:08Z You are probably right, Fixed. http://mathoverflow.net/questions/82661/book-recommendation-for-ergodic-theory-and-or-topological-dynamics/82670#82670 Comment by Blade Blade 2012-01-08T23:43:04Z 2012-01-08T23:43:04Z Thanks, This book perhaps is what I was looking for http://mathoverflow.net/questions/82661/book-recommendation-for-ergodic-theory-and-or-topological-dynamics Comment by Blade Blade 2011-12-05T09:51:11Z 2011-12-05T09:51:11Z @David Roberts, Yes, these are the books. Next time I'll post more specific bibliography. As for me, I am a grad student, but uexperienced with ergodic theory. http://mathoverflow.net/questions/72764/euler-characteristic-in-a-neighborhood-of-a-singularity-of-complex-curve-and-def Comment by Blade Blade 2011-08-18T10:57:30Z 2011-08-18T10:57:30Z regarding the torus holes, If I understand correctly, In the cubic deformation the torus appears 'globally'. when I look at some small neighborhood, I intersect the torus with some ball. what puzzles me Can I still somehow preserve the 'torus hole' under this local small intersection? I have milnor's book on my shelf :-) , I dont think he addresses these issues directly, but maybe the answer is hidden there some different form. I'll check it up again anyway http://mathoverflow.net/questions/72764/euler-characteristic-in-a-neighborhood-of-a-singularity-of-complex-curve-and-def Comment by Blade Blade 2011-08-17T13:29:52Z 2011-08-17T13:29:52Z by adding 'holes' I meant boundary components. (By the way, this now rises another issue I haven't thought of, can &quot;holes&quot; in the meaning of a torus appear in the deformation? I don't think so, but If you see a way these guys appear as well, please let me know) Thanks http://mathoverflow.net/questions/72764/euler-characteristic-in-a-neighborhood-of-a-singularity-of-complex-curve-and-def Comment by Blade Blade 2011-08-16T16:33:14Z 2011-08-16T16:33:14Z You are right, I should have probably looked at the normalization of the central fiber $C_0$ as well. now it gets separated into two discs and the characteristic is 2. I'll fix it in an edit. http://mathoverflow.net/questions/62627/intersection-of-curves-on-projective-toric-surface-and-some-enumerative-questions/62707#62707 Comment by Blade Blade 2011-04-23T12:57:07Z 2011-04-23T12:57:07Z Wow, Thanks :) You really cleared up many things for me :) But I still have some bits of things that puzzle me: I understand that the notion of a parametrization of a branch (with Puiseux series) is a local one, around a point. how do we use it to get some data not near the point around which we developed the series? By the way, could you refer me to some book/article/survey where this is discussed? I could'nt find it in Fulton's book (maybe I missed it due to different terminology or something?) Thanks again http://mathoverflow.net/questions/60430/what-information-hilbert-polynomial-encodes-other-than-dimension-degree-and-arit/60446#60446 Comment by Blade Blade 2011-04-04T22:05:05Z 2011-04-04T22:05:05Z Thank you, I liked your explanation as well, gave me good intuition. I thought that if someone stumbels upon this question he might be interested in the formulae in Donu's answer.