User aginensky - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T19:15:19Z http://mathoverflow.net/feeds/user/5100 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103186/intersecting-degree-0-divisors/103250#103250 Answer by aginensky for Intersecting Degree 0 Divisors aginensky 2012-07-27T00:05:06Z 2012-07-27T00:05:06Z <p>As has been mentioned, in general 'degree' is not so well/uniquely defined. However, suppose you take a smooth cubic surface in $\bf{ P}^3$ . There are 27 lines and they should all have degree 1. Take two lines $l_1$ and $l_2$ which are skew and let $D = l_1-l_2$ . This will be 'degree zero' with your definition. Clearly $deg(D|l_1) \neq 0$ Similarly on a quadric hypersurface in $\bf{P}^3$, letting $l_1$ and $l_2$ be two lines which meet , one can construct similar 'examples'. If $X$ is a variety with $Pic(X) = \bf{Z}$ then what you ask is true. I would suspect it fails any other time.</p> http://mathoverflow.net/questions/94030/cokernel-of-the-symmetric-product-of-an-injection cokernel of the symmetric product of an injection. aginensky 2012-04-14T13:53:24Z 2012-04-14T18:31:29Z <p>Clearly, my question can be asked more generally, but suppose for simplicity that $X$ is a smooth surface and that $0 \to E \to F \to K \to 0$ is exact with $E ,F$ rank two bundles on $X$ and $K$ a line bundle on a (smooth?) divisor $D \subset X$. Can one give information on the cokernel of the injection $Sym^n(E) \to Sym^n(F)$ ? For example is this cokernel some obvious sum of linear algebra constructions (various symmetric powers) of $E$ , $F$, and $K$ ? I can only say that I have thought about this question for a while and no answer has occurred to me. </p> <p>Adam</p> http://mathoverflow.net/questions/81322/classification-of-certain-algebraic-curves/81675#81675 Answer by aginensky for Classification of certain algebraic curves aginensky 2011-11-23T01:20:02Z 2011-11-23T01:20:02Z <p>I'm going to assume $L$ is base point free. I think it is clear how to change what I have written in the case where there are base points (numbers goes down by the degree of the base locus). In general, if $L$ is a line bundle of degree $d$ and $h^0(C,L)= r+1$, then the Clifford index of $L$, written Cliff(L) $= d-2r$. Cliffords theorem is Cliff(L) >= 0. Your line bundle satisfies Cliff(L) = 2. Two ways to achieve that if for the curve $C$ to have a $g^1_4$ or be a plane sextic. The Clifford index of a curve is the min{cliff(L)| $h^0(L)$ &amp; $h^1(L)$ >=2}. For a general curve $C$, Cliff(C) is the floor of (g-1)/2. So for a general curve this can't be done. It is absolutely true that A-C-G-H will have more details and I believe a careful perusal will give the classification of all curves with Clifford Index 2. I think it is only the cases I mention.</p> http://mathoverflow.net/questions/76815/normality-via-resolution-of-singularities/76872#76872 Answer by aginensky for Normality via resolution of singularities aginensky 2011-09-30T18:13:16Z 2011-09-30T18:13:16Z <p>Unfortunately it is not true that if $Y$ is smooth, $f$ is proper and the fibers of $f:Y \rightarrow X$ are reduced and connected that $X$ is normal. Unfortunately I am all to familiar with the following example, which has ended any optimism I have had about finding a criteria for normality using a desingularization. The question is local so I will just use the affine coordinates. Let $R = k[x^4,x^3y,xy^3, y^4]$. This ring is the cone of a projection of the quartic normal curve, so it is not normal at the origin. One checks easily using local coordinates that the resolution is just $\bf P^1$. I learned of this example from Mohan Kumar, but it also appears in one or more of Eisenbud's books, as well as a counter example to other questions asked here. It seems to be a universal counter example to any seemingly true facts about depth etc for which one cannot find a proof. </p> http://mathoverflow.net/questions/62695/is-there-a-fast-way-to-compute-matrix-multiplication-mod-p/62753#62753 Answer by aginensky for Is there a fast way to compute matrix multiplication mod p? aginensky 2011-04-23T15:43:27Z 2011-04-23T15:43:27Z <p>The improvement of matrix multiplication from $O(n^3)$ to $O(n^{2.4})$ is based on the Strassen equations for matrix multiplication. <a href="http://mathoverflow.net/questions/57725/strassen-algorithm-7-multiplications" rel="nofollow">http://mathoverflow.net/questions/57725/strassen-algorithm-7-multiplications</a> talks about this and gives further references. If your question is rephrased as "are there special equations for matrix multiplication in characteristic $p$ ?", then I think the answer, if known, will be in one of those references. If I were a betting man, I'd bet no. As a person not adverse to speculation, I am highly skeptical. Matrix multiplication feels to me to be characteristic independent and of the flavor if it was true in all large (positive) characteristics, then it would be true in characteristic zero. Also, almost all results about equations of determinantal varieties, secant varieties etc don't seem to be different in any characteristic. Of course, this is not true on the nose as higher syzygies can be different,and in my very feeble understanding of such matters, more complicated. Anyone in a position to terminate or validate my musings?</p> http://mathoverflow.net/questions/59018/equations-defining-a-subvariety equations defining a subvariety aginensky 2011-03-21T00:20:56Z 2011-03-27T02:51:54Z <p>The following question feels to me like a standard sort of 'fact' in birational geometry, but I can't seem to write down a correct set of details. Hopefully someone can point me to a reference and not a counter example!</p> <p>Suppose $X$ is a variety (reduced and irreducible over an algebraically closed field, perhaps of characteristic zero) and suppose that there exist a very ample line bundle $L$ and a linear system $V \subset H^0(X,L)$ such that $Y = Bs(V)$ is the singular set of $X$ scheme theoretically, that $Y$ is smooth of codimension at at least 2, and that $\tilde X$, the blow up of $X$ along $Y$ is smooth. Further assume that $\phi_{|V|} X--> S$ birationally maps $X$ onto a smooth variety $S$. Let $\tilde \phi$ be the map from $\tilde X \to S$ induced by $V$. Further assume that, denoting by $f$ the map $\tilde X \to X$, that $f^{-1}(Y) = T$ surjects onto $S$. Let $v_1, \dots v_s$ be $s = \dim(S)$ general sections of $V$ so that the intersection $Z(v_1) \cap \dots Z(v_s) \cap S$ consists of finitely many smooth points say $p_1, \dots p_m$. </p> <p>Also assume the $P = f( \tilde \phi^{-1}(\cup_{i=1:m} p_i))$ is a proper subset of $Y$. Then can one say that away from $P$, the sections $v_1 \dots v_s$ generate the ideal of $Y$ in $X$ ?</p> <p>The case I have in mind is where $Y$ is a smooth curve embedded in a sufficiently ample manner so that 1) $Y$ is defined by quadrics and 2) $X = Sec(X)$ is singular only along $Y$. Then $V$ would be the quadrics through $Y$. The point would be to use this sort of an argument to establish a minimum depth of $Sec(Y)$ along $Y$.</p> <p>This is my first question, so please feel free to correct etiquette with this question as well as the mathematics. </p> http://mathoverflow.net/questions/16087/defining-equations-for-secant-varieties/59205#59205 Answer by aginensky for defining equations for secant varieties aginensky 2011-03-22T17:08:40Z 2011-03-22T17:08:40Z <p>I'm very late to the conversation. In general nothing is known. For some cases of the Segre or Veronese variety, one can interpet the varieties as spaces of matrices and then the equations are determinants. In general this is not known. There is a large current literature on this topic. I would start at the arxiv with pretty much any current paper by J.M. Landsberg. It will contain loads of information and references on this topic. </p> http://mathoverflow.net/questions/1142/is-very-ampleness-of-a-divisor-on-a-curve-determined-entirely-by-degree-and-genus/57354#57354 Answer by aginensky for Is very ampleness of a divisor on a curve determined entirely by degree and genus? aginensky 2011-03-04T13:28:12Z 2011-03-04T13:28:12Z <p>I'm late to the game, but I would like to point out that the answer is systematically no. One class of examples. Suppose g>2 for simplicity. In that case any general line bundle of degree 2g is very ample and special ones are not. This can be seen by using the criteria that a line bundle is very ample iff for any effective divisor $D$ of degree 2, $h^0(L(-D) = h^0(L)-2$. One checks using R-R that this holds iff $L$ is not of the form $L= K_C(D)$ where $D$ is an effective divisor of degree 2. Line bundles of the form $L= K_C(D)$ are a 2 dimensional subset of the g dimensional (Picard) variety. This can be expanded upon.</p> http://mathoverflow.net/questions/22299/what-are-some-examples-of-colorful-language-in-serious-mathematics-papers/51813#51813 Answer by aginensky for What are some examples of colorful language in serious mathematics papers? aginensky 2011-01-12T03:02:49Z 2011-01-12T03:02:49Z <p>Milne's web page contains a number of amusing anecdotes- <a href="http://www.jmilne.org/math/apocrypha.html" rel="nofollow">http://www.jmilne.org/math/apocrypha.html</a></p> http://mathoverflow.net/questions/51531/theorems-that-are-obvious-but-hard-to-prove/51559#51559 Answer by aginensky for Theorems that are 'obvious' but hard to prove aginensky 2011-01-09T16:56:12Z 2011-01-09T16:56:12Z <p>Speaking of thesis advisers, mine said, "I think something should be called obvious only if it is obvious in the logical sense of if A implies B and if B implies C then A implies C". All else is subjective and hence capable of misuse. I have tried, but not necessarily succeeded, to follow this. I am constantly amazed/amused at how people coming at a problem from different points of views will find certain facts obscure or well known.</p> http://mathoverflow.net/questions/31655/statistics-for-mathematicians/50427#50427 Answer by aginensky for Statistics for mathematicians aginensky 2010-12-26T15:03:08Z 2010-12-26T15:03:08Z <p>For a very mathematical version of statistics, my favorite is on line lecture notes from two MIT courses. The instructor is named Panchenko and the course is called 'Statistics for Applications'. There are course notes that read like a book for the course in 2003 and 2006. I have enjoyed browsing through both of them. Here is a link: <a href="http://ocw.mit.edu/courses/mathematics/" rel="nofollow">http://ocw.mit.edu/courses/mathematics/</a>. </p> http://mathoverflow.net/questions/46768/how-many-independent-quadrics-should-one-intersect-to-get-the-canonical-curve/47364#47364 Answer by aginensky for How many independent quadrics should one intersect to get the canonical curve. aginensky 2010-11-25T19:18:06Z 2010-11-25T19:18:06Z <p>Amplifying on of Speyer's comments, if p is a point on a secant line of C, then the quadrics vanishing on C and p are of codimension one in the space of all quadrics vanishing on C. Such a quadric vanishes at 3 points of the secant line ( p and the two points of C defining the line as a secant) and hence vanished on L. Am I doing something silly?</p> http://mathoverflow.net/questions/41429/line-bundles-on-special-abelian-surfaces/46080#46080 Answer by aginensky for Line bundles on special abelian surfaces aginensky 2010-11-14T22:40:38Z 2010-11-14T22:40:38Z <p>Am I missing something? Doesn't this hold for arbitrary smooth pairs of varieties using the Kunneth decomposition?</p> http://mathoverflow.net/questions/32938/surfaces-in-mathbbp3-with-isolated-singularities/33238#33238 Answer by aginensky for Surfaces in $\mathbb{P}^3$ with isolated singularities aginensky 2010-07-24T22:21:22Z 2010-07-24T22:21:22Z <p>To the best of my knowledge this is a long standing open problem. I cannot recall a reference, as this is something I studied in the 1980's, but I recall this being phrased as an unsolved problem from the 19th century Italian school. The conjecture is that no normal surface in P^3 is birational to a smooth surface which has two dimensional image in it's Albanese. One specific case of this that has been studied more extensively are Zariski surfaces:z^n = f(x,y) where f is a polynomial of degree n with only cusps and nodes as singularities. There are lots of information about when such a surface is irregular, but beyond that not much is known. I believe that even if f is a sextic polynomial it is unknow whether or not the resulting surface can have 2 dimensional image in it's Albanese. I have heard Catanese ask about the case where S is an abelian surface. </p> http://mathoverflow.net/questions/24913/quick-proofs-of-hard-theorems/25451#25451 Answer by aginensky for Quick proofs of hard theorems aginensky 2010-05-21T02:25:01Z 2010-05-21T02:25:01Z <p>I was told by my (graduate school) teacher of functional analysis that originally the complex case of the Hahn-Banach theorem was considered a major open problem. It was eventually shown to be such a simple consequence of the real case, that now, no one knows who came up with the trick.</p> http://mathoverflow.net/questions/9734/on-the-clifford-index-of-a-curve/20245#20245 Answer by aginensky for On the Clifford index of a curve aginensky 2010-04-03T16:18:54Z 2010-04-03T16:18:54Z <p>when c= 0 Clifford's them includes the fact that any divisor with Clifford index 0 is a multiple of the hyperelliptic fiber, ie: a sum of fibers of the hyperelliptic map. If c=1 then the curve is either trigonal or a plane quintic- I believe that it is an exercise in A-C-G-H. Kind of a folk lore result. I have not heard of anyone explicating all the possible cases when c=2.</p> http://mathoverflow.net/questions/131315/embedded-associated-prime-and-non-zero-divisor Comment by aginensky aginensky 2013-05-21T15:45:39Z 2013-05-21T15:45:39Z You could have saved some time by saying &quot; please solve hw problem on page (insert page) of book (insert book name) http://mathoverflow.net/questions/130197/non-reducedness-in-linear-systems Comment by aginensky aginensky 2013-05-09T20:44:44Z 2013-05-09T20:44:44Z Unless I misunderstand what you are saying, pretty much anything is a counter example. On $P^n$ consider the linear system $O(2)$. A special member is a hyperplane with multiplicity two, but the general element will be a smooth conic. http://mathoverflow.net/questions/129086/borels-paris-lectures/129089#129089 Comment by aginensky aginensky 2013-04-30T23:31:40Z 2013-04-30T23:31:40Z If someone knows a deserving library, I am in possession of a copy- via the estate of Walter Baily*. I don't want money, but if the library was willing to make a nominal donation to the AMS in Walter's name, that would be great. It isn't signed, but on the inside cover there is a type written note saying &quot;with the compliments of the author. Borel and Baily were good friends. Feel free to delete this comment if deemed too commercial. http://mathoverflow.net/questions/128832/a-homeomorphism-betwen-two-topological-spaces-implies-that-the-n-th-homology-grou Comment by aginensky aginensky 2013-04-26T15:59:05Z 2013-04-26T15:59:05Z I don't know if this answers your question, but I think that one can construct a homotopy equivalence between this question and a hw problem. http://mathoverflow.net/questions/128688/gauss-mapping-in-finite-characteristic Comment by aginensky aginensky 2013-04-25T15:34:17Z 2013-04-25T15:34:17Z There are papers by Kleiman-Piene that discuss this question. My best recollection is that they tend to be inseparable, but finite. http://mathoverflow.net/questions/128100/non-singular-cubics-are-not-rational/128114#128114 Comment by aginensky aginensky 2013-04-21T14:33:28Z 2013-04-21T14:33:28Z @ voloch Good point. That is why I made it a comment and not an answer :) http://mathoverflow.net/questions/128100/non-singular-cubics-are-not-rational/128114#128114 Comment by aginensky aginensky 2013-04-20T16:14:41Z 2013-04-20T16:14:41Z I'm not sure if this is more elementary, but if the field was rational, one would have polynomials $h(t)$ and $g(t)$ s.t. $f(h(t),g(t))$ = 0. $f(x,y) = y^2-(x)(x-1)(x-\lambda)$ for an elliptic curve. It seems as if $\alpha$ is a root of $h(t)$ , that forces it to be a root of either $g$, $g-1$, or $g-\lambda$. But not a double root and hence a contradiction. I've not put in all the details, but I think that works and to my taste, it is more 'elementary' http://mathoverflow.net/questions/127538/clifford-index-of-curves-on-a-surface Comment by aginensky aginensky 2013-04-15T14:36:43Z 2013-04-15T14:36:43Z There is a paper of Green and Lazarsfeld in which they show that the Clifford index of a curve is constant in the case you mention if $X$ is a K-3 surface. I've never heard anyone suggest, and off-hand I can't think of any reason why, this is true on an arbitrary surface. My guess is that $P^3$ should already provide counterexamples. Namely take a curve $C$, then for $n&gt;&gt;0$ it should lie on a hypersurface that depends only on the gross numerical invariants of $C$ and not it's Clifford index. http://mathoverflow.net/questions/126861/non-proper-intersection-of-projective-schemes Comment by aginensky aginensky 2013-04-08T21:20:30Z 2013-04-08T21:20:30Z For the first question, maybe this is too obvious, but what about transversality ? http://mathoverflow.net/questions/120905/conditional-probability-with-percentages Comment by aginensky aginensky 2013-02-05T20:57:50Z 2013-02-05T20:57:50Z the probability is 100% that this is a hw question. http://mathoverflow.net/questions/120422/the-map-from-c4-to-theta-as-a-blow-up Comment by aginensky aginensky 2013-02-04T23:03:27Z 2013-02-04T23:03:27Z @ Jie Wang - Ch IV section 3 - starts with the sentence &quot; Our first goal, in this section, is easily stated. We fix a smooth genus g curve C and two non-negative integers r,d: we would then like to construct a subvariety $W^r_d$ of $Pic^d(C)$ whose support is the set of complete linear series of degree d and dimensionat least r. Probably one should start with the beginning of Ch. IV for background. They are determinantal and they also construct the resolution. http://mathoverflow.net/questions/120422/the-map-from-c4-to-theta-as-a-blow-up Comment by aginensky aginensky 2013-01-31T20:44:38Z 2013-01-31T20:44:38Z I am putting this in because you say 'any input is helpful'. As I recall, if you look in A-C-G-H, this is explained. The singularities of the theta divisor are a 'determinantal variety'. The construction is more general, it explains the algebraic structure of 'the set of line bundles with more sections than the general one'. Have a look. http://mathoverflow.net/questions/116831/tensor-rank-of-anti-symmetric-tensor Comment by aginensky aginensky 2013-01-17T17:17:37Z 2013-01-17T17:17:37Z Here is a link - <a href="http://arxiv.org/abs/1110.0745" rel="nofollow">arxiv.org/abs/1110.0745</a> . I think the rank of 'detrminant' considered as a symmetric tensor must be known, but I do't know it ! http://mathoverflow.net/questions/116831/tensor-rank-of-anti-symmetric-tensor Comment by aginensky aginensky 2013-01-17T17:15:30Z 2013-01-17T17:15:30Z For symmetric tensors, I think your problem is called 'Waring Problem for polynomials'. Specifically, identifying symmetric tensors with polynomials, the Waring problem asks- given a homogeneous polynomial of degree d, what is the minimum number of d-th powers of a linear polynomial that are needed to write the given polynomial. The generic number has been known for a while and is called (i hope i'm remembering correctly) the Alexander-Hirshowitz theorem. The problem of given a monomial, how many dth forms are needed to write it was just solved and is on the arxiv. http://mathoverflow.net/questions/117095/does-the-albanese-map-satisfy-torellis-theorem Comment by aginensky aginensky 2012-12-23T18:04:39Z 2012-12-23T18:04:39Z Any regular variety has trivial Albanese, so any flat family of regular varieties will fail- or did I misread the question. Less formally, it seems that except for curves, the Albanese is a very crude invariant and won't detect any part of the structure of a variety that is at all unirational.