User pmath - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T18:44:06Z http://mathoverflow.net/feeds/user/16409 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/126899/bounds-for-the-milnor-number-of-a-hypersurface-singularity Bounds for the milnor number of a hypersurface singularity pmath 2013-04-08T19:32:34Z 2013-04-09T00:08:38Z <p>I am having a hard time in finding an upper bound in terms of the degree and the dimension for the Milnor number of an isolated hypersurface singularity. I am mostly interested in surfaces on the projective space. Can some one please give me a hint on this?</p> <p>Thanks!</p> http://mathoverflow.net/questions/14489/limit-of-a-series-of-singularities/118699#118699 Answer by pmath for Limit of a series of singularities pmath 2013-01-12T03:32:47Z 2013-01-12T03:32:47Z <p>This is not an answer, but rather a long comment (grad student level, so please don't take it seriously). I use surfaces for simplicity. The answer must yes in some form. My belief is from the moduli space theory. It is known that the normal stable surfaces admit at worst log canonical isolated singularities. This includes $xyz+x^p+y^r+z^q$ singularities. However, to have a complete moduli space of surfaces, we must include no isolated singularities of the form $xyz$, $xyz+x^p$, and $xyz+x^p+y^r$ (among others). The resemblance of the equations must be more than a coincidence. So, I can imagine we can have an isolated singularity and consider all the deformations from it to non isolated ones. Then to look for the minimal "complete" family of such degenerations. </p> <p>I wish someone can say something more about all this. </p> http://mathoverflow.net/questions/107345/blowing-up-general-k-points-on-the-plane blowing up general k points on the plane pmath 2012-09-16T20:00:55Z 2012-09-16T22:43:43Z <p>Del Pezzo surfaces are obtained by blowing up $1 \leq k \leq 8$ points on general position in $\mathbb{P}^2$. What does it happen when the number of points is larger than nine? In this sense, Beauville's book in surfaces presents the topic in the context of linear system of cubics: Nine points in the plane determine a cubic curve, and del Pezzo surfaces $S_{9-k}$, with $k\leq 6$, are embedded into $\mathbb{P}^{9-k}$ by the linear system of cubics through the $k$ points. Is there a nice interpretation of the surfaces obtained by linear system of plane curves of degree $d$?</p> <p>I suppose this is well known but I cannot find a reference. Thanks!!</p> http://mathoverflow.net/questions/91839/commuting-the-resolution-of-1-dim-singular-locus-and-0-dim-singularities-in-a-non commuting the resolution of 1-dim singular locus and 0-dim singularities in a non isolated singularity of a surface pmath 2012-03-21T17:09:33Z 2012-03-22T11:28:58Z <p>Let $X$ be a surface with a non isolated singularity $C = Sing(X)$ such that the curve $C$ has singularities itself. We can solve $Sing(X)$ by blowing up close points and by normalizing. Indeed, we can first solve the 1-dimensional singularities with only normalizations $\pi_1: X_1 \to X$ . In the surface $X_1$ the preimage of $C$ has at most isolated singularities which we can solve by $\pi_2:X_2 \to X_1$. </p> <p>Another approach is to solve the 0-dimensional singularities by $\varphi_1:Y_1 \to X$, and then to finish with a one dimensional singular locus on $Y_1$ that we can be solve by normalization $\varphi_2:Y_2 \to Y_1$ such that $Y_2$ is smooth (Maybe some ADE singularities, but it would not matter). </p> <p>I am wondering if the second approach is always possible, and if "commutes" with the first one in some way. I have this idea that normalization remove the 1-dimensional singularity $C$ without affecting the 0-dimensional ones, even if they are supported on $C$. Is this true? In that fantasy, we can "commute" those processes of solving 0-dimensional singularities, and solving 1-dimensional singularities.</p> <p>I will appreciate any enlighting</p> http://mathoverflow.net/questions/86477/simple-question-in-the-representation-of-sl2-c Simple question in the representation of SL(2,C) pmath 2012-01-23T20:03:37Z 2012-01-26T04:00:18Z <p>Let $V$ the standard two dimensional representation of SL(2,C). The Fulton's book in representation theory say in pag 156 that $Sym^3(Sym^2V)=Sym^6(V) \oplus Sym^2(V)$. In the excercises 11.23, the books asks to prove the decomposition<br> $$Sym^3(Sym^3V) = Sym^9(V) \oplus Sym^5(V) \oplus Sym^3V$$</p> <p>In my work, I found $Sym^5(Sym^3V)$,and $Sym^k(Sym^3V)$, So I was looking for a similar decomposition. I am not familiar enought with the theory, and to study the subject will take a little to far from my current work. So I decided to ask here (sorry if the question is too simple). </p> <p>Thanks for any help!!</p> http://mathoverflow.net/questions/85512/how-can-we-find-a-surface-with-a-given-singularity How can we find a surface with a given singularity? pmath 2012-01-12T18:26:30Z 2012-01-13T17:12:59Z <p>I was surprised the first time I learned that a quintic plane curve can have an $A_{10}$ singularity i.e $x^2+y^{10}$. I am wondering if there is something about that phenomenon: Given a singularity on its normal form, and a fixed degree $d$. Is there a standard way to find an hypersurface of degree $d$ on $\mathbb{P}^n$ with the given singularity? Is there a lower bound for $d$ in terms of the singularity invariants?</p> <p>For example: Given the singularity $x^2+y^3+z^{13}$ and $d=5$, the surface $$(x+z^3)^2+(y-z^2)^3+x^3y^2+x^5$$ has that singularity at $(0,0,0)$. How can I define a surface for another singularity e.g. $x^2+y^4+z^{22}$ ?</p> <p>I am mostly thinking in surfaces and plane curves. Thanks for any hint or suggestion!</p> http://mathoverflow.net/questions/80979/what-is-it-known-about-4-modal-singularities what is it known about 4-modal singularities pmath 2011-11-15T13:17:16Z 2011-11-15T13:17:16Z <p>I am wondering what is it known about 4-modal singularities. C.T.C Wall has a lot of work on trimodal singularities, and Arnold did the cases $m=0,1,2$. However, I am having a hard time finding a references for $m=4$. My main motivation is to study some degenerations on algebraic surfaces. </p> <p>thanks!</p> http://mathoverflow.net/questions/78772/hyperplane-sections-of-isolated-hypersurface-singularities hyperplane sections of isolated hypersurface singularities. pmath 2011-10-21T15:33:58Z 2011-11-01T15:47:01Z <p>Given an isolated singularity $p$ in a hypersurface $Y$ of dimension $n$ (let say a surface in $\mathbb{P}^3$). I can intersect $Y$ with a hyperplane $H$ passing through $p$ such that it induces a singularity in a lower dimension. For example, if we start with a surface $Y$, we obtain a plane singularity on $H$. </p> <p>I want to use the singularities in the hyperplanes for classifying(?) the original singularity in the hypersurface. Using homeomorphism for defining an equivalence relation. There is a finite number of singularities types that I can obtained in the hyperplanes, and one of those types will be "generic". </p> <p>I want to believe that those hyperplane singularities + some "vicinity" data is enough for characterizing the original singularity. However, I cannot find related work, or theorems to start with. I appreciate any help or references.</p> <p>Thanks</p> http://mathoverflow.net/questions/79353/what-is-the-simplest-way-to-represent-a-d-5-singularity/79439#79439 Answer by pmath for What is the simplest way to represent a $D_5$ singularity? pmath 2011-10-29T00:42:51Z 2011-10-29T00:42:51Z <p>For classifying plane curves singularities, the "coordinate approach" is not always the better one*. For your question, the general case is in table 1, page 3, C.T.C Wall article" "sextic curves and quartic surfaces with higher multiplicity". For general methods, I recommend, C.T.C Wall article: "Notes in the classification of singularities"</p> <p>In general, the singularities $J_{r,i}$ or $E_{r,i}$ (the notation is not uniform) are given by $y^3+y^2x^r+x^{3r+i}$ with $r \geq 1$,and $i \geq 0$. In your case $r=i=1$ and the singularity is actually $D_5$. You can recognized it because it has two branches: one smooth, and another one with an $A_2$ singularity. Those branches separates after one blowing up. (see Table A, from the latest reference), and they are the factors that you see in your calculation. This "branch behavior" is the* definition of the $D_5$ singularity, and the normal form is deduced from it. A detailed discussion is in Barth's book in compact complex surfaces, page 79.</p> <p>I hope it helps!</p> <p>Psd: I don't see anything missing in your argument, but it is "simpler" by using $D_5$'s resolution. </p> <p>*to my knowledge/opinion</p> http://mathoverflow.net/questions/74444/applications-for-knowing-the-singularities-parametrized-by-the-boundary-of-a-modu Applications for knowing the singularities parametrized by the boundary of a moduli space pmath 2011-09-03T16:45:45Z 2011-09-04T14:29:17Z <p>Given a moduli space $M$ of some smooth algebraic geometric object such as curves, surfaces, etc. Let $\overline{M}$ be a compactification of $M$. Then, $\overline{M}\setminus M$ introduces singular objects in our moduli. The question is: What is the use of knowing the singularities parametrized by the boundary of the compactified moduli space i.e $\overline{M} \setminus M$. Usually $M$ has diferent compactifications, and so different " limit singular objects". Does this difference mean something?</p> <p>For example: the smooth genus $g=3$ have the $\overline{M_3}$ compactification with only stable curves in the boundary, but is possible to find another compactification by the GIT analysis of degree four plane curves. The singular curves present in the boundaries are quite different. What is the use of having an explicit descriptions of them?</p> http://mathoverflow.net/questions/74132/the-blowing-up-of-a-plane-curve-playing-me-tricks the blowing up of a plane curve playing me tricks. pmath 2011-08-31T03:18:47Z 2011-08-31T05:53:05Z <p>Sorry for the easy question but this is driving me crazy. Consider the blowing up of the curve $(y^2-x^3)^2+y^5$ at the origin. </p> <p>On the first blowing up, on the chart that intersects the exceptional divisor I have: $x^4(y^4-2xy^2+x^2+xy^5)$. </p> <p>The second blowing up, on the chart that intersects the exceptional divisor: $y^2(y^4x+y^2-2xy+x^2)$</p> <p>On the third blowing up, on the chart that intersects the exceptional divisor: $x^2(x^3y^4+y^2-2y+1)$. </p> <p>So the last strict transform is smooth at $(0,0)$ (the same for the other chart). I naively thought that this is the end, and I solved the singularity. </p> <p>However, the multiplicity sequence for the resolution is (4,2,2,2,1,1) , and the dual graph of the resolution is like a T, and it has six exceptional divisors. (the dual graph is here <a href="http://www.freeimagehosting.net/c2d80" rel="nofollow">link text</a>) So what am I missing? I tried several examples, and I am running into trouble specially when there is a node on my way. How can I keep going with the resolution all the way until the end?</p> <p>Ps: Calculations like the multiplicity sequence were done in Singular for avoiding trivial mistakes.</p> http://mathoverflow.net/questions/72328/about-maps-induced-for-a-divisor-d-in-p1 About maps induced for a divisor D in P^1 pmath 2011-08-08T04:16:35Z 2011-08-08T06:51:16Z <p>Let $D$ be an effective $\mathbb{Z}$-divisor on $\mathbb{P}^1$. Is there a form to associate a curve $C$, and morphism $C \to P^1$ to the divisor $D$ ? For example, let $Y$ be a singular plane curve, sometimes, we can built a cover of $\mathbb{P}^2$ that branches along $Y$. Is there a similar construction for covers of $\mathbb{P}^1$ ? I will appreciate any reference in this direction, or anything related? Thanks</p> http://mathoverflow.net/questions/72067/what-is-the-meaning-of-consecutive-triple-points what is the meaning of "consecutive triple points"? pmath 2011-08-04T06:21:13Z 2011-08-05T10:47:15Z <p>Hi, I am reading an the Shah's article " A complete moduli Space for K3 surfaces of degree 2" At some point, he analyses the singularities on plane curves of degree six. He uses the phrase: "Reduce sextics which have neither consecutive triple points.."[Th 2.4]. I am confuse for the sentence. What is the meaning of that? Are they fat points in some special configuration? Can anyone give me a polynomial that induces this kind of singularities? Thanks!</p> http://mathoverflow.net/questions/107345/blowing-up-general-k-points-on-the-plane Comment by pmath pmath 2012-09-16T22:10:45Z 2012-09-16T22:10:45Z thanks, I edited the question. http://mathoverflow.net/questions/107345/blowing-up-general-k-points-on-the-plane/107350#107350 Comment by pmath pmath 2012-09-16T22:10:08Z 2012-09-16T22:10:08Z For example, given 14-k points* on the plane, consider a d=4 plane curve passing through them, and let X be the blowing up of the 14-k points in P^2. The divisor on X obtained for pulling back such a quartic plane curve is ample, so I was wondering if its possible to repeat the del Pezzo case analysis but with this divisors instead of the anticanonical one? * for the k that makes sense. http://mathoverflow.net/questions/90862/what-are-some-examples-of-ingenious-unexpected-constructions/90916#90916 Comment by pmath pmath 2012-03-12T00:11:44Z 2012-03-12T00:11:44Z can you please add the one sentence proof? http://mathoverflow.net/questions/85512/how-can-we-find-a-surface-with-a-given-singularity Comment by pmath pmath 2012-01-13T01:26:57Z 2012-01-13T01:26:57Z I suppose that here by codimension you mean the codimension of the equisingular stratum in the base of the versal deformation of the isolated singularity. In that case, let $T$ be a singularity with codimension d &lt; 40. The quintic surfaces moves in a family of dimension 40. Does your argument implies that the quintic surfaces could have a $T$ singularity? In www.springerlink.com/content/vl707521j043m877/ singularities are classified by their &quot;jacobian algebra&quot; so I was thinking to find generators of this C-algebra on a given degree. Unfortunately, I dont know about that this technology! http://mathoverflow.net/questions/78772/hyperplane-sections-of-isolated-hypersurface-singularities/78776#78776 Comment by pmath pmath 2011-10-29T00:57:48Z 2011-10-29T00:57:48Z Thanks a lot for the answers and references. I was also wondering in the line of Arnolds's classification of singularities. Indeed, the Arnold's list, plus C.T.C Wall's work provide a complete classification up to modality ~3, and multiplcity ~6, but higher multiplicity singularities classification seems incomplete. However, I acknowledge those singularities are uglier than usual, and not that often exposed. Any case, thanks. http://mathoverflow.net/questions/74132/the-blowing-up-of-a-plane-curve-playing-me-tricks/74138#74138 Comment by pmath pmath 2011-08-31T06:37:26Z 2011-08-31T06:37:26Z Thanks for the answer. In the third blowup, the strict transform intersects the exceptional divisor at (0,0) and (0,1). So the point (0,0) is non-singular, but (0,1) is singular!!. If I move that point to the origin, I obtain $x^3(y+1)^4+y^2 = 0$. So, you are saying that I must follow the resolution by blowing at it?. http://mathoverflow.net/questions/72328/about-maps-induced-for-a-divisor-d-in-p1/72331#72331 Comment by pmath pmath 2011-08-08T07:08:37Z 2011-08-08T07:08:37Z Thanks a lot for answer and your reference. However, the main point is the case of a non reduce divisor. For example, if I have a cover $X \to P^1$ with two branching points, may I &quot;collapse&quot; those points for having a branch covering over my double point? Or it makes non sense the use of fat points in this context?