Small neighborhoods of singularities on varieties - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T05:45:26Z http://mathoverflow.net/feeds/question/33199 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/33199/small-neighborhoods-of-singularities-on-varieties Small neighborhoods of singularities on varieties Charles Staats 2010-07-24T13:56:55Z 2011-07-23T11:25:53Z <p>In <em>Singular points of complex hypersurfaces</em>, John Milnor proves the following theorem:</p> <p>Let $x \in V$ be a point on a variety $V$ in $\mathbb{R}^n$ or $\mathbb{C}^n$. Assume $x$ is either a smooth point or an isolated singularity. Let $D_{\epsilon}$ be the closed $\epsilon$-ball about $x$, $S_{\epsilon}$ its boundary (the sphere about $x$ of radius $\epsilon$), and $K = V \cap S_{\epsilon}$. Then for $\epsilon$ sufficiently small, the pair $(D_{\epsilon}, V \cap D_{\epsilon})$ is homeomorphic to the pair $(CS_{\epsilon}, CK)$, where $C$ denotes taking the cone. (Theorem 2.10)</p> <p>In Remark 2.11, Milnor observes that this theorem "likely" holds even if $x$ is a non-isolated singularity; in particular, it is known even in this case that "a suitably chosen neighborhood of any point is homeomorphic to the cone over something."</p> <p>This book was written in 1968. What is the current status of this problem?</p> http://mathoverflow.net/questions/33199/small-neighborhoods-of-singularities-on-varieties/33241#33241 Answer by Greg Kuperberg for Small neighborhoods of singularities on varieties Greg Kuperberg 2010-07-24T22:50:10Z 2010-07-25T06:20:09Z <p>There is a good paper of Goresky, "<a href="http://www.jstor.org/stable/2042563" rel="nofollow">Triangulation of Stratified Objects</a>", that I think reasonably quickly implies Milnor's result and its generalization to non-isolated singularities. The result is that any Whitney-stratified set, and in particular any algebraic variety in $\mathbb{C}^n$, is supported on a smooth triangulation. I think that you just need that and the inverse function theorem.</p> <p>As I meant to explain in the comments, this theorem has sometimes been regarded as a "chore" theorem. You can look at what Goresky says: "Triangulation theorems for stratified objects have been obtained independently by Hendricks (unpublished), Johnson (unpublished), and Kato (in Japanese)". When Goresky wrote his paper, it was a messy question that did not have a well-defined status. Now the situation is a bit better and I think that this generalization of Milnor's result can be called settled. Sometimes a good author not only proves a chore theorem, but also cleans it up an elevates it to non-chore status. But a lot of chore theorems are never proven in a clean form or are never proven at all.</p> http://mathoverflow.net/questions/33199/small-neighborhoods-of-singularities-on-varieties/71059#71059 Answer by A. Lerario for Small neighborhoods of singularities on varieties A. Lerario 2011-07-23T11:25:53Z 2011-07-23T11:25:53Z <p>Indeed the following theorem to me seems exactly you were looking for (see J. Bochnak, M. Coste, M-F. Roy, "Real algebraic geometry", Theorem 9.3.6 [Local conic structure]):</p> <p>Let $E$ be a semialgebraic susbet of $\mathbb{R}^n$ and $x$ be a nonisolated point of $E.$ Let also $D_\epsilon$ be the closed $\epsilon$-ball around $x$ and $S_\epsilon$ its boundary. Set $K=S_\epsilon \cap E$. Then there for $\epsilon>0$ small enough the pair $(D_\epsilon,E∩D_\epsilon)$ is semialgebraically homeomorphic to the pair $(CS_\epsilon,CK)$, where $C$ denotes taking the cone. Moreover the semialgebraic homeomorphism can be chosen as to preserve the distance from $x.$</p> <p>Two words of remarks on the previous statement: </p> <ol> <li>Every real or complex algebraic set in $\mathbb{R}^n$ or in $\mathbb{C}^n\simeq \mathbb{R}^{2n}$ is a semialgebraic set. </li> <li>The point $x$ is any nonisolated point of $E$ (no matter singular - in whatever meaning this word has for a general semialgebraic set - or regular).</li> </ol>