Timeline for Connectedness of Milnor fiber
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 22 at 6:18 | vote | accept | RKS | ||
| Jul 19 at 19:27 | answer | added | Jason Starr | timeline score: 9 | |
| Jul 19 at 11:08 | comment | added | Jason Starr | That is an “affine patch” of a projective hypersurface $F(z_1,\dots,z_n)-w^d=0$, where $w$ is a “homogenization “ variable, and where $F$ is the degree $d$ homogeneous polynomial that is the product of $d$ homogeneous linear polynomials, $L_1\cdots L_d$. Since your linear polynomials are pairwise linear independent, this is singular in codimension $2$. Hence the hypersurface is normal (by Serre), and thus irreducible. So the affine patch is connected. | |
| Jul 19 at 9:50 | comment | added | RKS | $\prod_{i < j}(z_i\pm z_j)\prod(z_1\pm z_2\pm \dots \pm z_n)$ (all possible combination of signs). This does not have isolated singularity at $(0,0,\dots ,0)$. | |
| Jul 19 at 9:43 | comment | added | RKS | Thanks a lot Dave and Don for your responses. I have a particular example in mind: | |
| Jul 19 at 6:34 | comment | added | Dave Benson | One relevant article is M. Oda, "On the connectivity of the Milnor fiber for mixed functions" in CONM742. | |
| Jul 19 at 5:59 | comment | added | Don | Not a complete answer but a helpful remark. If $f: (\mathbb{C}^{n+1},0) \to (\mathbb{C},0)$ defines an isolated hypersurface singularity at $0$, then in Milnor's book is proven that the Milnor fibre has the homotopy type of a wedge of spheres $S^n$. In particular, the Milnor fibre is connected. | |
| Jul 19 at 5:32 | history | edited | RKS | CC BY-SA 4.0 |
added 2 characters in body
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| Jul 19 at 5:20 | comment | added | RKS | It is known that that map is a locally trivial fibration (see Milnor's Singular points of complex hypersurfaces). Hence any two fibers are homeomorphic. My question is when is the fiber, e.g. $Q^{-1}(1)$ connected. | |
| Jul 19 at 2:30 | history | asked | RKS | CC BY-SA 4.0 |