Timeline for A nontrivial surface on which any two curves intersect
Current License: CC BY-SA 2.5
16 events
| when toggle format | what | by | license | comment | |
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| Oct 17, 2022 at 4:52 | review | Close votes | |||
| Oct 18, 2022 at 20:41 | |||||
| Oct 27, 2011 at 18:18 | answer | added | Sándor Kovács | timeline score: 11 | |
| Dec 15, 2010 at 10:52 | comment | added | user5117 | Related question: mathoverflow.net/questions/47895 | |
| Dec 15, 2010 at 3:49 | answer | added | roy smith | timeline score: 13 | |
| Sep 13, 2010 at 12:44 | comment | added | Arend Bayer | @Charles: I believe BCnrd is trying to maximize the "actual reputation/MO reputation score" quotient by posting most of his answers as comments :) | |
| Sep 13, 2010 at 10:15 | answer | added | JRoss | timeline score: 2 | |
| Jun 16, 2010 at 15:29 | comment | added | Charlie Frohman | Alas, it is a theorem of S. Mori in 1979, and of Siu and Yau in 1980 that if a complex manifold has positive holomorphic sectional curvature then it is a projective space. More generally, A. Howard and B. Smith classified algebraic surfaces with nonnegative holomorphic sectional curvature, and they are $CP(2)$, $CP(1)\times CP(1)$ and $CP(1)$-bundles over an elliptic curve. I looked it up after I had the idea. There are manifolds of nonnegative holomorphic curvature with negative sectional curvature, like the moduli spaces of semistable bundles over a nonsingular curve. | |
| Jun 16, 2010 at 14:06 | comment | added | Tim Perutz | Charlie, that's interesting. Do such surfaces exist (besides the projective plane)? I don't know anything about holomorphic curvature, but positive scalar curvature forces an alg surface to be rational or ruled (by Seiberg-Witten theory). | |
| Jun 16, 2010 at 3:19 | comment | added | Charlie Frohman | This is from a slightly different direction. If the surface has positive holomorphic curvature then any two nonsingular curves will intersect. The reason is if C and D are disjoint then there are points p in C and q in D that minimize distance. Let gamma be a geodesic path realizing the minimum distance. Apply Synges variational inequality to a suitably chosen pair of vector fields along the geodesic path to see that it is not distance minimizing. | |
| Jun 16, 2010 at 2:58 | comment | added | Charles Staats | I thought that your statement "to avoid blow-up examples, perhaps you intend to assume minimality" might indicate that it was easy to produce examples if you were allowed to use blow-ups, which did not make sense to me. And if your comment were an answer, I would have accepted it. (Perhaps you could consider posting such things as community wiki answers, so that the OP has the opportunity to indicate acceptance without having to deal with reputation points? It would also get around comment length restrictions.) | |
| Jun 16, 2010 at 2:22 | comment | added | BCnrd | Charles, as you noted yourself, when blowing up one tends to easily get curves that don't intersect: exceptional divisor and any curve missing the blow-up point. So one is led to consider your question only for minimal algebraic surfaces. Anyway, Tim and I have indicated a bunch of (minimal surface) examples that solve your question, along with a general technique to make more examples (provided you can control the Neron-Severi rank). | |
| Jun 16, 2010 at 2:12 | comment | added | Charles Staats | How does allowing blow-up examples make the problem any easier? | |
| Jun 16, 2010 at 1:43 | comment | added | Tim Perutz | (What I ought to have said is that any two curves have non-zero intersection number.) | |
| Jun 16, 2010 at 1:27 | comment | added | Tim Perutz | Some general type examples (over $\mathbb{C}$): en.wikipedia.org/wiki/Fake_projective_plane (for similar reasons to BCnrd's: any curve in a projective surface is non-trivial in rational homology, so when that has rank 1, all curves have non-zero self-intersection). | |
| Jun 16, 2010 at 1:03 | comment | added | BCnrd | Yep. To avoid blow-up examples, perhaps you intend to assume minimality? What makes the projective plane work is the Neron-Severi group having rank 1 (and trivial torsion), so this leads us to look for other minimal surfaces with NS-group $\mathbf{Z}$. Consider smooth quartics in $\mathbf{P}^3$. These are K3, and have NS-rank 1 precisely when every (integral closed) curve is a complete intersection with a hypersurface in $\mathbf{P}^3$, so all are ample and hence we win. A "generic" smooth quartic has NS-rank 1 (Deligne), and such examples even exist defined over $\mathbf{Q}$ (van Luijk). | |
| Jun 16, 2010 at 0:21 | history | asked | Charles Staats | CC BY-SA 2.5 |