Timeline for Smooth curves on smooth varieties
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| S May 27, 2016 at 14:14 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 3.0 |
Minor edits.
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| S May 27, 2016 at 14:14 | history | suggested | Sean Lawton | CC BY-SA 3.0 |
Minor edits.
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| May 27, 2016 at 14:04 | review | Suggested edits | |||
| S May 27, 2016 at 14:14 | |||||
| Nov 16, 2012 at 3:14 | comment | added | Olivier Benoist | @algori : yes, indeed ! The answer is positive if $X$ is separably rationally connected (over an algebraically closed field). Indeed, if $X$ is of dimension $\leq 2$, it is projective so there is no problem. And if it is of dimension $\geq 3$ it contains (lots of) smooth rational curves by Theorem IV 3.9 in Koll'ar's book "Rational curves on algebraic varieties". | |
| Nov 16, 2012 at 1:44 | comment | added | algori | Olivier -- it's worse than you think: I'm not sure how to prevent the curve from intersecting itself. | |
| Nov 16, 2012 at 1:22 | comment | added | Olivier Benoist | @algori : The problem with this strategy is that it is really possible to have a divisor $E$ on $X'$, exceptional over $X$, such that every curve meeting $E$ has singular image in $X$. This happens for instance if $X$ is a smooth surface, and $X'$ is constructed by blowing up a point, then blowing up a point on the exceptional divisor, then blowing up the intersection of the two exceptional divisors, and if you choose $E$ to be the third exceptional divisor. The problem is that the differential of $X'\to X$ is identically zero at every point of $E$. | |
| Nov 16, 2012 at 1:07 | comment | added | algori | Here is a strategy for showing that the answer is positive: blow up a proper smooth variety, call it $X$, to get a projective smooth $X'$ such that $X'\setminus D'$ is isomorphic to $X\setminus D$ with $D\subset X$ closed and $D'\subset X'$ a divisor with normal crossings; then take a curve in $X'$ that intersects transversally all the components of $D'$ and project back again. | |
| Nov 15, 2012 at 22:14 | history | edited | Antoine Ducros | CC BY-SA 3.0 |
edited body
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| Nov 15, 2012 at 21:44 | history | asked | Antoine Ducros | CC BY-SA 3.0 |