Timeline for Extending birational isomorphisms between planar curves to the P^2
Current License: CC BY-SA 3.0
12 events
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| May 30, 2012 at 23:33 | answer | added | Jérémy Blanc | timeline score: 4 | |
| May 11, 2012 at 20:00 | history | edited | Maarten Derickx | CC BY-SA 3.0 |
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| May 7, 2012 at 17:52 | comment | added | Jack Huizenga | OK, thanks! For anyone interested, the relevant exercise batch is Appendix A, Section 1 (page 56). More precisely, it is an exercise that if $\Gamma \subset \mathbb{P}^2$ is a degree $d$ nodal plane curve with $\delta < d-3$ nodes, then the induced $g_d^2$ on the normalization is unique. So this covers positively the case where the curves have the same degree and do not have too many nodes relative to the degree; in fact in these cases the curves are actually projectively equivalent. | |
| May 7, 2012 at 12:51 | comment | added | Felipe Voloch | @Jack. I don't have the book handy, but they prove in an exercise that the $g^2_d$ is unique on a smooth plane curve of degree $d$. | |
| May 7, 2012 at 11:21 | history | edited | R.P. | CC BY-SA 3.0 |
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| May 7, 2012 at 9:25 | vote | accept | Maarten Derickx | ||
| May 7, 2012 at 9:21 | history | edited | Maarten Derickx | CC BY-SA 3.0 |
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| May 7, 2012 at 9:08 | answer | added | Francesco Polizzi | timeline score: 12 | |
| May 7, 2012 at 2:25 | comment | added | Jack Huizenga | @Felipe: do you know where to find this in ACGH? I don't remember where to find a section discussing birational automorphisms. I certainly suspect this is very false for singular plane curves, which are somehow the "generic" case when viewed from the point of view of the moduli of curves instead of when viewed as zero loci of polynomials--most curves do not embed smoothly in P^2. I don't have a simple argument yet, though--the Cremona group of birational transformations is deceptively complicated. | |
| May 6, 2012 at 23:54 | comment | added | Felipe Voloch | True if both curves are smooth of the same degree at least four (see Geometry of Algebraic Curves by Arbarello et al). False in general, as Karl points out. | |
| May 6, 2012 at 23:08 | answer | added | Karl Schwede | timeline score: 3 | |
| May 6, 2012 at 21:27 | history | asked | Maarten Derickx | CC BY-SA 3.0 |