Timeline for Synthetic projective definition of cubic curves
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 3, 2021 at 4:35 | comment | added | brainjam | @MikeShulman, the synthetic approach to the complex plane was popular back in the day. There are lots of accounts, e.g. Hatton, Imaginary Geometry. | |
| Apr 27, 2015 at 7:54 | comment | added | Francesco Polizzi | Uhm, not that I'm aware of. But I'm not really an expert of synthetic projective geometry. | |
| Apr 27, 2015 at 7:14 | comment | added | Mike Shulman | Fraser's paper opens with a sketch of what appears to be a synthetic construction of the complex projective plane from the real one (I'm not sure what it would be in the general case, maybe the closure of a field under square roots?). This looks very cool, but there are a lot of details left out; is it written down carefully anywhere? | |
| Apr 24, 2015 at 18:44 | vote | accept | Mike Shulman | ||
| Apr 24, 2015 at 18:44 | comment | added | Mike Shulman | Fraser's paper seems to be exactly what I was looking for, thanks! (It's actually easier for me to read than your description, since I'm not an algebraic geometer but I've read some synthetic projective geometry.) | |
| Apr 22, 2015 at 8:36 | comment | added | Francesco Polizzi | I'm assuming that the ground field is any algebraically closed field $k$ of characteristic $0$ (for instance $k= \mathbb{C})$. The chapter on curves in Hartshorne's book is a good introduction for all this stuff. | |
| Apr 22, 2015 at 8:35 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
added 428 characters in body
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| Apr 22, 2015 at 8:30 | comment | added | Francesco Polizzi | It is a $1$-dimensional linear system of degree $2$ over a curve. By Riemann-Roch, since a smooth plane curve has genus $1$, every effective divisor of degree $2$ defines a $g^1_2$ on it. The classical term for such a object was "rational involution": in fact, every $g^1_2$ determines uniquely a degree $2$ morphism $C \to \mathbb{P}^1$. | |
| Apr 22, 2015 at 8:09 | comment | added | Dima Pasechnik | @MikeShulman - this is some particular kind of divisor, IIRC... | |
| Apr 22, 2015 at 7:36 | comment | added | Mike Shulman | What is a $g^1_2$? | |
| Apr 22, 2015 at 6:43 | history | answered | Francesco Polizzi | CC BY-SA 3.0 |