Timeline for Why there are two point at infinity on certain elliptic curve
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 4, 2014 at 17:41 | comment | added | Michael Stoll | As to your second comment, the singuarity shows up because you do not take the "right" projective closure. If you have a smooth affine curve, then there is always a smooth projective curve (sitting in ${\mathbb P}^n$ for some $n$) that is birational to it; this smooth projective model is unique up to isomprphism. The "points at infinity" then are the points one has to add to the affine curve to obtain the smooth projective model. | |
| Dec 4, 2014 at 17:37 | comment | added | Michael Stoll | ... singular points (which are not on the curve above, however). But, as Joe Silverman points out, it is possible to avoid the weighted projective plane here and work with a collection of affine patches instead. | |
| Dec 4, 2014 at 17:35 | comment | added | Michael Stoll | To get the points of ${\mathbb P}^2_{1,2,1}$ you identify coordinates $(x,y,z)$ and $(\lambda x, \lambda^2 y, \lambda z)$ (with $\lambda \neq 0$ as usual). The degree of a polynomial (w.r.t. the weights) is then defined by using $\deg(x) = 1$, $\deg(y) = 2$, $\deg(z) = 1$, so that the equation in my answer is homogeneous of degree 4 and therefore defines a subvariety of ${\mathbb P}^2_{1,2,1}$. The advantage of using these weighted spaces is that one can get smooth models in lower-dimensional projective spaces. A disadvantage is that weighted projective spaces can have ... | |
| Dec 4, 2014 at 16:20 | comment | added | Maxim | One more question: there was smooth curve and then we map it to curve with singularity at infinity. So we created singularity in some case. What is the general approach to such situations? | |
| Dec 4, 2014 at 16:16 | comment | added | Maxim | Thanks! Where is it possible to read about weighted projective plane? It is quite new notion for me. | |
| Dec 4, 2014 at 16:01 | history | answered | Michael Stoll | CC BY-SA 3.0 |