Timeline for Tricks to produce examples of hypersurfaces with index greater than $1$
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Apr 7, 2013 at 18:54 | answer | added | Mohan | timeline score: 1 | |
| Apr 7, 2013 at 0:22 | comment | added | R.P. | Thanks! I realized that you can indeed take any curve and 'pinch' some effective zero-cycle into a rational point. | |
| Apr 6, 2013 at 23:08 | comment | added | Jason Starr | @Pannekoek: My comment above (perhaps a bit cryptic) is an example of a birational, projective morphism between non-normal varieties for which Lang-Nishimura fails. | |
| Apr 6, 2013 at 23:07 | comment | added | Jason Starr |
@Pannekoek: Consider, for instance, a plane conic $X$ with affine equation $au^2 + bv^2 = 1$ for $a,b\in K$ such that $X$ has no degree $1$ zero-cycle (easy to determine using Legendre's theorem). Consider the morphism $f:X\to \mathbb{A}^3_K$ given by $f(u,v) = (au^2-1,v,uv)$. This maps the degree $2$ zero-cycle $Z(au^2-1,v)$ to a $K$-point $(0,0,0)$. On the open complements, this map is an isomorphism.
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| Apr 6, 2013 at 22:49 | comment | added | R.P. | Right. But since the second part of my answer was incorrect, I have deleted it. The first part read: "A non-trivial Severi-Brauer variety of dimension $2$ over $K$ has index $3$ and is birational to a smooth cubic surface over $K$: just blow up a zero-cycle of degree $6$." | |
| Apr 6, 2013 at 21:42 | comment | added | Jason Starr | If you are looking for examples of varieties with index > 1, I recommend you look up the phrase "normic forms". I believe Lang produced normic forms for p-adic fields. As Pannekoek mentions below, you can use central simple algebras / Severi-Brauer varieties to produce normic forms (e.g., look at the reduced norm of the algebra), and local class field theory characterizes central simple algebras over p-adic fields. | |
| Apr 6, 2013 at 18:51 | history | asked | Jana | CC BY-SA 3.0 |