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Recall, index of an algebraic scheme $X$ is defined to be the greatest common divisor of the degrees of the space of zero cycles on $X$. I am interested in examples of hypersurfaces in $\mathbb{P}^n_K$ where $K$ is a p-adic field (for example $\mathbb{Q}_p$) whose index is greater than $1$. If $n=2$ such examples are not very difficult to construct. In general what is known about hypersurfaces with index greater than $1$? Is there any standard trick to produce examples of such hypersurfaces? Any idea/reference in this direction will be most helpful.

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  • $\begingroup$ 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. $\endgroup$
    – Jason Starr
    Commented Apr 6, 2013 at 21:42
  • $\begingroup$ 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 $6$." $\endgroup$
    – R.P.
    Commented Apr 6, 2013 at 22:49
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    $\begingroup$ @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. $\endgroup$
    – Jason Starr
    Commented Apr 6, 2013 at 23:07
  • $\begingroup$ @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. $\endgroup$
    – Jason Starr
    Commented Apr 6, 2013 at 23:08
  • $\begingroup$ Thanks! I realized that you can indeed take any curve and 'pinch' some effective zero-cycle into a rational point. $\endgroup$
    – R.P.
    Commented Apr 7, 2013 at 0:22

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Here is a set of examples I constructed (see `Stably free modules', Amer. Journal, Vol 107, 1985). I will briefly describe them, since these may not be of your interest. Let $k$ be any field and let $f(x)$ be a degree $p$ polynomial over $k$ for a prime $p$, with $f(0)\neq 0$ and $f(x^{p^{n-1}})$ irreducible over $k$ and some $n$. Using such an $f$, you can construct a hypersurface in $\mathbb{P}^n_k$ of index $p$. My main interest in the aforementioned paper was for $k$, rational functions over another field, where such $f$ are easy to construct.

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  • $\begingroup$ @Mohan: In your paper as far as I understand, to prove that every point in $S_n$ (notations as in the paper) has degree a multiple of degree $p$ you produce a point which is of degree $p$. Since this point is the intersection of $S_n$ with $n-1$ general hyperplane sections, this gives the degree of $S_n$. Do I understand this correctly? $\endgroup$
    – Jana
    Commented Apr 8, 2013 at 20:48
  • $\begingroup$ @Jana, I do not think the degree $p$ point is the intersection of $S_n$ with a general linear subspace. It is rather special. The degree of the hypersurface $S_n$ is $p^n$. $\endgroup$
    – Mohan
    Commented Apr 8, 2013 at 22:13

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