Are there any computer algebra systems (e.g. Macaulay2 og singular) that allows one to compute the Picard number (i.e. the rank of the NeronSeveri group) of a given variety?
1 Answer
A more basic question is whether there even exists an algorithm to compute this number. I've wondered this for a long time, and I honestly don't know what to expect. Any algorithm would have to be quite subtle. In the early 1980's Shioda had to work quite hard to construct explicit examples of surfaces in $\mathbb{P}^3$, defined over $\mathbb{Q}$, with Picard number 1.
Added: Of course, I should have said Shioda's example have degree >1. As further evidence of subtlety of the problem: one can decide whether an elliptic curve $E$ has CM by computing the Picard number of $E\times E$.

$\begingroup$ I really like the elliptic curve example! $\endgroup$– S. Carnahan ♦May 30, 2010 at 16:59

$\begingroup$ Assuming the Tate conjecture, one can imagine an algorithm computing ladic cohomology. $\endgroup$ May 30, 2010 at 17:20

$\begingroup$ Are you saying that there may be an algorithm to compute the Galois invariant part? I would be interested to see this spelled out if you have some thoughts on this. $\endgroup$ May 30, 2010 at 18:07

$\begingroup$ @Donu: I don't claim to have thought this through but I don't see an obstacle for a simpleminded approach, as long as one doesn't care about efficiency. Chebotarev should provide a finite set of matrices that describe the action and then it's just linear algebra (?). $\endgroup$ May 30, 2010 at 20:00

$\begingroup$ I'm probably showing my ignorance, but is the first step involving Chebotarev effective? $\endgroup$ May 31, 2010 at 13:57