Is there an algorithm for calculating the Chow groups of a variety over a finite field?
It is know that $H^{2i,i}_\mathrm{mot}(X,\mathbf{Z}) = CH^i(X)$. In how many cases does this help us?
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$\begingroup$ Are there any algorithms for varieties over, say, C or Q? $\endgroup$– Kevin H. LinCommented Jan 10, 2010 at 17:21
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$\begingroup$ @Kevin: A^1 is isomorphic to Pic. Since there is no known algorithm for computing Pic of an elliptic curve over Q, there can't be a known algorithm for computing Chow groups over Q either. $\endgroup$– David E SpeyerCommented Jan 10, 2010 at 20:08
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3$\begingroup$ @David: there is an algorithm for computing Pic of an elliptic curve over Q. We just haven't proved it terminates yet! :-) $\endgroup$– Kevin BuzzardCommented Jan 10, 2010 at 20:52
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$\begingroup$ Fair enough. As you say, the right statement is that there is an algorithm which is conjectured to always terminate and, when it terminates, it computes Pic of an elliptic curve over Q. $\endgroup$– David E SpeyerCommented Jan 20, 2010 at 22:02
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$\begingroup$ Hey, have you had any luck with this question? $\endgroup$– Dror SpeiserCommented Jun 11, 2011 at 12:51
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1 Answer
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I am not an expert, but let me point out that computing $CH^0(X)$ (which is freely generated by the irreducible components) is already quite hard. Algorithms do exist in this case, see page 206 of "Ideals, varieties and algorithms" by Cox, Little, O'Shea for references. I know of no way to compute the class groups (which can be identified with $CH^1(X)$ for smooth $X$) in general, but I will be very interested in what other people have to say about this.
Of course, in special situations, more is known. For example, the total Chow group of quadric hypersurfaces (at least up to tensoring with $\mathbb Q$).
answered Jan 10, 2010 at 17:03