Let X_n be the moduli space of n-tuples of points in A^2 such that no three are collinear. The comment section of this blog computed the number of components of X_n(R) back in January. Skorobogatov asked what I could say about the cohomology of X_n(C). Well, not a lot! But if I were going to make a good guess, I’d start by trying to estimate the number of points on X_n over a finite field F_q.
So here’s a question: can you estimate the number of degree-n 0-dimensional subschemes S of A^2/F_q which have no three points collinear? It seems very likely to me that the answer is of the form
for some power series P.
One way to start, based on the strategy in Poonen’s Bertini paper: given a line L, work out the probability P_L that S doesn’t have three points on L. Now your first instinct might be to take the product of P_L over all lines in A^2; this will be some version of a special value of the zeta function of the dual P^2. But it’s not totally clear to me that “having three points on L_1” and “having three points on L_2” are independent.