# Equidistribution relative to the Zariski topology

I'm interested in measuring the distribution of an ordered set of points relative to the Zariski topology. Possibly this is a standard idea (with different terminology?) in algebraic geometry. Any pointers to the literature would be appreciated. Here's what I have in mind.

Let $X\subset\textbf{P}^N$ be a projective variety defined over an algebraically closed field $k$. For any finite set of points of $X$, define the $X$-degree of the set to be the minimal degree of a homogeneous polynomial $F\in k[X_0,\ldots,X_n]$ that does not vanish identically on $X$, but vanishes at all of the points in the set. (In other words, the minimal degree of a hypersurface that goes through the points, but does not contain $X$.)

Now take a sequence of points $S=(P_1,P_2,P_3,\ldots)\subset X$, and for each $n\ge1$, let $S[n]=\{P_1,P_2,\ldots,P_n\}$. Intuitively, if $\deg_X S[n]$ grows quickly, then the set $S$ is well distributed relative to the Zariski topology. A rough guess is that one might define equidistribution by the condition $$\lim_{n\to\infty} \frac{\log\deg_XS[n]}{\log n} = \frac{1}{\dim X}.$$ (It is not hard to check that the limsup of the left-hand side is at most $\frac{1}{\dim X}$.)

[Edit: value of limit fixed as suggested by JSE]

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First of all, I think the limit above would like to be 1/(dim X) rather than 1/N -- given d^N points on X, there will indeed be a degree-d polynomial passing through them, but this polynomial might well vanish on all of X (e.g. consider the case where X is a line in P^N -- in order to pass through d points on the line and not vanish on the line, your polynomial had better have degree d.)

I would say this is closer to a weak notion of "general position" than of "equidistribution" -- for points on the plane, aren't you asking for some coarse version of "not too many of the first few points on a line, not too many of the first few points on a conic, ..... etc?"

Anyway, as for where this kind of thing appears in the literature, maybe the argument of Heath-Brown about uniform bounds for points on curves? (Huayi Chen has some appealing-looking recent papers extending this approach to more general varieties.) You could describe these results as saying that if you have a long list of rational points whose height grows very slowly, they DO tend to line up on unexpectedly low-degree hypersurfaces; so in your context, this would be the opposite of the usual situation in which slow growth of height enforces equidistribution!

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@JSE: Thanks, Jordan. I'll have to think about the limit, probably my "back of the envelope" calculation was completely wrong. I agree this is a notion of general position, but maybe it's reasonable to say that "uniformly in general position" is a good way to define "equidistributed in the Zariski topology". (I don't know of another def for the latter.) Good thought about Heath-Brown et.al., although in principle, the points in S need not have small height, nor indeed do they have to be algebraic. But in my application, they are algebraic, and maybe they have bounded height. –  Joe Silverman Jan 22 '11 at 3:49