Bourqui on spaces of rational curves and motivic Batyrev-Manin

David Bourqui just posted a really nice paper, “Asymptotic Behavior of Rational Curves,” notes from a lecture series he gave last summer at the Institut Fourier at Grenoble.  I’ll try to sum up here what it’s about and hopefully entice people to have a look!

Let N_X(B) net of rational points of height at most B on a Fano variety X endowed with an ample line bundle L over a global field K.  By now we are used to the idea (summed up in greatest generality by the Baytrev-Manin conjecture, refined by Peyre) that N_X(B) satisfies an asymptotic regularity — it approaches

$c B^a (log B)^{b-1}$

for some constants a and b (which are integers) and c (a real number.)  Batyrev and Manin gave predictions for a and b, Peyre pinned down c.  To fix ideas, let’s suppose that the projective embedding of X is given by the anticanonical divisor.  Then a is 1 and b is the Picard rank of X (over K.)  So for instance when X = P^1, you can see immediately that the number of points of height B should be linear in B, and that’s true (remembering that the canonical height is the square of the usual Weil height on P^1.)

Now these conjectures are not exactly right.  There is the problem of accumulating subvarieties, like the lines on a cubic surface, which have way too many rational points; you have to strip these out before you can expect to get down to the expected asymptotic.  And there are more subtle counterexamples, like the one produced by Batyrev and Tschinkel, where the number of rational points is too high by some power of log B.  But the conjecture has been proved for many classes of varieties (toric, homogeneous, very low degree relative to dimension…)

Bourqui’s approach starts from the consideration of these conjectures over the global function field K = k(t), where k = F_q is a finite field.  For simplicity let’s take X to be defined over the constant field k.  Now N_K(B) has two meanings.  You can think of it as a set of K-points of X of bounded height — or you can think it as the number of k-points on the space C_d(X) of degree-d rational curves on X, where q^d = B.  The Batyrev-Manin conjecture, as we phrased it here, is about the first interpretation.  But you can also read it as a statement about the varieties C_d, and it turns out what it says is that

|C_d(k)| / |A^(k)|^d

approaches a limit as d goes to infinity.

Doesn’t this seem a rather astonishing claim at first glance?  These are higher and higher-dimensional varieties over a fixed finite field; the Weil conjectures offer us no useful control.  Why shouldn’t their point-counts fluctuate wildly?

To get some idea of what might be gone, think of a family of varieties that’s easily seen to display this behavior:  the projective spaces P^d.  Evidently,

$\lim_{d \rightarrow \infty} (|\mathbf{P}^d(k)| / |\mathbf{A}^1(k)|^d) = (1-1/q)^{-1}$.

Why is there a limit?  I can think of two reasons.

First reason:  projective spaces have stable cohomology;  the compactly supported cohomology has one dimension in degrees 2d,2d-2, … 0 and is empty in the odd degrees, and all the cohomology is of Tate type.  Note that (apart from the location of the top cohomology group) this description is independent of d, and it follows that the point count provided by the Lefschetz trace formula is (apart from a prepended power of q) independent of d as well.

Second reason:  Forget about finite fields — the expression

$\lim_{d \rightarrow \infty} ([\mathbf{P}^d] / [\mathbf{A}^1]^d) = (1-1/[A^1])^{-1}$

remains true motivically (in the suitably completed version of the Grothendieck ring of varieties.)  To be fair, this alone doesn’t quite imply that the point-counting limits hold.  (For instance, the sequence $1 + latex 2^{2^d} [\mathbf{A}^1]^{-d}$ converges motivically to 1, while none of its point-counts converge)  But the motivic convergence is highly suggestive.

My own work in this circle of ideas is mostly concerned with stable cohomology.  What Bourqui is interested in, on the other hand, is whether one has a motivic Batyrev-Manin conjecture; is it the case that

$[C_d] / [A^1]^d$

approaches a limit in d, and what is this limit?  (This is question 1.11.2 in Bourqui’s paper — to be precise, Bourqui asks for something more precise where one breaks up C_d according to the numerical equivalence class of the rational curve.)  Bourqui proves this is indeed the case when X is a smooth projective toric variety over a field of characteristic 0.  This is by no means a straightforward imitation of the proof of Batyrev-Manin for toric varieties over global fields:  proving motivic identities is hard!

In the case of toric varieties, by the way, both routes to Batyrev-Manin are available; the fact that spaces of rational curves on toric varieties have stable cohomology was proved by Martin Guest.  Guest shows that the cohomology stabilization holds for all smooth projective toric varieties and some of the singular ones as well — the main tool is a diffeomorphism between this space of rational curves and a certain kind of decorated configuration space on the sphere.  I wonder, is Guest’s configuration space implicitly present in Bourqui’s proof?

Motivic puzzle: the moduli space of squarefree polynomials

As I’ve mentioned before, the number of squarefree monic polynomials of degree n in F_q[t] is exactly q^n – q^{n-1}.

I explained in the earlier post how to interpret this fact in terms of the cohomology of the braid group.  But one can also ask whether this identity has a motivic interpretation.  Namely:  let U be the variety over Q parametrizing monic squarefree polynomials of degree d.  So U is a nice open subvariety of affine n-space.  Now the identity of point-counts above suggests the question:

Question: Is there an identity [U] = [A^n] – [A^{n-1}] in the ring of motives K_0(Var/Q)?

I asked Loeser, who seemed to feel the answer was likely yes, and pointed out to me that one could also ask whether the two classes were identical in the localization K_0(Var/Q)[1/L], where L is the class of A^1.  Are these questions different?  That is, is there any nontrivial kernel in the natural map K_0(Var/Q) -> K_0(Var/Q)[1/L]?  This too is apparently unknown.

Here, I’ll start you off by giving a positive answer in the easy case n=2!  Then the monic polynomials are parametrized by A^2, where (b,c) corresponds to the polynomial x^2 + bx + c.  The non-squarefree locus (i.e. the locus of vanishing of the discriminant) consists of solutions to b^2 – 4c = 0; the projection to c is an isomorphism to A^1 over Q.  So in this case the identity is indeed correct.

Update:  I totally forgot that Mike Zieve sent me a one-line argument a few months back for the identity |U(F_q)| = q^n – q^{n-1} which is in fact a proof of the motivic identity as well!  Here it is, in my paraphrase.

Write U_e for the subvariety of U consisting of degree-d polynomials of the form a(x)b(x)^2, with a,b monic, a squarefree, and b of degree e.  Then U is the union of U_e as e ranges from 1 to d/2.  Note that the factorisation as ab^2 is unique; i.e, U_e is naturally identified with {monic squarefree polynomials of degree d-2e} x {monic polynomials of degree e.}

Now let V be the space of all polynomials (not necessarily monic) of degree d-2, so that [V] = [A^{n-1}] – [A^{n-2}].  Let V_e be the space of polynomials which factor as c(x)d(x)^2, with d(x) having degree e-1.  Then V is the union of V_e as e ranges from 1 to d/2.

Now there is a map from U_e to V_e which sends a(x)b(x)^2 to a(x)(b(x) – b(0))^2, and one checks that this induces an isomorphism between V_e x A^1 and U_e, done.

But actually, now that I think of it, Mike’s observation allows you to get the motivic identity even without writing down the map above:  if we write $U^d_e$ for the space of monic squarefrees of degree d in stratum e, then $U^d_e = U_{d-2e} \times \mathbf{A}^e$, and then one can easily compute the class $U^d_0$ by induction.

“Le Groupe Fondamental de la Droite Projective Moins Trois Points” is now online

The three papers that influenced me the most at the beginning of my mathematical career were “Rational Isogenies of Prime Degree,” by my advisor, Barry Mazur; Serre’s “Sur les représentations modulaires de degré 2 de $\text {Gal}({\overline {\Bbb Q}}/{\Bbb Q})$;” and Deligne’s 200-page monograph on the fundamental group of the projective line minus three points.  The year after I got my Ph.D. I used to carry around a battered Xerox of this paper wherever I went, together with a notebook in which I recorded my confusions, questions, and insights about what I was reading.  This was the paper where I learned what a motive was, or at least some of the things a motive should be; where I first encountered the idea of a Tannakian category; where I first learned the definition of a Hodge structure, and what was meant by “periods.” Most importantly, I learned Deligne’s philosophy about the fundamental group:  that the grand questions proposed by Grothendieck in the “Esquisse d’un Programme” regarding the action of Gal(Q) on the etale fundamental group $\pi := \pi_1^{et}(\mathbf{P}^1/\overline{\mathbf{Q}} - 0,1,\infty)$ were simply beyond our current reach, but that the nilpotent completion of $\pi$ — which seems like only a tiny, tentative step into the non-abelian world! — nonetheless contains a huge amount of arithmetic information.  My favorite contemporary manifestation of this philosophy is Minhyong Kim’s remarkable work on non-abelian Chabauty.

Anyway:  Deligne’s article appears in the MSRI volume Galois Groups over Q, which is long out of print; I bought a copy at MSRI in 1999 and I don’t know anyone who’s gotten their hands on one since.  Kirsten Wickelgren, a young master of the nilpotent fundamental group, asked me the obvious-in-retrospect question of whether it was possible to get Deligne’s article back in print.  I talked to MSRI about this and it turns out that, since Springer owns the copyright, the book can’t be reprinted; but Deligne himself is allowed to make a scan of the article available on his personal web page.  Deligne graciously agreed:  and now, here it is, a publicly available .pdf scan of “Le Groupe Fondamental de la Droite Projective Moins Trois Points.”

Enjoy!

Positive motivic measures are counting measures

A new, very short paper with Michael Larsen, “Positive motivic measures are counting measures” is up on the arXiv today.  I thought I’d say a bit here about where the problem came from, since we don’t do so in the paper.

In the project with Akshay that I talked about at the recent Columbia conference on rational curves on varieties, one thing you do is compute estimates for |M_n(F_q)|, where M_n is the moduli space of algebraic maps of degree n from P^1 to some fixed target variety X, and F_q is a finite field.  These inequalities turn out to be very nicely uniform in q, which leads one naturally to ask; do the proofs actually give “motivic estimates” for the class [M_n] in the Grothendieck ring K_0(Var_K), for various non-finite fields K?

Well, what does it mean for one element r of a ring R to “estimate” another element s?  It might mean that r-s is rather deep in some natural filtration on R.  Those don’t seem to be the kind of estimates our methods provide; rather, they say something more like

(r-s)^2 <= B

where B is some fixed element of K_0(Var_K).  But what does “<=” mean?  Well, it means that B – (r-s)^2 is nonnegative.  And what does “nonnegative” mean?  That’s the question.  What the proof really gives is that B – (r-s)^2 lies in a certain semiring N of “nonnegative motives” in K_0(Var_K).  Let’s not be too precise about what N is; let’s just say that it includes [V] for every variety V, and it has the property that |n(F_q)| >=0 for all q, whenever n lies in N.  In particular, that means that

(|r(F_q)| – |s(F_q)|)^2 <= B(F_q)

so that, on the level of counting points, s(F_q) really is a good estimate for r(F_q).

So one might ask:  are there other interesting positive motivic measures — that is, homomorphisms

f: K_0(Var_K) -> reals

which take N to nonnegative reals?  If so, f(s) would be a good estimate for f(r).

And the point of this note with Larsen is to say, with some regret, no — any motivic measure which assigns nonnegative values to the classes of varieties is in fact just counting points over some finite field.  Which sort of kills in its crib the initial hope of some exciting world of “motivic inequalities.”

Of course, the reals are not the only ordered ring!  As Bjorn Poonen pointed out to me, for a general field K you can find an ordered ring A and a measure

g: K_0(Var_K) -> A

which is positive in the sense that g sends every variety to an element of A greater than or equal to 0; these come from big ultraproducts of counting measures of different finite fields.  Whether these measures are “interesting” I’m not sure.