## Braid monodromy and the dual curve

Nick Salter gave a great seminar here about this paper; hmm, maybe I should blog about that paper, which is really interesting, but I wanted to make a smaller point here.  Let C be a smooth curve in P^2 of degree n. The lines in P^2 are parametrized by the dual P^2; let U be the open subscheme of the dual P^2 parametrizing those lines which are not tangent to C; in other words, U is the complement of the dual curve C*.  For each point u of U, write L_u for the corresponding line in P^2.

This gives you a fibration X -> U where the fiber over a point u in U is L_u – (L_u intersect C).  Since L_u isn’t tangent to C, this fiber is a line with n distinct points removed.  So the fibration gives you an (outer) action of pi_1(U) on the fundamental group of the fiber preserving the puncture classes; in other words, we have a homomorphism

$\pi_1(U) \rightarrow B_n$

where B_n is the n-strand braid group.

When you restrict to a line L* in U (i.e. a pencil of lines through a point in the original P^2) you get a map from a free group to B_n; this is the braid monodromy of the curve C, as defined by Moishezon.  But somehow it feels more canonical to consider the whole representation of pi_1(U).  Here’s one place I see it:  Proposition 2.4 of this survey by Libgober shows that if C is a rational nodal curve, then pi_1(U) maps isomorphically to B_n.  (OK, C isn’t smooth, so I’d have to be slightly more careful about what I mean by U.)

## Hain-Matsumoto, “Galois actions on fundamental groups of curves…”

I recently had occasion to spend some time with Richard Hain and Makoto Matsumoto’s 2005 paper “Galois actions on fundamental groups and the cycle C – C^-,” which I’d always meant to delve into.  It’s really beautiful!  I cannot say I’ve really delved — maybe something more like scratched — but I wanted to share some very interesting things I learned.

Serre proved long ago that the image of the l-adic Galois representation on an elliptic curve E/Q is open in GL_2(Z_l), so long as E doesn’t have CM.  This is a geometric condition on E, which is to say it only depends on the basechange of E to an algebraic closure of Q, or even to C.

What’s the analogue for higher genus curves X?  You might start by asking about the image of the Galois representation G_Q -> GSp_2g(Z_l) attached to the Tate module of the Jacobian of X.  This image lands in GSp_{2g}(Z_l).  Just as with elliptic curves, any extra endomorphisms of Jac(X) may force the image to be much smaller than GSp_{2g}(Z_l).  But the question of whether the image of rho must be open in GSp_2g(Z_l) whenever no “obvious” geometric obstruction forbids it is difficult, and still not completely understood.  (I believe it’s still unknown when g is a multiple of 4…?)  One thing we do know in general, though, is that when X is the generic curve of genus g (that is, the universal curve over the function field Q(M_g) of M_g) the resulting representation

$\rho^{univ}: G_{Q(M_g)} \rightarrow GSp_{2g}(\mathbf{Z}_\ell)$

is surjective.

Hain and Matsumoto generalize in a different direction.  When X is a curve of genus greater than 1 over a field K, the Galois group of K acts on more than just the Tate modules (or l-adic H_1) of X; it acts on the whole pro-l geometric fundamental group of X, which we denote pi.  So we get a morphism

$\rho_{X/K}: G_K \rightarrow Aut(\pi)$

What does it mean to ask this representation to have “big image”?

## Disks in a box, update

In April, I blogged about the space of small disks in a box.  One question I mentioned there was the following:  if C(n,r) is the space of configurations of n non-overlapping disks of radius r in a box of sidelength 1, what kind of upper bounds on r assure that C(n,r) is connected?  A recent preprint of Matthew Kahle gives some insight into this question: he produces configurations of disks which are stable (each disk is hemmed in by its neighbors) with r on order of 1/n.  (In particular, the density of such configurations goes to 0 as n goes to infinity.)  Note that Kahle’s configurations are not obviously isolated points in C(n,r); it could be, and Kahle suggests it is likely to be, that his configurations can be deformed by moving several disks at once.

Also appearing in Kahle’s paper is the stable 5-disk configuration at left; this one is in fact an isolated point in C(5,r).

More Kahle: another recent paper, with Babson and Hoffman, features the theorem that a random 2-complex on n vertices, where the edges are all present and each 2-face appears with probability p, transitions from non-simply-connected to simply connected when p crosses n^{-1/2}.  This is in sharp contrast with the H_1 of the complex with Z/ ell Z coefficients, which disappears almost surely once p exceeds 2 log n / n, by a result of Meshulam and Wallach.  So in some huge range, the fundamental group is almost surely a big group with no nontrivial abelian quotient!  (I guess this doesn’t formally follow from Meshulam-Wallach unless you have some reasonable uniformity in ell…)

One naturally wonders:  Let pi_1(n,p) be the fundamental group of a random 2-complex on n vertices with facial probability p.   If G is a finite simple group, what is the expected number of surjections from pi_1(n,p) to G?  Does it sharply transition from nonzero to zero?  Is there a range of p in which pi_1(n,p) is almost certainly an infinite group with no finite quotients?

## Anabelian puzzle 5: isogenies between Jacobians and metabelian fundamental groups

Another question that came up at the Newton Insitute:  can two different curves X,Y over F_q have the same geometrically metabelian pro-l fundamental group?

I would think not, and here’s why.  First of all, the actions of Frob_q on H_1(X,Z_ell) and on H_1(Y,Z_ell) agree.  This already implies that Jac(X) and Jac(Y) are isogenous.  Can this actually happen in large genus?  Yes:  a recent arXiv preprint by Ben Smith gives lots of explicit examples of pairs of hyperelliptic curves with isogenous Jacobians.  From Smith’s paper I learned about the recent construction by J. F. Mestre of pairs of hyperelliptic curves in every genus with isogenous Jacobians.

In other words, the geometrically abelian fundamental group need not distinguish X from Y.

But the fact that the geometrically metabelian pro-l fundamental groups agree implies the following much stronger fact.  Let X_n be the maximal abelian cover of X/F_qbar whose Galois group has exponent l^n, and define Y_n similarly.  Then X_n and Y_n have isogenous Jacobians for all n.  I would think this would be impossible if X and Y were not isomorphic; but I don’t have the slightest idea for a proof.

Baby version of this question:  do there exist non-isomorphic curves X and Y of large genus (say, for the moment, over C) whose Jacobians are isogenous, and such that each Prym of X is isogenous to a Prym of Y?

## In which I attend a conference on fundamental groups in arithmetic geometry from the comfort of my own home

I don’t watch videotaped lectures — in general I’ve found the difficulty of seeing the board and hearing the lecturer makes it impossible for me to maintain enough focus to engage with the mathematics and take good notes.  In fact, I think the only online video lecture I’ve ever viewed all the way through was one of my own, because I somehow lost the notes I’d used and needed to generate a new set so I could give the talk again.

But I was really sorry not to be able to make last week’s introductory workshop for the Newton Institute’s special semester on non-abelian fundamental groups in arithmetic geometry — so sorry that I decided to try watching the recorded lectures on my laptop.  And they’re great!  Crisp sound and visuals, appropriately timed close-ups on the board, and even a camera pointed at the audience so you can see the people asking questions.  And you can download the talks to your iPod!   Three cheers for the A/V team at the Newton Institute.

As of tonight just the Monday and Tuesday talks are up, which is already plenty to keep me busy.  I just watched Deligne talk about counting l-adic local systems on curves over finite fields; highly recommended.

When I was first giving public lectures, someone gave me the hoary advice that I should quell nervousness by imagining the members of the audience in their underwear.  Strange to think that, in this new broadband world, most of them actually are.

One-second precis of Deligne’s talk:  starting with Drinfel’d in the early 80s, you can count the number of l-adic local systems on a curve over F_q by applying whatever version of the Langlands correspondence you have available and then using an appropriate trace formula to count automorphic forms.  It turns out that the number of rank-d l-adic local systems “defined over F_{q^n}” seems to behave as if it were governed by a Lefschetz fixed point formula, i.e. as if it were the number of F_{q^n}-rational points on some variety.  But what variety?  Not the moduli space of rank-d vector bundles with connection on the curve; that has dimension twice as large as the dimension of the purported variety suggested by the result of the counting problem.  But one still may hope — bolstered to some extent by recent work of Arinkin and Flicker — that the point count is reasonably legible and has something to do with the hyperkahler geometry of that moduli space.  I don’t think that summary made tons of sense — so watch the video!