How many points does a random curve over F_q have?

So asks a charming preprint by Achter, Erman, Kedlaya, Wood, and Zureick-Brown.  (2/5 Wisconsin, 1/5 ex-Wisconsin!)  The paper, I’m happy to say, is a result of discussions at an AIM workshop on arithmetic statistics I organized with Alina Bucur and Chantal David earlier this year.

Here’s how they think of this.  By a random curve we might mean a curve drawn uniformly from M_g(F_q).  Let X be the number of points on a random curve.  Then the average number of points on a random curve also has a geometric interpretation: it is

|M_{g,1}(\mathbf{F}_q)|/|M_{g}(\mathbf{F}_q)|

What about

|M_{g,2}(\mathbf{F}_q)|/|M_{g}(\mathbf{F}_q)|?

That’s just the average number of ordered pairs of distinct points on a random curve; the expected value of X(X-1).

If we can compute all these expected values, we have all the moments of X, which should give us a good idea as to its distribution.  Now if life were as easy as possible, the moduli spaces of curves would have no cohomology past degree 0, and by Grothendieck-Lefschetz, the number of points on M_{g,n} would be q^{3g-3+n}.  In that case, we’d have that the expected value of X(X-1)…(X-n) was q^n.  Hey, I know what distribution that is!  It’s Poisson with mean q.

Now M_g does have cohomology past degree 0.  The good news is, thanks to the Madsen-Weiss theorem (née the Mumford conjecture) we know what that cohomology is, at least stably.  Yes, there are a lot of unstable classes, too, but the authors propose that heuristically these shouldn’t contribute anything.  (The point is that the contribution from the unstable range should look like traces of gigantic random unitary matrices, which, I learn from this paper, are bounded with probability 1 — I didn’t know this, actually!)  And you can even make this heuristic into a fact, if you want, by letting q grow pretty quickly relative to g.

So something quite nice happens:  if you apply Grothendieck-Lefschetz (actually, you’d better throw in Kai Behrend’s name, too, because M_g is a Deligne-Mumford stack, not an honest scheme) you find that the moments of X still agree with those of a Poisson distribution!  But the contribution of the tautological cohomology shifts the mean from q to q+1+1/(q-1).

This is cool in many directions!

  • It satisfies one’s feeling that a “random set,” if it carries no extra structure, should have cardinality obeying a Poisson distribution — the “uniform distribution” on the groupoid of sets.  (Though actually that uniform distribution is Poisson(1); I wonder what tweak is necessary to be able to tune the mean?)
  • I once blogged about an interesting result of Bucur and Kedlaya which showed that a random smooth complete intersection curve in P^3 of fixed degree had slightly fewer than q+1 points; in fact, about q+1 – 1/q + o(q^2).  Here the deviation is negative, rather than positive, as the new paper suggests is the case for general curves; what’s going on?
  • I have blogged about the question of average number of points on a random curve before.  I’d be very interested to know whether the new heuristic agrees with the answer to the question proposed at the end of that post; if g is a large random matrix in GSp(Z_ell) with algebraic eigenvalues, and which multiplies the symplectic form by q, and you condition on Tr(g^k) > (-q^k-1) so that the “curve” has nonnegatively many points over each extension of F_q, does this give something like the distribution the five authors predict for Tr(g)?  (Note:  I don’t think this question is exactly well-formed as stated.)

 

Tagged , , , , , , , , , ,

What’s so great about JetBlue?

JetBlue is changing its practices to run more like other airlines, and people are going nuts.

While all other airlines save Southwest switched to charging for checked bags, JetBlue made a free bag part of its unique selling proposition

What does United charge for checked bags?  $25, I think.  According to Google Flights, and a NYC-Chicago round trip costs $303 on JetBlue, $250 on United.  So you get your choice — pay your fifty bucks in extra airfare, or pay it in bag fees.  I’m not sure why the former is more customer-friendly.  It’s certainly not more friendly to me, since I rarely check.

Then there’s the seat issue:

Starting in 2016, JetBlue will stuff 15 more chairs on its Airbus A320s, bringing the seat count per aircraft to 165 from 150. That means JetBlue will fly its A320 at a higher seating density than many major competitors, including United Airlines (138-150 seats), US Airways (150) and Virgin America (146-149).

In other words:  customer-friendly JetBlue is now operating at the same density as USAir and a higher density than United and Virgin Atlantic!

I sympathize with JetBlue here.  People seem to want to pay a low base fare and then pay for things a la carte.  Food in the airport is much better than it used to be and people would rather pay 10 bucks for a sandwich (or bring food from home) than buy a more expensive ticket and eat airline food.  People would rather watch their own stuff on a tablet than buy a more expensive ticket and watch the airline’s stuff on a seatback.  And etc. and etc.  I think the one thing people do want is more room to sit.  But if you want to pay $50 extra for that, you can do it on the big three by buying a premium coach seat at checkin.

OK, to be fair, this actually costs more like $50 each way.  On the other hand, United Economy Plus gives you 37 inches of seat pitch; a standard JetBlue seat is 34, and a standard United seat 33.

 

Tagged , ,

Loudly and bravely

Wallace Shawn:

As I write these words, in New York City in 1985, more and more people who grew up around me are making this decision; they are throwing away their moral chains and learning to enjoy their true situation:  Yes, they are admitting loudly and bravely, We live in beautiful homes, we’re surrounded by beautiful gardens, our children are playing with wonderful toys, and our kitchen shelves are filled with wonderful food.  And if there are people out there who are envious of us and who might even be tempted to break into our homes and take what we have, well then, part of our good fortune is that we can afford to pay guards to protect us.  And if those who protect us need to hit people in the face with the butts of their rifles, or if they need perhaps even to turn around and shoot, they have our permission, and we only hope they’ll do what they do with diligence and skill.

The amazing thing I’ve noticed about these friends of mine who’ve made that choice is that as soon as they’ve made it, they begin to blossom, to flower, because they are no longer hiding, from themselves or anyone else, the true facts about their own lives.

Tagged

So you think you can shop

CJ and I, on the plane back from Thanksgiving today, had a good idea for a reality show:  somebody has to buy every item in the Skymall catalogue and then make use of all of them in one day’s time.

 

Tagged ,

Drowned out

Conservative blogger Glenn Reynolds is mad mad mad mad mad about .. well, I’ll let him tell it:

After years of effort, the European Space Agency’s lander Philae landed on a comet 300 million miles away. At first, people were excited. Then some women noticed that one of the space scientists, Matt Taylor, was wearing a shirt, made for him by a female “close pal,” featuring comic-book depictions of semi-naked women.  And suddenly, the triumph of the comet landing was drowned out by shouts of feminist outrage about … what people were wearing.

Drowned out!

Let’s sit with that a minute.  I just searched for “Philae” on Twitter and you know how many tweets I had to scroll through before I found one that mentioned Matt Taylor and his shirt?  32.  That sounds about right — I’d say 3% of the coverage I saw of the comet landing had to do with Matt Taylor’s shirt, and 97% had to do with the fact that we awesomely landed a robot on a comet.

But for Reynolds, the 3% drowns out the 97%.  3% is too much.  1% is too much!  Any little speck of feminist content is like the pea under the mattress for these guys. They can’t rest because the 3% is digging into them, it keeps them up all night, the feminism is still there, I can feel it, make it stop make it stop!

Tagged , ,

Let us now praise Delta Airlines

I flew home from Montreal yesterday via Minneapolis.  MSP was still kind of messy, recovering from a snowstorm, and my Montreal-Minneapolis leg was delayed.  Delta told me I wasn’t going to make the last flight back to Madison, gave me a hotel voucher for Minneapolis, and rebooked me for the first flight the next morning.  But when we landed in Minneapolis, there were still 5 minutes left until the Madison flight was departing.  The gate agent got a guy in a motorized cart to take me all the way from the end of concourse C to the beginning of concourse F.  Those things can go pretty fast when there’s nobody in the airport!  Even so, we got there two minutes after departure time and the gate was shut.  But I could see the plane still parked at the end of the jetway.  So the agent opened the gate back up, took me down, and got the pilot to re-open the main boarding door so I could get on the plane.  The whole thing was awesome, I was extremely happy to sleep at home, and I’m feeling very warm towards the people at Delta for making it happen.

Tagged ,

Wisconsin is not a blue state

Another Wisconsin election day!  By the polls — and I trust the polls, absent any reason not to — incumbent governor Scott Walker is likely to squeeze by with a narrow win.  If you don’t live in Wisconsin, how much should you care about this?  A lot, says Slate’s Betsy Woodruff, who calls this race “The Most Important Race in America.”

Winning statewide as a conservative Republican in Wisconsin isn’t easy. Even though five of its eight congressmen are Republicans and the GOP controls its statehouse, Wisconsin is a very blue state. It’s historically been a union stronghold, and it hasn’t gone Republican in a presidential race since 1984. For progressives, the Republicans’ fragile hold on state government is an insult, an affront that should be corrected.

Wisconsin is not a very blue state.  In those 30 years since 1984, a Republican has been governor for 19 of them.  In both 2000 and 2004, the Democratic candidate won Wisconsin’s electoral vote by less than half a percentage point.  In 2012, Obama won Wisconsin by 7 points, in a year he won nationally by 4 points.  So Wisconsin, in Obama’s home turf of the Upper Midwest, was slightly bluer than the country that year.

But it’s not California or Maryland.  It’s not even New Jersey.  It’s a state that’s half Republican and half Democratic.  (See also:  “It’s a recall, not an omen.”)  That’s why elections here are close.  Despite what Woodruff writes, neither liberals nor conservatives think they have a right to own the state.  Walker has the advantage of incumbency and he’s probably going to win.  That’s important for his dreams of a Presidential run; but I don’t think it has much to say about national politics.

Tagged , ,

Bumgarner 2014, Ortiz 2013, and the World Series OVP

I was wondering about the question of whether Madison Bumgarner was not only the MVP of the 2014 World Series, but the MVP of all recent World Series.  I mean, here’s something startling:  Bumgarner’s ERA for the series was 0.43, over 21 innings.  The rest of the Giants staff recorded a 5.71 ERA in their 40 innings of work.  Bumgarner wasn’t just the most valuable player — he was, on the pitching side, the only valuable player.

I asked Daniel Erman how far back you had to go to find a comparable performance, and he pointed out that the answer is “One year.”  David Ortiz had an insane 1.948 OPS for the series.  The next-highest mark on the Red Sox?  Jacoby Ellsbury, at 0.599.  That is amazing.  At least the Giants had Affeldt, who was effective in four games of middle relief.  Boston really had no hitter other than Ortiz who was any good at all.  The Red Sox had 62 baserunners in the series.  19 of those were David Ortiz.

Are there any other World Series OVPs?

2014 World Series stats

2013 World Series stats

Tagged , , , , ,

Why I’m voting no on the Wisconsin transportation referendum

All attention is focused on Mary Burke and Scott Walker, so I didn’t even realize there’s a state ballot proposition in next week’s election.  And it’s not a trivial one, either.

Question 1: “Creation of a Transportation Fund. Shall section 9 (2) of article IV and section 11 of article VIII of the constitution be created to require that revenues generated by use of the state transportation system be deposited into a transportation fund administered by a department of transportation for the exclusive purpose of funding Wisconsin’s transportation systems and to prohibit any transfers or lapses from this fund?”

Mary Burke supports this.  So does Governor Walker.  The bill to put the referendum on the ballot was passed by large majorities of both houses.  “Yes on 1″ has an organized campaign and a snappy website; as far as I can tell, there is no such thing as “No on 1.”

But I’m voting no.  I don’t expect every dime of people’s property taxes to support upkeep of residential infrastructure.  I don’t think the sales tax should be restricted to promoting Wisconsin retail.  I think money is money and it’s the job of the legislature, not the constitution, to decide how money can best be raised and where in the state it’s most needed.

The amendment prevents gas taxes and vehicle registration fees from being used to fund schools and hospitals and police, but it doesn’t prevent other revenue sources from being raided to fund our highways and bridges.  And that’s what’s actually happening right now; the current administration takes $133 million from the general fund to fund transportation in the current budget.  I’m not sure why transportation, out of all state projects, ought to enjoy a special status:  allowed to draw money from the general fund, but constitutionally prohibited from releasing any back.

The Yes on 1 FAQ points out that many states around the country have constitutional language enforcing segregation of the the transportation fund.  I looked at a few of these, and it’s true!  But those provisions are of a rather different nature.  California’s constitutional provision requires that 25% of the money go to public transportation.  In Minnesota, it’s 40%.  Our referendum has no such restriction, requiring only that the money go to things funded by the DoT.  The Yes on 1 FAQ points out, correctly, that “Wisconsin’s segregated transportation fund is the sole source of state funding for the entire transportation system – highways, air, rail, transit, harbors, bicycle and pedestrian facilities.”  Pretty weak sauce — the fund will not be prohibited from funding other forms of transportation.  Unless an enterprising governor splits off transit into a separate department, that is.  (Ohio’s Constitution, by the way, already forbids gas taxes and license fees from aiding mass transit.)

The amendment establishes one class of spending and taxing as privileged above all the rest.  It shouldn’t be part of our state constitution.

Links:

Tagged , , , ,

1. Produce defective cars 2. ? 3. Double profit!

“Profit Doubles at G.M., as It Strives to Move Past Its Litany of Recalls”:

General Motors’ quarterly earnings report on Thursday was noteworthy mostly for what it lacked: another big financial charge for safety recalls.

After running up special charges of nearly $3 billion in the first half of the year for safety problems, G.M., the nation’s biggest automaker, avoided additional charges for recalls in the third quarter.

While G.M. did incur $700 million in costs for fixing recalled vehicles during the quarter, the company had already booked those charges in previous periods….

By accounting for the bulk of its recall costs in the first half of the year, G.M. has turned a corner — at least financially — in its struggle to move beyond the worst safety crisis in its history.

So let me make sure I understand this:  GM is still blowing trainloads of cash fixing its mistakes, but they decided to declare that the money they’re spending now was actually spent earlier in the year, so that their official profit in the first half is below the real figure, and their official profit for the third quarter is above the real figure, and then they get a sunny headline in the New York Times saying they “doubled their profit?”

My grandfather the CPA would not approve.

 

 

Tagged , , ,
Follow

Get every new post delivered to your Inbox.

Join 588 other followers

%d bloggers like this: