# Some Mathoverflow challenges

Mathoverflow has become a roaring success! The site is getting about 20-25 questions a day, and about 95% of them get answered. But one of the problems with being so successful is that a question rarely remains on the front page for more than a day. Here are some which I believe are interesting, challenging and solvable.

Why do flag manifolds, in the $P(V_{\rho})$ embedding, look like products of $\mathbb{P}^1$‘s.
$\mathcal{F}\ell_n$ and $(\mathbb{P}^1)^{\binom{n}{2}}$ have the same multi-graded Hilbert functions. Is there any reason for this?

Smooth proper schemes over $\mathbb{Z}$ with points everywhere locally,
Non-simply-connected smooth proper scheme over $\mathbb{Z}$?.
The condition of being smooth and proper over $\mathbb{Z}$ appears to be very restrictive. Can you construct examples with these properties?

Pencils with many completely decomposable fibers
Does there exist a pencil of hypersurfaces in $\mathbb{P}^4$, which is not a cone over a pencil in $\mathbb{P}^3$, but has three fibers which are unions of hyperplanes?

Are submersions of differentiable manifolds flat morphisms?
This is basically a commutative algebra question: Is $C^{\infty}(\mathbb{R}^{n+1})$ flat over $C^{\infty}(\mathbb{R}^{n})$? Several people made progress, but no one finished it off.

Five Front Battle
Two generals, with armies of equal sizes, must apportion their troops between five fronts. On each front, the army with more troops will win; the nation which wins three fronts will win the war. What is the optimal strategy?

## 3 thoughts on “Some Mathoverflow challenges”

1. GS says:

Maybe in future some great open problems may debut as MO questions.

2. Is MathOverflow based on StackExchange? If so, who is funding it, do you know? I REALLY like the idea, I’d just like it to keep going!

3. David Speyer says:

Yes, the underlying software for MathOverflow is StackExchance. It is funded by Ravi Vakil. See the MO FAQ for slightly more detail.