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61
votes
6answers
6k views

Is there an underlying explanation for the magical powers of the Schwarzian derivative?

Given a function $f(z)$ on the complex plane, define the Schwarzian derivative $S(f)$ to be the function $S(f) = \frac{f'''}{f'} - \frac{3}{2} (\frac{f''}{f'})^2$ Here is a somewhat more conceptual ...
14
votes
2answers
955 views

Why do all incidence theorems follow from Pappus' theorem?

In Hilbert and Cohn-Vossen's ``Geometry and the Imagination," they state in the last paragraph of Chapter 20 that "Any theorems concerned solely with incidence relations in the [Euclidean ...
5
votes
1answer
551 views

Top self-intersection of exceptional divisors

Let $Y\subset\mathbb{P}^n$ be a smooth variety of codimension two. Consider the blow-up $X = Bl_Y\mathbb{P}^n$ of $\mathbb{P}^n$ along $Y$, and let $E$ be the exceptional divisor over $Y$. Then $E$ ...
7
votes
2answers
500 views

Can projective hypersurfaces contain linear spaces? How big?

I am in this, rather friendly, situation: I have a complex projective space $\mathbb{P}^n$, and there i have a (possibly non-smooth) hypersurface $S$ defined by one irreducible polynomial $P$ of ...
2
votes
0answers
256 views

Enumerating certain types of permutation polynomials

Given a prime power $q$, I would like to enumerate (preferably up to isomorphism*) all the permutation polynomials $f(x)$ on $K = GF(q^3)$ satisfying the following conditions: $f(ax) = af(x)$ for ...
1
vote
2answers
252 views

Möbius transformation by 3 points in the Minkowski model

Goal I'm interested in describing Möbius transformations in the plane, and I'd like to define them in terms of three points and their images. What I have tried I know that a projective ...
3
votes
2answers
213 views

Proper subgroups of $\rm{SU}(d)$ that act transitively on $\rm{CP}^{d-1}$?

The special unitary group $\rm{SU}(d)$ has a canonical action on the Hilbert space of dimension $d$, and this action induces a canonical action on the projective space $\rm{CP}^{d-1}$, which is ...
1
vote
0answers
155 views

What is the Birkhoff norm of a Perron vector?

Let $A$ be a positive matrix. What is known about the Birkhoff norm of its Perron vector? By the Birkhoff norm of a vector $x$ I refer to the quantity $\frac{\max{x}}{\min{x}}$. P.S. This is ...