Hyperbolic Coxeter polytopes and Del-Pezzo surfaces

Question

Added. In the following link there is a proof of the observation made in this question: http://dl.dropbox.com/u/5546138/DelpezzoCoxeter.pdf

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I would like to find a reference for a beautiful construction that associates to Del-Pezzo surfaces hyperbolic Coxeter polytopes of finite volume and ask some related questions.

Recall that a hyperbolic Coxeter polytope is a domain in $\mathbb H^n$ bounded by a collection of geodesic hyperplanes, such that each intersecting couple of hyperplanes intersect under angle $\frac{\pi}{n}$ ($n=2,3,...,+\infty$). Del Pezzo surface is a projective surface obtained from $\mathbb CP^2$ by blowing up (generically) at most $8$ points.

Now, the construction Del Pezzo $\to$ Coxeter polytope goes as follows.

Consider $H_2(X,\mathbb R)$, this is a space endowed with quadratic form of index $(1,n)$ (the intersection form), and there is a finite collection of vectors $v_i$ corresponding complex lines on $X$ with self-intersection $-1$. It is well known, for example that on a cubic surface in $\mathbb CP^3$ there are $27$ lines and this collection of lines has $E_6$ symmetry (if you consider it as a subset in $H_2(X,\mathbb Z)$). Now we just take the nef cone of $X$, or in simple terms the cone in $H_2(X,\mathbb R)$ of vectors that pair non-negatively with all vectors $v_i$. This cone cuts a polytope from the hyperbolic space corresponding to $H_2(X,\mathbb R)$, and it is easy to check that this polytope is Coxeter, with angles $\frac{\pi}{2}$ and $0$ (some points of this polytope are at infinity, but its volume is finite). Indeed, angles are $\frac{\pi}{2}$ and $0$ since $v_i^2=-1$, $v_i\cdot v_j=0 \;\mathrm{or}\; 1$.

Example. If we blow up $\mathbb CP^2$ in two points this construction produces a hyperbolic triangle with one angle $\frac{\pi}{2}$ and two angles $0$.

The connection between algebraic surfaces an hyperbolic geometry is very well-known, and exploited all the time but for some reason I was not able to find the reference to this undoubtedly classical fact (after some amount of googling). So,

Question 1. Is there a (nice) reference for this classical fact?

This question is motivated in particular by the following article http://maths.york.ac.uk/www/sites/default/files/Preprint_No2_10_0.pdf where the polytope corresponding to the cubic surface is used. The authors mention the relation of the polytope to 27 lines on the cubic, but don't say that the relation is in fact almost canonical.

Question 2. The group of symplectomorhpisms (diffeos) of each Del-Pezzo surface $X$ is acting on $H_2(X,\mathbb R)$, let us denote by $\Gamma$ its image in the isometries of corresponding hyperbolic space. What is the relation between $\Gamma$ and the group generated by reflections in the faces of the corresponding Coxeter polytope?

PS It one considers rational surfaces with semi-ample anti-canonical bundles, i.e. surfaces that can have only rational curves with self-intersection $-1$ and $-2$ one gets more examples of Coxeter polytopes; the faces of such polytopes intersect under angles $(\frac{\pi}{2}, \frac{\pi}{3}, \frac{\pi}{4}, 0)$.

Here is a reference on "Algebraic surfaces and hyperbolic geometry" by Burt Totaro (but I don't think that the answer to question one is contained there).

Answer 1

Perhaps the nice article "Reflection groups in Algebraic Geometry" of Dolgachev describes this: link http://www.math.lsa.umich.edu/~idolga/reflections.pdf

Other places to look: papers of Vinberg and Nikulin, and Manin's book on "Cubic forms".

Answer 2

I tried to investigate the question for some time and sent emails to several experts in the field. For the moment the conclusion seems to be the following:

The statement is well known to experts, was used many times, but was never written as a lemma, propostition, or a theorem in any article, or book.

The only relatively unambiguous reference that I managed to find are lines 14-0 from the bottom of page 4 in the article of Viacheslav V. Nikulin "On algebraic varieties with finite polyhedral Mori cone" that you can find here http://arxiv.org/abs/math/0305040

It might be, that looking better one would be able to find an actual reference, and I would beg anyone who knows such a reference to give the answer to the question...

Meanwhile I would like to formulate a beautiful (and apparently strongest) known result in this direction, that is omnipresent in the book of Alexeev and Nikulin (I would like to thank both authors for pointing out this to me), but again never stated explicitly.The name of the book is Classification of log del Pezzo surfaces of index $\le 2$, you can find it here: http://arxiv.org/abs/math/0406536

Theorem, Alexeev, Nikulin. Let $S$ be a Log del Pezzo surface of index 2. Then $S$ admits a particular (possibly non-minimal) resolution of singularities so that the set of unit vectors in the Nef cone of the resolution is a hyperbolic Coxeter polytope of finite volume.

Here is an explanation of the terminology. log-del Pezzo surface is a (possibly singular) rational surface with ample anti-canonical bundle. The index of such surface is the minimal number $n$ such that $nK$ is a Cartier divisor. Log del Pezzo surface of index two have only quotient singularities, isomorphic to $\mathbb C^2/\Gamma$ where $\Gamma$ is a finite group with elements with determinants $\pm 1$.

The proof of the theorem is a combination of the Cone theorem (this gives the finiteness of the volume) and the following simple

Observation. Suppose we have vectors $v_i$ in $\mathbb R^{1,n}$ such that $v_i\cdot v_j=0,1$, $v_i\cdot v_i=-1,-2,-4$, and $v_i \cdot v_j=0$ if $v_i\cdot v_i=-2$, $v_j\cdot v_j=-4$. Then the angles between hyperplanes orthogonal to vectors $v_i$ can be only $\pi/2$, $\pi/3$, $\pi/4$, or $0$.

Finally one just needs to find an appropriate resolution of singularities of $S$.

Answer 3

For $Q2$, in my paper joint with T.-J. Li http://arxiv.org/abs/1012.4146 there's a description by reflections of $\Gamma$, and Shevchishin proved the same conclusion in http://arxiv.org/abs/0904.0283 but his language is more Coxeter. We actually dealt with all rational surfaces.