MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

6 A link to a one page proof is added

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

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 much 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" (but I don't think that the answer to question one is contained there) : http://www.dpmms.cam.ac.uk/~bt219/algebraic.pdf

5 added 505 characters in body

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 mention 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 much more examples of Coxeter polytopes; the faces of such polytopes intersect under angles $(\frac{\pi}{2}, \frac{\pi}{3}, 0)$.

Here is a reference on "Algebraic surfaces and hyperbolic geometry" (but I don't think that the answer to question one is contained there) : http://www.dpmms.cam.ac.uk/~bt219/algebraic.pdf

4 added 338 characters in body; edited tags

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 mention that 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?

3 added 409 characters in body
2 added 1 characters in body
1