Enriques surfaces

Enriques surfaces
Pinch points
Other singularities
The parameter map
The movie
The hard work

Enriques surfaces

An Enriques surface

X is a smooth compact complex surface with irregularity

q(X)=0 and nontrivial canonical sheaf

K_X satisfying

K_X²= O_X. Such surfaces have been studied first by the italian mathematician Federigo Enriques (1871-1946) in 1914. Enriques surfaces cannot be embedded into projective threespace

P₃(C). However there exist birational morphisms onto singular surfaces in

P₃(C). Such a singular surface is called a singular model of an Enriques surface. There is a well known family of such singular models, which has been constructed by Enriques himself. It is the family of surfaces degree 6 going doubly through the edges of a tetrahedron. Here we present a subfamily with tetrahedral symmetry. It is the two parameter family of surfaces

X_r,c given by

X_r,c= {f_rx₀x₁x₂x₃+ c( x₀²x₁²x₂²+ x₀²x₁²x₃²+ x₀²x₂²x₃²+ x₁²x₂²x₃²)=0}

parameterized by the complex numbers r and c. The polynomial f_r defines a sphere with radius r

f_r= (3-r) (x₀²+ x₁²+ x₂²+ x₃²)- 2(1+r) (x₀x₁+x₀x₂+x₀x₃+ x₁x₂+x₁x₃+x₂x₃)

and the faces of the tetrahedron are just the planes defined by the linear forms x₀=x₁=x₂=x₃=0. We give an overview how the surface X_r,c degenerates for real values of r and c. As an example, the following pictures exhibit the real affine part of X_2,4. The second pictures also shows the edges of the tetrahedron.

Pinch points

For general real values of

r and

c the surface

X_r,c has two pairs of pinch points on each edge of the tetrahedron. However they are not always real. The four lines

L₁={c=2(r-1)}

L₂={c=2(1-r)}

L₃={c=4}

L₄={c=-4}

divide the real plane with coordinates c and r into 10 regions. The different shaded regions in the following picture are the regions where one pair of pinch points is real. For every point $($ r,c) on L₁ or L₂ one pair of pinch points of X_r,c coincides at the midpoints of the edges of the tetrahedron. For every point $($ r,c) on L₃ or L₄ one pair of pinchponts of X_r,c coincides at infinity.

[600x400 pixel pinchpoint map]

Other singularities

There are some other curves in the parameter space where degenerations occur. The equation of

X_r,c shows that the surface is reducible on the line

L₅={c=0}, it splits into a sphere of radius

r and four planes. On the line

L₆={r=3} the surface has fourfold points in the edges of the tetrahedron. On the line

L₇={r=4c} the surface has a isolated singularity in the origin. Along the cubic curve

C={ (7c-36)r²+ (2c²-18c+24)r- c³+ 6c²- 9c-4=0}

the surface has four double points on the four diagonals of the tetrahedron.

[600x400 pixel map of singularities]

The parameter map

The following map shows the (real part of) surfaces which occur if

r and

c are real numbers. Click on any small surface to get a larger image.

The movie

It is very interesting to watch how the surface

X_r,c deforms when the point

(

r,c) moves along the rational cubic curve

C. We present a computer animation which exhibits this deformation. It is a 53 MB fli-animation of 500 frames at a size of 500 x 500 pixels. It can be played using xanim on most unix machines. Download as gzipped file (17 MB).

The hard work

All calculations to find the above map have been performed by W. Barth. All images and the movie have been generated with the computer program surf by myself.

Stephan Endraß

Last update: Wednesday, 16-Apr-2003 10:40:30 CEST

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