Enriques Surfaces

An Enriques surface $X$ is a smooth compact complex surface having irregularity $q(X)=0$ and nontrivial canonical sheaf $K_X$ such that $K_X^2=O_X$ (Endraß). Such surfaces cannot be embedded in projective 3-space, but there nonetheless exist transformations onto singular surfaces in projective 3-space. There exists a family of such transformed surfaces of degree six which passes through each edge of a Tetrahedron twice. A subfamily with tetrahedral symmetry is given by the two-parameter ($r,c$) family of surfaces

$$f_rx_0x_1x_2x_3+c({x_0}^2{x_1}^2{x_2}^2+{x_0}^2{x_1}^2{x_3}^2+{x_0}^2{x_2}^2{x_3}^2+{x_1}^2{x_2}^2{x_3}^2=0$$

and the polynomial $f_r$ is a sphere with radius $r$,

$f_r=(3-r)({x_0}^2+{x_1}^2+{x_2}^2+{x_3}^2)$
$-2(1+r)(x_0x_1+x_0x_2+x_0x_3+x_1x_2+x_1x_3+x_2x_3)$

(Endraß).

References

Angermüller, G. and Barth, W. ``Elliptic Fibres on Enriques Surfaces.'' Compos. Math. 47, 317-332, 1982.

Barth, W. and Peters, C. ``Automorphisms of Enriques Surfaces.'' Invent. Math. 73, 383-411, 1983.

Barth, W. P.; Peters, C. A.; and van de Ven, A. A. Compact Complex Surfaces. New York: Springer-Verlag, 1984.

Barth, W. ``Lectures on K3- and Enriques Surfaces.'' In Algebraic Geometry, Sitges (Barcelona) 1983, Proceedings of a Conference Held in Sitges (Barcelona), Spain, October 5-12, 1983 (Ed. E. Casas-Alvero, G. E. Welters, and S. Xambó-Descamps). New York: Springer-Verlag, pp. 21-57, 1983.

Endraß, S. ``Enriques Surfaces.'' http://www.mathematik.uni-mainz.de/AlgebraischeGeometrie/docs/enriques.shtml.

Enriques, F. Le superficie algebriche. Bologna, Italy: Zanichelli, 1949.

Enriques, F. ``Sulla classificazione.'' Atti Accad. Naz. Lincei 5, 1914.

Hunt, B. The Geometry of Some Special Arithmetic Quotients. New York: Springer-Verlag, p. 317, 1996.