Coble rational surfaces.

*(English)*Zbl 1056.14054From the text: A Coble surface is a smooth rational projective surface such that its anti-canonical linear system is empty while the anti-bicanonical linear system is nonempty. In this paper we shall classify Coble surfaces and consider the finiteness problem of the number of negative rational curves on it modulo automorphisms.

First of all we divide Coble surfaces into two major classes. For a surface of the first class (elliptic type) there exists a birational morphism onto a surface \(Y\) such that the anticanonical linear system \(| -K_Y|\) has only one member, and a general member of the mobile part of \(| -2K_Y|\) is a smooth elliptic curve. Surfaces of the second class (rational type) admit a similar birational morphism only this time the mobile part of \(| -2K_Y|\) consists of divisors of arithmetic genus \(0\). We show that Coble surfaces of elliptic type are obtained as either blow-up of singular points and their infinitely near points of a nonmultiple fiber on a minimal rational elliptic surface with one multiple fiber of multiplicity 2 (Halphen type), or as blow-downs of some disjoint sections and maybe components of one fiber of a nonminimal rational elliptic surface with a section (Jacobian type). We also give a construction for surfaces of rational types as blow-ups of minimal rational surfaces. It turns out that surfaces of elliptic type always admit a birational morphism to \(\mathbb{P}^2\). However, for any given \(n\) there are Coble surfaces of rational type which do not admit a birational morphism to a minimal ruled surface. We prove that Coble surfaces of elliptic type are obtained by blowing up \(\mathbb{P}^2\) with centers at singular points of certain plane curves \(\Gamma\) of degree 6 (Coble sextics), but the center of the very last blow-up may not be on \(\Gamma\). We describe such sextics.

An important class of Coble surfaces \(X\) to which the original example of Coble belongs is the one where the linear system \(| -2K_X|\) contains a reduced divisor. In this case \(X\) admits a double cover which is a \(K3\)-surface with at most ordinary double points. We prove, under an appropriate condition of generality, that surfaces of this kind contain only finitely many smooth rational curves with negative intersection modulo automorphisms of the surface. We show an example that this statement cannot be extended to all Coble surfaces.

First of all we divide Coble surfaces into two major classes. For a surface of the first class (elliptic type) there exists a birational morphism onto a surface \(Y\) such that the anticanonical linear system \(| -K_Y|\) has only one member, and a general member of the mobile part of \(| -2K_Y|\) is a smooth elliptic curve. Surfaces of the second class (rational type) admit a similar birational morphism only this time the mobile part of \(| -2K_Y|\) consists of divisors of arithmetic genus \(0\). We show that Coble surfaces of elliptic type are obtained as either blow-up of singular points and their infinitely near points of a nonmultiple fiber on a minimal rational elliptic surface with one multiple fiber of multiplicity 2 (Halphen type), or as blow-downs of some disjoint sections and maybe components of one fiber of a nonminimal rational elliptic surface with a section (Jacobian type). We also give a construction for surfaces of rational types as blow-ups of minimal rational surfaces. It turns out that surfaces of elliptic type always admit a birational morphism to \(\mathbb{P}^2\). However, for any given \(n\) there are Coble surfaces of rational type which do not admit a birational morphism to a minimal ruled surface. We prove that Coble surfaces of elliptic type are obtained by blowing up \(\mathbb{P}^2\) with centers at singular points of certain plane curves \(\Gamma\) of degree 6 (Coble sextics), but the center of the very last blow-up may not be on \(\Gamma\). We describe such sextics.

An important class of Coble surfaces \(X\) to which the original example of Coble belongs is the one where the linear system \(| -2K_X|\) contains a reduced divisor. In this case \(X\) admits a double cover which is a \(K3\)-surface with at most ordinary double points. We prove, under an appropriate condition of generality, that surfaces of this kind contain only finitely many smooth rational curves with negative intersection modulo automorphisms of the surface. We show an example that this statement cannot be extended to all Coble surfaces.