Kulikov surfaces form a connected component of the moduli space

Abstract

We show that the Kulikov surfaces form a connected component of the moduli space of surfaces of general type with p g = 0 and K 2 = 6. We also give a new description for these surfaces, extending ideas of Inoue. Finally we calculate the bicanonical degree of Kulikov surfaces, and prove that they verify the Bloch conjecture.

Moduli of Weighted Hyperplane Arrangements

“…These surfaces are smooth, have ample K X , K 2 X = 6, and p g = q = 1. The moduli space together with its compactification was considered in [16], from a different point of view. We give it here to illustrate our methods.…”

Enumerating exceptional collections of line bundles on some surfaces of general type

“…For details on the Kulikov surface (first described in [34]), its torsion group and moduli space, see [19]. The Kulikov surface X is a (Z/3) 2 -cover of the del Pezzo surface Y of degree 6.…”

“…The exceptional curves are denoted E i . By results of [19], the torsion group Tors X is isomorphic to (Z/3) 3 , so the maximal abelian cover ψ : A → Y has group G ∼ = (Z/3) 5 . Let g i generate G, and write g * i for the dual generators of G * .…”

“…Hence HH k (A) = HH k (X) for k ≤ 2, and HH 3 (A) ⊃ HH 3 (X). By the Hochschild-Kostant-Rosenberg isomorphism, the dimensions of HH k (X) follow from the infinitesimal deformation theory of the Kulikov surface, which was studied in [19]: H 1 (T X ) = 1 and H 2 (T X ) = 3.…”