Double covers and Calabi-Yau varieties

Abstract

Introduction. In the present paper we study examples of double coverings of the projective space P branched over an octic surface. A double covering of P branched over a smooth octic is a Calabi-Yau threefold. If the octic is singular then so is the double covering and we study its resolution of singularities. In this paper we restrict our considerations to the case of octics with only non-isolated singularities of a special type, namely looking locally like plane arrangements. Our research was inspired by a paper of Persson [5] where K3 surfaces arising as double covers of P branched over curves of degree six are studied. In this note we also adopt some methods introduced in [3] by Hunt in studying Fermat covers of P branched over plane arrangements. The main results of this note are Theorem 2.1 and Theorem 3.5 which can be formulated together as follows Theorem. Let S ⊂ P be an octic arrangement with no q-fold curve for q ≥ 4 and no p-fold point for p ≥ 6. Then the double covering of P branched along S has a non-singular model Y which is a Calabi-Yau threefold. Moreover if S contains no triple elliptic curves and l3 triple lines then the Euler characteristic e(Y ) of Y is given as follows