Low Codimension Fano–enriques Threefolds

Introduction In the 1970s, Reid introduced the graded rings method for the explicit classification of surfaces, which he used to produce a list of 95 K3 quasi-smooth hypersurfaces in weighted projective spaces (which were proved to be the only ones). Later, Fletcher used this method to create more lists of different weighted complete intersections. From the K3 surfaces he developed two lists of anticanonically polarised Fano threefolds that have the K3s as hyperplane section. These two lists for Fano threefolds of codimension one and two can be found in [F]. Later on, Altınok has developed in [A] a formula to calculate the Hilbert series of a Fano threefold (which is very important for the graded rings method) and has written a list of codimension three K3 surfaces (which produces a list of codimension three Fano threefolds). All lists are also in [GRDW]. In this paper we deal with Fano–Enriques threefolds (Fano threefolds with a torsion divisor σ). These are quotients of Fano threefolds under an action by a Z/(r) group, where r is the order of σ. We use the above lists of codimension 1, 2 and 3 to give in this paper all possible Fano–Enriques quotients that can be obtained from these lists. To find these quotients, we extend Reid’s graded rings method and we find restrictions for the covers. After that, we test all members of the lists and, for all those members that satisfy the restrictions, we calculate a Hilbet series to apply the extended graded rings method and search for a quotient. The distribution of this paper is as follows. In a first section, we give some preliminaries of Fano–Enriques threefolds. We describe Altınok’s method to compute the Hibert series for the anticanonical ring of a Fano threefold and the graded rings method in section 2. In section 3 we give an extension of these methods (Altınok’s and graded rings) to Fano–Enriques threefolds. Finally, in section 4, we see the complete way to obtain the lists of Fano–Enriques quotiens and give these lists. Most of this work in this paper was developed during a stay, supported by Maria Sklodowska-Curie graduate school Threefolds in algebraic geometry (3-fAG) reference number MCFH-1999 00687 grants, at