Hilbert Series of Polarised Varieties [HB 95]

New Features:

• The K3 database now contains 24,099 objects (an increase from around 400 in previous versions). In a precise sense, this now represents all possible (infinitely many) Hilbert series that may be associated to a polarised K3 surface.

• The machinery used to generate the database has been lifted to the user level. This includes new datatypes for (the numerical data associated to) polarised singularities and polarised varieties.

• There is a small database of Fano 3-folds. A major point of this work has been to contribute to the (hopefully imminent) classification of Fano 3-folds. This small database is the first step of what may become the comprehensive classification of (Hilbert series of) Fano 3-folds.

• New Riemann-Roch formulas used to compute Hilbert series have been included. This allows one to make graded ring calculations for K3 surfaces, Fano 3-folds, Calabi-Yau 3-folds and curves polarised by a subcanonical divisor. It is possible for a user to generate many families of these varieties interactively--a typical application of these methods is to generate large numbers of examples.

• A facility for users to write and read their own graded ring databases has been added. This makes genuine binary data files together with various indexing methods, as is usual with Magma databases, and it is essential when one generates large numbers of examples.