Introduction to the theory of weighted projective spaces (Q1091442)

The content of this well-written expository paper was the subject of talks given by the second author in West Germany in 1984-85. After a short historical introduction the following subjects are discussed: (1) quotient varieties under actions by finite groups, in particular the case of a finite group acting on a variety in such a way that the translations are morphisms. - (2) Graded rings and quasicones. - (3) Weighted projective spaces: \({\mathbb{P}}(Q)\) is the scheme Proj(S(Q)), where S(Q) is the polynomial ring \(k[T_ 0,...,T_ r]\) graded by \(\deg (T_ i)=q_ i\), \(Q=(q_ 0,...,q_ r)\) being a set of positive integers. - (4) Structure and homological properties of the relevant projecting quasicones. - (5) Examples.- (6) The sheaves \({\mathcal O}^{(n)}_{{\mathbb{P}}(Q)}\), sections, ampleness criteria. - (7) The regular locus of Mori. - (8) Dualizing sheaves, differentials. - (9) Picard groups and weighted complete intersections. - Finally: An appendix on reflexive modules and Weil divisors and a well-documented bibliography.