Graded rings and varieties in weighted projective space, 2002. unpublished chapter from an upcoming book on surfaces, available at http://www.warwick.ac.uk/∼masda/surf/more/grad.pdf

This chapter is a first introduction to weighted projective spaces (wps) P = P(a0,..., an) and the Proj correspondence projective variety X ⊂ P ←→ graded ring R = k[x0,..., xn]/I (1) The correspondence (1) between geometry and algebra is a minor but very fruitful generalisation of the usual idea of varieties in straight projective space Pn = P(1,..., 1). The simple device of working with varieties contained in the ready-made ambient spaces P = P(a0,..., an) allows us in many cases to by-pass the definition of abstract variety (or more general schemes) at the cost of a bit of messing around with weighted homogeneous polynomials, so that projective varieties in wp