Paul Hacking,
Working in weighted projective spaces
Abstract: Given a variety X and an ample divisor D, we have X=Proj R(X,D)
where R(X,D) is the graded ring with nth graded piece H^0(nD). By picking
generators of the ring
R(X,D) we obtain an embedding of X in a weighted projective space . (Note
that we do not assume R(X,D) is generated in degree 1). It is often much
easier to study X using
such an embedding instead of via an embedding in a regular projective
space using a very ample divisor (e.g. a multiple of D), the point being
that the former embedding has much
lower codimension. For example, there are 95 families of K3 surfaces
which can be realised as a hypersurface in weighted projective space (and
of course only one family in P^3,
namely the quartics). I will define weighted projective spaces, review
some of their basic properties, and give some examples of the
construction above.