Farida Shahata, Department of Pure Mathematics, University of Waterloo
"More projective geometry"
Projective schemes make a comeback! Following our previous discussion on closed immersions and closed subschemes, the next natural step is to study closed embeddings of projective schemes.
As for affine schemes, we will see that a surjection of graded rings induces a closed embedding of projective schemes. Conversely, every closed subscheme of a projective scheme arises from a graded ideal, hence, is projective itself.
Moreover, we will see that we can embed, quite naturally, a projective scheme into projective space as a closed subscheme.
A special case of such closed immersions into projective space is the Veronese embedding.
We start with the classical construction of the Veronese embedding of projective varieties, whose image is isomorphic to projective space. With the use of Veronese subrings, we extend this construction to projective schemes.
MC 5417