Symplectic geometry seminar

Monday, 2 March 2009, 2:10pm Bahen 6183

PINAKI MONDAL

University of Toronto

Projective completions of affine varieties via degree-like functions

Given a ‘degree like’ function δ, i.e. a map of the coordinate ring A of an affine variety X to integers such that δ(f + g) ≤ max{δ(f), δ(g)} (with strict inequality implying δ(f) = δ(g)) and δ(fg) ≤ δ(f) + δ(g), one can associate to it a projective completion of X^{δ} of X. We show that the ideal I_{∞} of the hypersurface of ‘points at infinity’ is radical iff δ is the maximum of finitely many semidegrees (i.e. degree like functions δ′ such that δ′(fg) = δ′(f) + δ′(g) for all f, g), which are then in a 1-1 correspondence with the prime components of I_{∞}.

Given a projective completion Z of X determined by a degree like function δ, we define a ‘normalized’ degree like function δ̄ which is a maximum of finitely many semidegrees. When X is normal, the corresponding completion is isomorphic to the normalization of Z. The construction of δ̄ from δ generalizes the construction of the normal toric completion X_{P} of (C^{∗})^{n} determined by a convex integral polytope P from the toric variety determined by an arbitrary finite subset S of integral points in P whose convex hull is P. I will describe this construction and the properties of toric completions these are so far known to preserve.