Abstract

• keeps track of a “boundary ” in X, or • contains infinitesimal information about a family, of which X is a fiber and involves monoids ( → toric geometry, combina-torics, etc.). Example: start with a smooth non-proper variety Y/k, compactify it to Y ⊆ X, and say D = X \ Y is a NC (normal crossings) divisor. X is nicer to work with, but what we actually care about is Y. Idea: encode the stuff that we added (the divisor D) in the geometry. One way to do it: take M(U) = {f ∈ OX(U) | f is invertible outside of D}. This is a subsheaf of monoids of OX containing O×X, that somehow keeps track of the boundary D.