Recently Goodman and
Hartshorne have considered the question of characterizing those divisors in a
complete linear equivalence class whose support has an affine complement. However
their characterization is not clearly “linear”, and in fact we have to resort to Serre’s
characterization of affine schemes to prove that, indeed, the condition “the support of
an effective divisor has an affine complement” is, in the language of Italian
geometry, expressed by linear conditions. In the language of Weil this means that
the set of effective divisors, in a complete linear equivalence class, whose
supports have affine complements is a linear system. This is our first result.
Subsequently we study the intersection of all such affine-complement supports of
effective divisors in the multiples of a given linear equivalence class, and prove
the following: if the ambient scheme is a surface or a threefold, or if the
characteristic of the groundfield is 0, (or assuming that we can resolve singularities!)
then a minimal intersection cannot have zero-dimensional components, nor
irreducible components of codimension 1, whose associated sheaf of ideals is
invertible.
