Vol. 31, No. 3, 1969

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Affine complements of divisors

Mario Borelli

Vol. 31 (1969), No. 3, 595–607

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.

Mathematical Subject Classification
Primary: 14.05
Received: 30 January 1969
Published: 1 December 1969
Mario Borelli