Reducible vector bundles on a quadric surface

Extract

0. Introduction. In a paper (1) with the same title, one of us attempted to classify the reducible k²-bundles on a quadric surface Q = P₁ × P₁ defined over an algebraically closed field k. This attempt suffered from two defects: first, the classification was given by a one-one correspondence rather than by an algebraic parameter variety, and, secondly, there is a mistake in the proof of Proposition 6 of (1), as a result of which Proposition 6, Proposition 7 and the exceptional clause of the Theorem in (1) are false. To correct these defects we use the definitions, notations and numbering of (1) without comment. Finally, we show how the classification of reducible k²-bundles on P₁ × … × P₁ (n factors) is determined by that of reducible k²-bundles on P₁ × P₁. The net effect of these corrections is to underline the moral of (1): while the classification of reducible k²-bundles (but with a distinguished factor of the product variety P₁ × … × P₁) is as nice as it could possibly be, the classification of reducible k²-bundles alone is as nasty as it could possibly be. Since every decomposable k²-bundle is just a sum of two line bundles, we consider only indecomposable reducible k²-bundles.