On Higher Rank Globally Generated Vector Bundles over a Smooth Quadric Threefold

We give a complete classification of globally generated vector bundles of rank 3 on a smooth quadric threefold withc1≤ 2 and extend the result to arbitrary higher rank case. We also investigate the existence of globally generated indecomposable vector bundles, and give the sufficient and necessary conditions on numeric data of vector bundles for indecomposability.

The configuration determined by five generators of a quadric threefold

Through four generally placed lines in space of four dimensions there passes a doubly infinite system of quadric primals, but through five lines there pass in general no quadrics. It therefore follows that there must exist some special relationship between five lines in order that they may be generators of a quadric. This problem has been discussed by Richmond,1 who gives a condition which is in a restricted sense an extension of Pascal's Theorem. The five lines being taken in order certain points may be obtained which lie in a space. In Section I we state Richmond's criterion and show that it is sufficient as well as necessary. Section II is concerned with the twelve spaces which arise if all the different possible orders of the lines are considered. They cut by pairs in six planes whose configuration is developed. In Section III other lines connected with the configuration are introduced. It is shown that by taking crossers of the lines of our original figure in a certain manner five further generators are obtained, and that the same entire configuration of generators arises whether we begin with the five original or the five final lines. Furthermore, though the twelve spaces analogous to Pascal lines obtained from the final five are new, yet the six planes, their intersections by pairs, and the configuration dependent from them, are the same as those constructed from the original five.

On Five Lines, in Space of Four Dimensions, which lie upon a, Quadric Threefold and the Normal Quartic Curves of which they are Chords

The connexion between the conditions for five lines of S4(i) to lie upon a quadric threefold,and (ii) to be chords of a normal quartic curve,leads to an apparent contradiction. This difficulty is explained in the first paragraph below and, subsequently, two investigations are given of which the first uses, mainly, properties of space of three dimensions.