On congruences of lines in the projective space.

Abstract

This well-written paper contains the thesis of Arrondo, written under the supervision of Sols. The topic is the study of smooth congruences (i.e. surfaces in the Grassmannian G=Gr(1,3) ), showing their parallelism with surfaces in P 4 . The authors give a simple proof of the fact that the only indecomposable bundles on G with vanishing intermediate cohomology are the line bundles and the twists of the spinor bundle. This fact is needed in order to introduce and study the good notion of linkage for congruences, called spinorial linkage. Some results in the spirit of the paper of A. P. Rao [Math. Ann. 258 (1981/82), no. 2, 169–173] are proved. Moreover, the Hilbert schemes of all smooth congruences of degree at most nine are described, improving a paper of the authors [J. Reine Angew. Math. 393 (1989), 199–219;] and a paper of A. Verra [Manuscripta Math. 62 (1988), no. 4, 417–435]. The most original result is the classification, in the flavor of Severi's theorem, of the smooth congruences that can be obtained as a projection from another surface in Gr(1,4) . There are five classes, all described. The proof is geometrical and is completely different from the case of P 4 . In the last chapter, done in collaboration with M. Pedreira, the authors prove that there are finitely many components of the Hilbert scheme consisting of smooth congruences not of general type. The analogous result for surfaces in P 4 was proved by G. Ellingsrud and C. Peskine [Invent. Math. 95 (1989), no. 1, 1–11]. Some technical lemmas which extend to curves in Q 3 the Gruson-Peskine bound for curves in P 3 are needed.