RT Journal Article
T1 On congruences of lines in the projective space.
A1 Sols, Ignacio
A1 Arrondo Esteban, Enrique
AB 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.
PB Société Mathématique de France
SN 0249-633X
YR 1992
FD 1992
LK https://hdl.handle.net/20.500.14352/58413
UL https://hdl.handle.net/20.500.14352/58413
LA eng
NO Chapter 6 written in collaboration with M. Pedreira (On congruences of lines in the projective space);Mémoire de la Société Mathématique de France. Nouvelle Série. Supplément au Bulletin de la Société Mathématique de France.
DS Docta Complutense
RD 3 may 2024