manuscripta math. 90, 239 - 266 (1996) manumcripta mathematica Sprlnger-Vet'la8 1996 On the sectional irregularity of congruences Luis Giraldo and Ignacio Sols* Departamento de Algebra, Fac. Matem~tticas Universidad Complutense de Madrid 28040 Madrid, Spain Received March 21, 1996 0 Introduction Let G = G(2, 4) C ps be the grassmann variety of lines in the projective space pa over an algebraically closed field of characteristic 0. A congruence is an integral surface X C G. The Chow groups are: Aa( G) = rJ3Z G) = ,j2Z AI(G) = rllZ The cycle qa is a special linear complex, i.e. parametrizes all lines in pa meet- ing a given line; the cycle r h is an c~-plane, i.e. parametrizes all lines in pa passing through a given point; the cycle r/~ is an a'-plane, i.e. parametrizes all lines of pa contained in a given plane; the cycle r h is a line pencil, i.e. parametrizes all lines contained in a given plane and passing through a given point of it. Therefore ! 7]3 2 =r]2+~2 ! q3'r/2 =q3"r/2 =rh q3 " ql = pt zl2~ =zl; 2=pt ! rl2 9rl~ = 0 and the Chow class of a complex, i.e. threefold of G, is given by its degree or intersection number with a generic line pencil; the Chow class of a surface of G, is given by its bidegree (d, d'), i.e. the intersection number d with a generic *Partially supported by CICYT PB90-0637