Order 1 congruences of lines with smooth fundamental scheme

Abstract

In this note we present a notion of fundamental scheme for Cohen- Macaulay, order 1, irreducible congruences of lines. We show that such a congruence is formed by the k-secant lines to its fundamental scheme for a number k that we call the secant index of the congruence. If the fundamental scheme X is a smooth connected variety in PN, then k = (N-1)/(c-1) (where c is the codimension of X) and X comes equipped with a special tangency divisor cut out by a virtual hypersurface of degree k-2 (to be precise, linearly equivalent to a section by an hypersurface of degree (k-2) without being cut by one). This is explained in the main theorem of this paper. This theorem is followed by a complete classification of known locally Cohen-Macaulay order 1 congruences of lines with smooth fundamental scheme. To conclude we remark that according to Zak's classification of Severi Varieties and Harthsorne conjecture for low codimension varieties, this classification is complete.