Order 1 Congruences of Lines with smooth Fundamental Scheme
In this note we present a notion of fundamental scheme for Cohen-Macaulay, order I, 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 FN, then k = (N — l)/(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 Hartshome conjecture for low codimension varieties, this classification is complete.