Order 1 congruences of lines with smooth fundamental scheme

TL;DR

This paper introduces a fundamental scheme concept for Cohen-Macaulay, order 1, irreducible congruences of lines, linking them to k-secant lines of a smooth fundamental scheme and providing a classification of such congruences.

Contribution

It defines the fundamental scheme for these congruences, relates it to secant lines, and classifies all known cases with smooth fundamental schemes.

Findings

Congruences are formed by k-secant lines to their fundamental scheme.

For smooth fundamental schemes, the secant index k is explicitly determined.

Complete classification of known order 1 congruences with smooth fundamental schemes.

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.