Curves of degree two and ropes on a line: their ideals and even liaison classes

Abstract

In this paper we investigate degree two curves of arbitrary codimension. This requires to study also ropes supported on a line. After establishing several characterizations of ropes supported on a line we describe their homogeneous ideals and Hartshorne–Rao modules; in particular we characterize the arithmetically Buchsbaum ropes. Then we describe the even Gorenstein liaison classes of the non-arithmetically Buchsbaum, non-degenerate ropes generalizing a well-know result of Migliore on double lines of codimension two. The result relies on the explicit description of certain arithmetically Gorenstein curves with maximal tangent spaces at each point. As a consequence, we can decide if two curves of degree two belong to the same Gorenstein liaison class. Finally, we show as an application how ropes can be used in order to construct quasi-extremal curves.