The first systematic approach to the construction of multiple structures was given by Ferrand in 1975. He made a construction for doubling curves in projective three-space, inspired by the theory of liaison by Peskine and Szpiro. Later, his techniques were generalized by Banica and Forster to produce structures of other multiplicities on curves in three-folds. Higher dimensions were treated by Banica, and also by Manolache. Holme considered e particular class of multiple structures, namely multiple structures on a linear subspace of codimension two in projective space. His work was not published, partly because Manolache published the results that Holme needed in the intended applications of these structures. This is the starting point for my interest in the subject: my Cand.Scient. thesis should clarify Holme's methods. Holme has also been my advisor both for the Cand.Scient. and for the Dr.Scient. work.

In my Cand.Scient. thesis I used the techniques introduced by Holme, but a bit refined, to push the bound for classification of multiple structures on linear spaces of codimension two from multiplicity four to multiplicity five (with an extra condition). It was submitted in 1998, and accepted for the degree Cand.Scient. at the University of Bergen in July 1998. It is written in norwegian, and a (gzipped postscript) version can be found here. Unfortunately, the title page is missing. Also, it is written in norwegian.

In my doctoral work these ideas were developed further. You can download the thesis, "Towards a Classification of Multiple Structures" (the title page is in a separate file). "Errata." The abstract is as follows :

We study non-reduced projective schemes whose reduced subspace is a smooth connected projective variety, with emphasis on (locally) Cohen-Macaulay schemes. To such a scheme we associate a filtration of subschemes with consecutively milder nilpotencies. Following Manolache, we define a scheme to be of type I if this sequence has an especially nice form. Using these techniques we get lists of representatives for type I multiple schemes of multiplicity less than or equal to five with support a linear subspace of codimension two in projective space, lists of representatives for double structures on the twisted cubic, and lists of examples of non-type I schemes on complete intersections surfaces in projective four-space. Some specific results: we show that any chern classes of rank two bundles on \$\mathbb{P}^3\$ (up to twist), with the sole requirement that \$c_2\$ is even, can be realized by bundles with sections vanishing doubly on smooth rational curves. We reformulate questions concerning splittings and extensions of bundles in terms of the geometry of the first infinitesimal neighbourhood of the reduced subspace in various projective embeddings. In particular Hartshorne's Conjecture on complete intersections in codimension two is reformulated entirely in terms of the existence of certain schemes of degree three. Monomial Cohen-Macaulay ideals with radical that is the ideal of a linear subspace are thoroughly examined, especially in codimension two. We show that all such ideals with a given Hilbert function live in the same component of the Hilbert scheme. We also give an operation on pairs of such schemes generalizing a well-known construction for points in the plane.

The thesis was defended in March 2002.

After the defence, I put two parts of the thesis on the arxiv: Multiple Structures and Monomial Multiple Structures.

More recently (June 2005), I've reorganized the material in four shorter parts: Multiple Structures and Hartshorne's Conjecture, Multiple structures of low degree on a linear subspace of codimension two, Monomial Multiple Structures and Double Structures on Rational Space Curves. These are all submitted

A related question is whether the powers of a given ideal are saturated, or more precisely: what is the differende between the Hilbert functions of powers of an ideal and the saturations of the same powers. This was studied in joint work with Alessandro Arsie, "A note on symbolic and ordinary powers of ideals" , which appeared in Ann. Univ. Ferrara.