Abstract. In this paper we determine the irreducible components of the Hilbert schemes H 4;g of locally Cohen-Macaulay space curves of degree four and arbitrary arithmetic genus g: there are roughly $ ðg 2 =24Þ of them, most of which are families of multiplicity structures on lines. We give deformations which show that these Hilbert schemes are connected. For g4À 3 we exhibit a component that is disjoint from the component of extremal curves and use this to give a counterexample to a conjecture of Aı¨t-Amrane and Perrin.

We study in detail locally Cohen-Macaulay curves in P 3 which are contained in a double plane 2H, thus completing the classification of curves lying on surfaces of degree two. We describe the irreducible components of the Hilbert schemes H d,g (2H) of locally Cohen-Macaulay curves in 2H of degree d and arithmetic genus g, and we show that H d,g (2H) is connected. We also discuss the Rao module of these curves and liaison and biliaison equivalence classes.

The uniform position principle states that, given an irreducible non-degenerate curve C in P^r (projective r-space over the complex numbers), a general(r-2)-plane L in P^r is uniform, that is, projection from L induces a rational map C -> P^1 whose monodromy group is the full symmetric group. In this paper we first show the locus of non-uniform r-2-planes has codimension at least two in the Grassmannian. This result is sharp because, if there is a point x in P^r such that projection from x induces a map C -> P^{r-1} that is not birational onto its image, then the Schubert cycle \sigma(x) of (r-2)-planes through x is contained in the locus of non-uniform (r-2)-planes. For a smooth curve C in P^3, we show that any irreducible surface of non-uniform lines is a cycle \sigma(x) as above, unless C is a rational curve of degree three, four, or six

We study the lowest dimensional open case of the question whether every arithmetically Cohen–Macaulay subscheme of P^N is glicci, that is, whether every zeroscheme in P^3 is glicci. We show that a general set of n ≥ 56 points in P^3 admits no\ud strictly descending Gorenstein liaison or biliaison. In order to prove this theorem, we establish a number of important results about arithmetically Gorenstein zero-schemes in P^3

Let C be an ACM (projectively normal) nonsingular curve in ‫ސ‬ 3 ‫ރ‬ not contained in a plane, and suppose C is general in its Hilbert scheme -this is irreducible once the postulation is fixed. Answering a question posed by Peskine, we show the gonality of C is d − l, where d is the degree of the curve and l is the maximum order of a multisecant line of C. Furthermore l = 4 except for two series of cases, in which the postulation of C forces every surface of minimum degree containing C to contain a line as well. We compute the value of l in terms of the postulation of C in these exceptional cases. We also show the Clifford index of C is equal to gon(C) − 2.