Questions of Connectedness of the Hilbert Scheme of Curves in P3{\mathbb P}^3

We review the present state of the problem, for each degree $d$ and genus $g$, is the Hilbert scheme of locally Cohen--Macaulay curves in ${\mathbb P}^3$ connected?Comment: 10 page

Some Examples of Gorenstein Liaison in Codimension Three

Gorenstein liaison seems to be the natural notion to generalize to higher codimension the well-known results about liaison of varieties of codimension~2 in projective space. In this paper we study points in ${\mathbb P}^3$ and curves in ${\mathbb P}^4$ in an attempt to see how far typical codimension~2 results will extend. While the results are satisfactory for small degree, we find in each case examples where we cannot decide the outcome. These examples are candidates for counterexamples to the hoped-for extensions of codimension~2 theorems.Comment: 29 page

Publication history of von Staudt's Geometrie der Lage

From a census of forty copies, we can distinguish three different editions of von Staudt's Geometrie der Lage: the first of 1847 and two undated ones from the 1870's.Comment: 4 page

Clifford Index of ACM Curves in P3{\mathbb P}^3

In this paper we review the notions of gonality and Clifford index of an abstract curve. For a curve embedded in a projective space, we investigate the connection between the \ci of the curve and the \gc al properties of its \emb. In particular if $C$ is a curve of degree $d$ in ${\P}^3$, and if $L$ is a multisecant of maximum order $k$, then the pencil of planes through $L$ cuts out a $g^1_{d-k}$ on $C$. If the gonality of $C$ is equal to $d-k$ we say the gonality of $C$ can be computed by multisecants. We discuss the question whether the \go of every smooth ACM curve in ${\P}^3$ can be computed by multisecants, and we show the answer is yes in some special cases.Comment: 13 page

On Rao's Theorems and the Lazarsfeld-Rao Property

Let $X$ be an integral projective scheme satisfying the condition $S_3$ of Serre and $H^1({\mathcal O}_X(n)) = 0$ for all $n \in {\mathbb Z}$. We generalize Rao's theorem by showing that biliaison equivalence classes of codimension two subschemes without embedded components are in one-to-one correspondence with pseudo-isomorphism classes of coherent sheaves on $X$ satisfying certain depth conditions. We give a new proof and generalization of Strano's strengthening of the Lazarsfeld--Rao property, showing that if a codimension two subscheme is not minimal in its biliaison class, then it admits a strictly descending elementary biliaison. For a three-dimensional arithmetically Gorenstein scheme $X$, we show that biliaison equivalence classes of curves are in one-to-one correspondence with triples $(M,P,\alpha)$, up to shift, where $M$ is the Rao module, $P$ is a maximal Cohen--Macaulay module on the homogeneous coordinate ring of $X$, and $\alpha: P^{\vee} \to M^* \to 0$ is a surjective map of the duals.Comment: 17 page

Experiments with Gorenstein Liaison

We give some experimental data of Gorenstein liaison, working with points in ${\mathbb P}^3$ and curves in ${\mathbb P}^4$, to see how far the familiar situation of liaison, biliaison, and Rao modules of curves in ${\mathbb P}^3$ will extend to subvarieties of codimension 3 in higher ${\mathbb P}^4$.Comment: 13 page

Geometry of arithmetically Gorenstein curves in P4{\mathbb P}^4

We characterize the postulation character of arithmetically Gorenstein curves in ${\mathbb P}^4$. We give conditions under which the curve can be realized in the form $mH-K$ on some ACM surface. Finally, we strengthen a theorem of Watanabe by showing that any general arithmetically Gorenstein curve in ${\mathbb P}^4$ can be obtained from a line by a series of ascending complete-intersection biliaisons.Comment: 15 page

Liaison with Cohen-Macaulay Modules

We describe some recent work concerning Gorenstein liaison of codimension two subschemes of a projective variety. Applications make use of the algebraic theory of maximal Cohen-Macaulay modules, which we review in an Appendix.Comment: 16 page

Curves in the double plane

We study locally Cohen-Macaulay curves in projective three-space 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 of locally Cohen-Macaulay curves in 2H of given degree and arithmetic genus. We show that these Hilbert schemes are connected. We also discuss the Rao modules of these curves, and liaison and biliaison equivalence classes.Comment: 20 page