arXiv subjects
Publications and source records attributed to Robin Hartshorne.
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 ${\mathbb P}^4$.
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?
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 $¶^3$, 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 $¶^3$ can be computed by multisecants, and we show the answer is yes in some special cases.
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.
Let $A$ be a noetherian ring and $R_A$ be the graded ring $A[X,Y,Z,T]$. In this article we introduce the notion of a triad, which is a generalization to families of curves in ${\bf P}^3_A$ of the notion of Rao module. A triad is a complex of graded $R_A$-modules $(L_1 \to L_0 \to L_{-1})$ with certain finiteness hypotheses on its cohomology modules. A pseudo-isomorphism between two triads is a morphism of complexes which induces an isomorphism on the functors $ h_0 (L\otimes .)$ and a monomorphism on the functors $h_{-1} (L\otimes .)$. One says that two triads are pseudo-isomorphic if they are connected by a chain of pseudo-isomorphisms. We show that to each family of curves is associated a triad, unique up to pseudo-isomorphism, and we show that the map $\{\hbox{families of curves}\}\to \{\hbox{triads}\}$ has almost all the good properties of the map $\{\hbox{curves}\}\to \{\hbox{Rao modules}\}$. In a section of examples, we show how to construct triads and families of curves systematically starting from a graded module and a sub-quotient (that is a submodule of a quotient module), and we apply these results to show the connectedness of $H_{4,0}$.
Let $A$ be a local noetherian ring and $N$ be a locally sheaf on the projective space $P^3_A$ : one proves easily that there exists a family $C$ of (smooth connected) curves contained in $P^3_A$, flat over $A$, and an integer $h$ such that the ideal sheaf $J$ of $C$ has a resolution $0\to P\to N\to J\to 0$ where $P$ is a direct sum of invertible sheaves $O_P(-n_i)$. In this paper we determine, for a given sheaf $N$, all the families of curves with such a resolution, especially the minimal ones (corresponding to the minimum value of $h$). It gives a description of the biliaison class related to $N$, and a tool for constructing families of space curves.
The aim of this paper is to prove a generalization of a theorem of Rao for families of space curves, which caracterizes the biliaison classes of curves. First we introduce the concept of pseudo-isomorphism of coherent sheaves, which generalizes the concept of stable isomorphism. An N-type resolution for a family of curves $C$ over the local ring $A$, defined by an ideal $J$, is an exact sequence $0\to P\to N\to J\to 0$ where $N$ is a locally free sheaf on $P^3_A$ and $P$ is a direct sum of invertible sheaves $O_P(-n_i)$. We prove the two following results, when the residual field of $A$ is infinite : 1. Let $C$ and $C'$ be two flat families of space curves over the local ring $A$. Then $C$ and $C'$ are in the same biliaison class if and only if their ideals $J$ and $J'$ are pseudo-isomorphic, up to a shift. 2. Let $C$ and $C'$ be two flat families of space curves over the local ring $A$, with N-type resolutions, involving sheaves $N$ and $N'$. Then $C$ and $C'$ are in the same biliaison class if and only if $N$ and $N'$ are pseudo-isomorphic, up to a shift.