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

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

Positivity of Chern Classes for Reflexive Sheaves on P^N

It is well known that the Chern classes $c_i$ of a rank $n$ vector bundle on \PP^N, generated by global sections, are non-negative if $i\leq n$ and vanish otherwise. This paper deals with the following question: does the above result hold for the wider class of reflexive sheaves? We show that the Chern numbers $c_i$ with $i\geq 4$ can be arbitrarily negative for reflexive sheaves of any rank; on the contrary for $i\leq 3$ we show positivity of the $c_i$ with weaker hypothesis. We obtain lower bounds for $c_1$, $c_2$ and $c_3$ for every reflexive sheaf \FF which is generated by H^0\FF on some non-empty open subset and completely classify sheaves for which either of them reach the minimum allowed, or some value close to it.Comment: 16 pages, no figure

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

Simple D-module components of local cohomology modules

For a projective variety V in P^n over a field of characteristic zero, with homogeneous ideal I in A = k[x0,x1,...,xn], we consider the local cohomology modules H^i_I(A). These have a structure of holonomic D-module over A, and we investigate their filtration by simple D-modules. In case V is nonsingular, we can describe completely these simple components in terms of the Betti numbers of V.Comment: 22 page