Higher Dimensional Projective Spaces Research Articles

A remark on the geometry of elliptic scrolls and bielliptic surfaces

The union of two quintic elliptic scrolls in P^4 intersecting transversally along an elliptic normal quintic curve is a singular surface Z which behaves numerically like a bielliptic surface. In the appendix to the paper [W. Decker et al.: Syzygies of abelian and bielliptic surfaces in P^4, alg-geom/9606013] where the equations of this singular surface were computed, we proved that Z defines a smooth point in the appropriate Hilbert scheme and that Z cannot be smoothed in P^4. Here we consider the analogous situation in higher dimensional projective spaces P^{n-1}, where, to our surprise, the answer depends on the dimension n-1. If n is odd the union of two scrolls cannot be smoothed, whereas it can be smoothed if n is even. We construct an explicit smoothing.

On the Hilbert scheme of curves in higher-dimensional projective space

In this paper we prove that, for anyn≥3, there exist infinitely manyr∈N and for each of them a smooth, connected curveC r in ℙ r such thatC r lies on exactlyn irreducible components of the Hilbert scheme Hilb(ℙ r ). This is proven by reducing the problem to an analogous statement for the moduli of surfaces of general type.

Arguesian Identities in Invariant Theory

Having been motivated by an example of Doubilet, Rota, and Stein [Stud. Appl. Math.56(1976), 185–216], we present a technique for constructing geometric identities in a Grassmann–Cayley algebra. Each identity represents a projective invariant closely related to the Theorem of Desargues in the plane and its generalizations to higher dimensional projective space. The construction employs certain combinatorial properties of matchings in bipartite graphs. We also prove a dimension independence result for Arguesian identities, thereby connecting the identities with lattice theory.

Noether-Lefschetz Locus for Surfaces

We generalize M. Green’s Explicit Noether-Lefschetz Theorem to the family of smooth complete intersection surfaces in the higher dimensional projective spaces. Moreover, we give a new proof of the Density Theorem due to C. Ciliberto, J. Harris, and R. Miranda [5].

Noether-Lefschetz locus for surfaces

We generalize M. Green's Explicit Noether-Lefschetz Theorem to the family of smooth complete intersection surfaces in the higher dimensional projective spaces. Moreover, we give a new proof of the Density Theorem due to C. Ciliberto, J. Harris, and R. Miranda [5].

Thomas Gerald Room, 10 November 1902 - 2 April 1986

Thomas Gerald Room, Professor of Mathematics in the University of Sydney from 1935 until his retirement in 1968, died at his home near Sydney on 2 April 1986, at the age of 83. Born and educated in England, he came to Sydney in his early thirties and made it his adopted home. He was a classical geometer with unusual gifts of intuition and combinatorial skill, who will be particularly remembered for his profound insight into configurations in higher dimensional projective space. He and his slightly older contemporary, T. M. Cherry, were the elder statesmen of Australian mathematics in their time. He exerted enormous influence on the course of mathematics in New South Wales at both school and university levels and will be remembered by mathematicians, mathematics teachers and generations of students.

Vector bundles on complex projective spaces and systems of partial differential equations. I

This paper establishes and investigates a relationship between the space of solutions of a system of constant coefficient partial differential equations and the cohomology ( H 1 {H^1} in particular) of an associated vector bundle/reflexive sheaf on complex projective space. Using results of Grothendieck and Shatz on vector bundles over projective one-space, the case of partial differential equations in two variables is completely analyzed. The final section applies results about vector bundles on higher-dimensional projective spaces to the case of three or more variables.

Singularities of Projective Embedding (Points of order n on an Elliptic Curve)

In the Plücker formula for a curve embedded in a higher dimensional projective space, one encounters the notion of stationary point (cf, [B], [W]). W. F. Pohl gave new view point about it in terms of vector bundles and he defined “the singularities of embedding” (cf. [P]). At first, we shall give dual formulation of Pohl’s one by means of the sheaf of principal parts of order n, and next we shall prove the following: If an elliptic curve is embedded in (n — l)-dimensional projective space as a curve of degree n, singularities of projective embedding of order n — 1 are exactly the points of order n with suitable choice of a neutral element on the curve which is an abelian variety of dimension one. The proof is given by making use of the relation between and Schwarzenberger’s secant bundle which we shall also give.