Daniel Pedoe

On the Virtual Grade of Curves on an Algebraic Surface

The theoretical method for computing the grade of a curve on an algebraic surface is well known. In practice difficulties arise which are not considered in the theory; so that it seems worth while to describe a practical method. This is done in § 1 of this paper. The method is then applied to some examples with the object of discovering whether Noether exceptional curves are necessarily exceptional curves†. In particular, a certain quintic surface with three tacnodes is studied, and our examination leads us to results which differ from those which have been accepted up till now. Another example illustrates the limitations of a practical method for computing grades, because of the possible presence of infinitesimal curves, and leads to the transformation of the quintic surface with two tacnodes into a double plane of order ten of a certain type, which has the singularity known as a (5, 5) point on its branch curve. Light is thrown on the Noether composition of this singularity by the transformation, which also shows the relation between two well-known types of surface for which the Noether relation p (2) = p (1) − 1 does not hold.

In Love with Geometry

Dan Pedoe was born in London, England, in 1910, and after serving Magdalene College, Cambridge, as Scholar and Bye-fellow, received a Ph.D. from the University of Cambridge in 1937. He held posts at the University of Southhampton, the University of Birmingham, the University of London, Khartoum University (Sudan), the University of Singapore, Purdue University, and the University of Minnesota, where he is Professor Emeritus. He held a Senior Fulbright Fellowship for Australian travel and received an MAA Lester Ford Award. He has written 50 or so research and expository papers in geometry and is the author or co-author of many books on various aspects of geometry, some translated into several languages. The monumental three-volume Methods of Algebraic Geometry with W. V. D. Hodge was recently republished by the Cambridge University Press. His other books are Circles: A Mathematical View, The Gentle Art of Mathematics, A Geometric Introduction to Linear Algebra, An Introduction to Projective Geometry, Geometry: A Comprehensive View, Geometry and the Visual Arts, and, most recently, Japanese Temple Geometry Problems, with H. Fukagawa.

On the canonical systems of certain algebraic surfaces

A complete linear system of curves on an algebraic surface may have assigned base points. The canonical system, from its definition, has no assigned base points at simple points of the surface. But we may construct surfaces on which, all the same, the canonical system has “accidental base points” at simple points of the surface. The classical example, due to Castelnuovo, is a quintic surface with two tacnodes. On this surface the canonical system is cut out by the planes passing through the two tacnodes. These planes also pass through the simple point in which the join of the two tacnodes meets the surface again. This point is the accidental base point of the canonical system on the quintic surface.

On a Class of Irregular Surfaces

1. Let P = 0, Q = 0, U = 0, V = 0 represent four independent quadric surfaces, and consider the equation