Reading Jessop

PREFACE

THE purpose of the present treatise is to give a brief account of the leading properties, at present known, of quartic surfaces which possess nodes or nodal curves.

A surface which would naturally take a prominent position in such a book is the Kummer surface, together with its special forms, the tetrahedroid and the wave surface, but the admirable work written by the late R. W. H. T. Hudson, entitled Kummer’s Quartic Surface, renders unnecessary the inclusion of this subject. Ruled quartic surfaces have also been omitted.

For the convenience of readers, a brief summary of all the leading results discussed in this book has been prefixed in the form of an Introduction.

I have to express my great obligation to Prof. H. F. Baker, Sc.D., F.R.S., who has given much encouragement and valuable criticism. Finally I feel greatly indebted to the staff of the University Press for the way in which the printing has been carried out.

C. M. JESSOP.

March, 1916.

INTRODUCTION

Ch. I. Quartic surfaces with isolated singular points.

This chapter, which is based on the results of Cayley* and Rohn, gives a method of classification of quartic surfaces which possess a definite number of isolated nodes and no nodal curves. The number of such nodes cannot exceed sixteen. Rohn has given a mode of classification for the surfaces having more than seven nodes, based on the properties of a type of seven-nodal plane sextic curves.

• Recent researches, etc., Proc. Lond. Math. Soc. (1869–71).

Twelve nodes on the quartic surface give rise to eleven nodes on the sextic, which must therefore break up into simpler curves; this process of decomposition goes on until we arrive at six straight lines, which case corresponds to the sixteen-nodal or Kummer surface.

Preface

The purpose of this treatise is to give a brief account of the principal properties then known of quartic surfaces with nodes or double curves.

The Kummer surface, including its special forms the tetrahedroid and the wave surface, would naturally have a prominent place. Jessop omits a detailed treatment because R. W. H. T. Hudson’s Kummer’s Quartic Surface already provides one. Ruled quartic surfaces are also omitted.

An Introduction summarizes the principal results of the book. Jessop acknowledges H. F. Baker’s encouragement and criticism and thanks the University Press for the printing.

C. M. Jessop, March 1916.

Introduction: Chapter I

Drawing on Cayley and Rohn, this chapter classifies quartic surfaces by their isolated nodes, assuming that they have no double curves. There can be at most sixteen such nodes. Rohn’s method for more than seven nodes uses seven-nodal plane sextics.

Cayley: “Recent researches,” Proceedings of the London Mathematical Society, 1869–71.

Twelve nodes on the surface produce eleven nodes on a plane sextic. The sextic must then be reducible. The successive decompositions end with six lines, corresponding to the sixteen-nodal Kummer surface.

Surfaces, nodes and projections

A quartic surface in $\mathbf P^3_{\mathbf C}$ is the zero locus of a homogeneous polynomial of degree four. A point is singular when all four first partial derivatives vanish there. An ordinary node is an isolated singularity whose quadratic term in three local coordinates is nondegenerate. A double curve is a one-dimensional singular locus; it is outside Chapter I’s classification.

We work over the complex numbers. This makes intersection counts geometric counts with multiplicities, without asserting that every point is real. The later real shapes discussed by Jessop require additional work.

Jessop’s Preface excludes a detailed account of Kummer surfaces and ruled quartics. His Introduction nevertheless surveys all nine chapters, including double conics, cyclides, double lines, rational surfaces and determinant surfaces. These later topics are context for the present chapter, not prerequisites silently imported into it.

What the projection records

Projection from a node replaces the surface by a generically two-to-one map to a plane. The two residual intersections of a line with the quartic merge along a sextic curve. The local quadratic cone at the node determines a conic in that plane, tangent to the sextic at six points in the ordinary situation.

Two different double covers will appear. The first is the surface projecting to a plane, branched over the sextic. The second is a smooth quadric projecting to the same plane, branched over the conic. Keeping them separate is essential: the second cover is a tool for lifting the sextic, not another description of the first surface.

On an irreducible sextic with $\delta$ ordinary nodes, separate the two branches at each node to obtain a smooth curve $\widetilde C$, its normalization. Its genus is $10-\delta$. Cubics through the nodes then encode differential forms on $\widetilde C$. This makes Jessop’s paired points and tangent-cubic counts statements about maps of smooth curves.

Classification and its limits

Once the surface has twelve nodes, its projected sextic has eleven; an irreducible sextic cannot have that many ordinary nodes. The classification must therefore track components and their intersections. A decomposition seen from one node need not be the decomposition seen from another. Counting plane decompositions alone does not classify surfaces.

The following pages develop the mechanisms across §§1–12. Jessop’s detailed incidence deductions and enumerative multiplicities remain identifiable source inputs where a replacement proof is not supplied. The selected Original passages preserve their wording and are not a complete transcription of the chapter.