viii PREFACE

Cox. Because of the growing current interest in performing explicit calculations in

algebraic geometry, we hope that our description of duality theory in terms of

differential forms and their residues will prove to be useful.

The residues

Resx

ω

f1, . . . , fd

can be considered as intersection invariants, and by a suitable choice of the regular

d-form ω, a residue can have many geometric interpretations, including intersection

multiplicity, angle of intersection, curvature, and the centroid of a zero-dimensional

scheme. The residue theorem then gives a global relation for these local invariants.

In this way, classical results of algebraic geometry can be reproved and generalized.

It is part of the culture to relate current theories to the achievements of former

times. This point of view is stressed in the present notes, and it is particularly sat-

isfying that some applications of residues and duality reach back to antiquity (the-

orems of Apollonius and Pappus). Alicia Dickenstein gives applications of residues

and duality to partial differential equations and problems in interpolation and ideal

membership.

Since the book is introductory in nature, only some aspects of duality the-

ory can be covered. Of course the theory has been developed much further in

the last decades, by Lipman and his coworkers among others. At appropriate

places, the text includes references to articles that appeared after the publication

of Hartshorne’s Residues and Duality [38]; see for instance the remarks following

Corollary 11.9 and those at the end of § 12. These articles extend the theory of the

book considerably in many directions. This leads to a large bibliography, though it

is likely that some important relevant work has been missed. For this, I apologize.

The students in my course were already familiar with commutative algebra,

including K¨ ahler differentials, and they knew basic algebraic geometry. Some of

them had profited from the exchange program between the University of Regensburg

and Brandeis University, where they attended a course taught by David Eisenbud

out of Hartshorne’s book [39]. Similar prerequisites are assumed about the reader

of the present text. The section by David Cox requires a basic knowledge of toric

geometry.

I want to thank the students of my lectures who insisted on clearer exposition,

especially Reinhold H¨ ubl, Martin Kreuzer, Markus N¨ ubler and Gerhard Quarg, all

of whom also later worked on algebraic residue theory, much to my benefit and the

benefit of this book. Thanks are also due to the referees for their suggestions and

comments and to Ina Mette for her support of this project.

July 2008 Ernst Kunz

Fakult¨ at f¨ ur Mathematik

Universit¨ at Regensburg

D-93040 Regensburg, Germany

David A. Cox

Department of Mathematics & Computer Science

Amherst College

Amherst, MA 01002, USA

Alicia Dickenstein

Departamento de Matem´ atica, FCEN

Universidad de Buenos Aires

Cuidad Universitaria-Pabell´ on I

(C1428EGA) Buenos Aires, Argentina