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 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 ubl, Martin Kreuzer, Markus 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 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
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