Table Of ContentRecent Titles in This Series
127 F . L. Zak, Tangents and secants of algebraic varieties, 1993
126 M . L. Agranovskii, Invariant function spaces on homogeneous manifolds of Lie groups
and applications, 1993
125 Masayosh i Nagata, Theory of commutative fields, 1993
124 Masahis a Adachi, Embeddings and immersions, 1993
123 M . A. Akivis and B. A. Rosenfeld, Elie Cartan (1869-1951), 1993
122 Zhan g Guan-Hou, Theory of entire and meromorphic functions: Deficient and
asymptotic values and singular directions, 1993
121 LB . Fesenko and S. V. Vostokov, Local fields and their extensions: A constructive
approach, 1993
120 Takeyuk i Hida and Masuyuki Hitsuda, Gaussian processes, 1993
119 M . V. Karasev and V. P. Maslov, Nonlinear Poisson brackets. Geometry and
quantization, 1993
118 Kenkich i Iwasawa, Algebraic functions, 1993
117 Bori s Zilber, Uncountably categorical theories, 1993
116 G . M. Fel'dman, Arithmetic of probability distributions, and characterization problems
on abelian groups, 1993
115 Nikola i V. Ivanov, Subgroups of Teichmuller modular groups, 1992
114 Seizolto , Diffusion equations, 1992
113 Michai l Zhitomirskii, Typical singularities of differential 1-forms and Pfaffian
equations, 1992
112 S . A. Lomov, Introduction to the general theory of singular perturbations, 1992
111 Simo n Gindikin, Tube domains and the Cauchy problem, 1992
110 B . V. Shabat, Introduction to complex analysis Part II. Functions of several variables,
1992
109 Isa o Miyadera, Nonlinear semigroups, 1992
108 Take o Yokonuma, Tensor spaces and exterior algebra, 1992
107 B . M. Makarov, M. G. Golnzina, A. A. Lodkin, and A. N. Podkorytov, Selected problems
in real analysis, 1992
106 G.-C . Wen, Conformal mappings and boundary value problems, 1992
105 D . R. Yafaev, Mathematical scattering theory: General theory, 1992
104 R . L. Dobnishin, R. Kotecky, and S. Shlosman, Wulff construction: A global shape from
local interaction, 1992
103 A . K. Tsikh, Multidimensional residues and their applications, 1992
102 A . M. II'in, Matching of asymptotic expansions of solutions of boundary value
problems, 1992
101 Zhan g Zhi-fen, Ding Tong-ren, Huang Wen-zao, and Dong Zhen-xi, Qualitativ e
theory of differential equations, 1992
100 V . L. Popov, Groups, generators, syzygies, and orbits in invariant theory, 1992
99 Nori o Shimakura, Partial differential operators of elliptic type, 1992
98 V . A. Vassiliev, Complements of discriminants of smooth maps: Topology and
applications, 1992
97 Itir o Tamura, Topology of foliations: An introduction, 1992
96 A . L Markushevich, Introduction to the classical theory of Abelian functions, 1992
95 Guangchan g Dong, Nonlinear partial differential equations of second order, 1991
94 Yu . S. Il'yashenko, Finiteness theorems for limit cycles, 1991
93 A . T. Fomenko and A. A. Tuzhilin, Elements of the geometry and topology of minimal
surfaces in three-dimensional space, 1991
(Continued in the back of this publication)
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Tangents an d Secant s
of Algebraic Varietie s
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10.1090/mmono/127
Translations of
MATHEMATICAL
MONOGRAPHS
Volume 12 7
Tangents an d Secant s
of Algebraic Varietie s
F. L. Zak
American Mathematical Society
§ Providence , Rhode Island
$EHOP JIA3APEBH H 3AK
KACATEJItHLIE H CEKYHIHE
AJirEBPAH^ECKHX MHOrOOBPA3H H
Translated b y the author from a n original Russian manuscrip t
Translation edited by Simeon Ivano v
2000 Mathematics Subject Classification. Primar y 14Jxx .
ABSTRACT. This book is devoted to geometry of algebraic varieties in projective spaces. Amon g
the objects considered in some detail are tangent and secant varieties, Gauss maps, dual varieties,
hyperplane sections, projections, an d varieties of small codimension. Emphasi s is made on the
study of interplay between irregular behavior of (higher) secant varieties and irregular tangencies to
the original variety. Classification of varieties with unusual tangential properties yields interesting
examples many of which arise as orbits of representations of algebraic groups.
Library of Congress Cataloging-in-Publication Dat a
Zak, F. L.
[Kasatel'nye i sekushchie algebraicheskikh mnogoobrazii. English ]
Tangents and secants of algebraic varieties/F. L. Zak
p. cm. — (Translations of mathematical monographs; v. 127)
Includes bibliographical references.
ISBN 0-8218-4585-3 (hard cover)
ISBN 0-8218-3837-7 (soft cover)
1. Algebraic varieties. I . Title. II . Series.
QA564.Z3513 199 3 93-1750 2
516.3'5—dc20 CI P
Copying an d reprinting . Individua l reader s of this publication, an d nonprofi t librarie s
acting for them, are permitted to make fair use of the material, such as to copy a chapter for use
in teaching or research. Permissio n is granted to quote brief passages from this publication i n
reviews, provided the customary acknowledgment of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publication
is permitted onl y under licens e from th e America n Mathematica l Society . Request s fo r suc h
permission should be addressed to the Acquisitions Department, American Mathematical Society,
201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by
e-mail to [email protected] .
Copyright © 199 3 by the American Mathematical Society. All rights reserved.
Reprinted by the American Mathematical Society, 2005.
Translation authorized by the
All-Union Agency for Authors' Rights, Moscow
The American Mathematical Society retains all rights
except those granted to the United States Government .
Printed in the United States of America.
@ Th e paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
Visit the AMS home page at http://www.ams.org /
10 9 8 7 6 5 4 3 2 1 1 0 09 08 07 06 05
Contents
Introduction 1
Chapter I. Theore m on Tangencies and Gauss Maps 1 5
§1. Theorem on tangencies and its applications 1 5
§2. Gauss maps of projective varieties 2 0
§3. Subvarieties of complex tori 2 8
Chapter II. Projection s of Algebraic Varieties 3 7
§1. An existence criterion for good projections 3 7
§2. Hartshome's conjecture on linear normality and its relative
analogs 4 2
Chapter III. Varietie s of Small Codimension Corresponding to Orbits
of Algebraic Groups 4 9
§1. Orbits of algebraic groups, null-forms, and secant varieties 4 9
§2. //F-varieties of small codimension 5 5
§3. i/F-varieties as birational images of projective spaces 6 6
Chapter IV. Sever i Varieties 7 1
§1. Reduction to the nonsingular case 7 1
§2. Quadrics on Severi varieties 7 4
§3. Dimension of Severi varieties 8 0
§4. Classification theorems 8 5
§5. Varieties of codegree three 9 1
Chapter V. Linea r Systems of Hyperplane Sections on Varieties of
Small Codimension 1 0 5
§ 1. Higher secant varieties 1 0 5
§2. Maximal embeddings of varieties of small codimension 11 2
Chapter VI. Scorz a Varieties 12 1
§ 1. Properties of Scorza varieties 12 1
§2. Scorza varieties with S = 1 12 5
§3. Scorza varieties with 5 = 2 12 9
§4. Scorza varieties with 3 = 4 13 5
§5. The end of the classification of Scorza varieties 14 9
References 15 5
Index of Notations 16 1
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10.1090/mmono/127/01
Introduction
During the last twenty years algebraic geometry has been experiencing a
remarkable shift of interest from development of abstract theories to investi-
gation of concrete properties of projective varieties. Many problems of clas-
sical algebraic geometry concentrated around the notions of linear systems,
projections, (embedded) tangent spaces, etc. By using modern techniques it
has lately become possible to make considerable progress toward the solution
of some of these problems.
Among recent achievements in the field of multidimensional projective ge-
ometry we mention results of Hironaka, Matsumura, Ogus, and Hartshorne
on formal neighborhoods and local cohomology, theorems of Barth, Goresky,
and MacPherson on the topology of projective varieties, classification of Fario
varieties given by Iskovskih, Mori, and others, and various versions of Schu-
bert's enumerative geometry. On e of the most important results of the last
decade is the connectedness theorem of Fulton and Hansen (cf. [26], [27]),
which is essentially used in Chapter I of this book. A powerful incentive to
further research was given by Hartshorne's report [33], where several conjec-
tures concerning properties of projective varieties of small codimension were
put forward (some of them are proven in this book), and the relationship of
this circle of problems with local algebra and the theory of vector bundles
was considered. The main thrust of Hartshorne's conjectures is that the lower
the codimension of a nonsingular projective variety X n c P^ th e closer are
its properties to those of complete intersections (som e results of this type
were known before, e.g., Barth and Larsen showed that for i <2n - N th e
homology groups H.(X , Z ) an d the homotopy groups n (X) ar e the same
t
as those of complete intersections (and of P ^ )).
Our approach to this circle of problems is based on the study of the relative
position of (embedded) tangent spaces at various points of X . A special role
of tangent spaces in the study of geometric properties of projective varieties
was already discovered at the dawn of algebraic geometry with the introduc-
tion of enumerative geometry, theory of polar loci, dual varieties, etc. Later
the use of tangent spaces was essentially reduced to computations with Chern
classes; for example, in this way it was shown that many particular varieties
I
