Algebraic Surfaces in Positive Characteristics

455 Pages·2021·14.198 MB·English

by  Masayoshi Miyanishi, Hiroyuki Ito

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World Scientific
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Published by
World Scientific Publishing Co. Pte. Ltd.
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UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Names: Miyanishi, Masayoshi, 1940– author. | Ito, Hiroyuki, 1966– author.
Title: Algebraic surfaces in positive characteristics : purely inseparable
phenomena in curves and surfaces / Masayoshi Miyanishi, Hiroyuki Ito.
Description: New Jersey : World Scientific, [2021] | Includes bibliographical references and index.
Identifiers: LCCN 2020020098 | ISBN 9789811215209 (hardcover) | ISBN 9789811215216 (ebook) |
ISBN 9789811215223 (ebook other)
Subjects: LCSH: Surfaces, Algebraic. | Curves, Algebraic. | Characteristic classes.
Classification: LCC QA571 .M573 2021 | DDC 516.3/52--dc23
LC record available at https://lccn.loc.gov/2020020098
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
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May27,2020 10:35 AlgebraicSurfaces—11690 9789811215209 pagev
Preface
Recent progress in algebraic geometry is very fast if it is confined to the
case of characteristic zero. Knowledge on curves and surfaces is one of the
basic grounds, and the theory of higher-dimensional algebraic varieties is
built on these grounds.
Meanwhile, if we consider algebraicvarieties defined overa field of pos-
itive characteristic, we have a good amount of knowledge on the theory of
algebraic curves due to E. Artin and C. Chevalley, Castelnuovo-Enriques-
KodairaclassificationofalgebraicsurfacesduetoE.BombieriandD.Mum-
ford and the theory of rational double singularities due to M. Artin and J.
Lipman. Notwithstanding all these contributions, one often and easily en-
counters or steps into an unexplored domain which causes phenomena not
parallel to the case of characteristic zero.
Paradoxically, one can say that these theories have been built to check
howfarthetheoriesestablishedinthecaseofcharacteristiczeroholdinthe
caseofpositivecharacteristic. Phenomenaparticulartothecaseofpositive
characteristic can occur.
To give a few examples, a fibration on a smooth projective surface can
havemovingsingularitiesongeneralfiberslikeasurfacewithaquasi-elliptic
fibration, and a group scheme is not necessarily reduced like infinitesimal
finite groupschemes α andμ . Evena discrete finite groupZ/pZ behaves
p p
like a unipotent group. Hence there appear algebraic surfaces obtained as
the quotients of infinitesimal group scheme actions or wild actions of finite
p-groups.
Inthecaseofasurfacewithmovingsingularitiesalongfibers,thegeneric
fiber of the fibration is a normal algebraic curve defined over the function
field K of the base curve and has hidden singular points. Namely, if one
changesthebasefieldK toitsalgebraicclosureK,thenthesingularpoints
v
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vi Algebraic surfaces inpositive characteristics
are visualized and move along the fibration. In the case of a quasi-elliptic
fibration, the generic fiber with the hidden singular point is a form of the
affine line defined over K. If the base curve is a rational curve, the sur-
face with a quasi-elliptic fibration is a unirational surface. One of the
mostnotablefactsisthatthereareplentyofunirationalsurfacesofvarious
(classification) invariants.
The present book explains the details of these situations by taking, as
examples, purely inseparable k-forms of the affine line, unirational quasi-
elliptic fibrations or quasi-hyperelliptic fibrations, Zariski surfaces, Artin-
Schreier coverings and rational double points.
AplanesandwichisanalgebraicsurfaceV whichdecomposestheFrobe-
nius endomorphism F of the projective plane P2 by rational mappings f
and g:
F : P2 −f→V −g→P2 ,
where degf = degg = p. Replacing P2 by other surfaces like abelian
surfaces, etc., we canconsiderother kindsofsandwiches. Inthe caseofP2,
since the function field k(V) of V is contained in a purely transcendental
field extension f∗ : k(V) (cid:3)→ k(x,y), V is a unirational surface, hence q :=
h1(V,O ) = 0 in particular. However, there is a vast world of unirational
V
surfaces which is mostly unexplored. It contains many interesting surfaces
of general type.
Extendingunirationalsurfaces,weexplainZariskisurfacesV definedby
zp =f(x,y), whicharepurelyinseparablecoveringsofdegreepofP2. The
curvef(x,y)=0 behaveslike the branchlocus inthe caseofcharacteristic
zero. If f(x,y) = 0 has a singular point, the point of V lying over it
hasasurfacerationalsingularityasoneofhypersurfacesingularitieswhose
analysisprovidesapeculiarfeatureinthecaseofpositivecharacteristic. We
also consideran Artin-Schreiercoveringof analgebraicsurfaceW which is
defined in terms of function fields by k(V)=k(W)(ξ), where ξp−ξ =f ∈
k(W). Even if W is smooth and V is normal, there appear singular points
in V, including rational double points, which are mostly hard to analyze.
To be short, if one takesa purely inseparablecoveringor anArtin-Schreier
covering of a smooth surface, analysis of rational double singular points
will be crucial in understanding the structure of such covering surfaces.
Study of these surfaces is not complete, and there are many problems left
unsolved.
We often consider actions on a smooth surface V of infinitesimal group
schemes α ,μ or a finite simple group Z/pZ and consider the quotient
p p
May27,2020 10:35 AlgebraicSurfaces—11690 9789811215209 pagevii
Preface vii
surfaces, which will have againsingularpoints to be analyzed. If one looks
the given surface V from the quotient surface W as a purely inseparable
covering or an Artin-Schreier covering, analysis of singular points on W is
different in a subtle manner.
We are motivated by these problems in writing the present book. We
intend it to be a comprehensive book but our emphasis tends naturally to
beputonananalysisofsingularity. Wetryalsotobegeneralintheorybut
hasto be concentratedinspecific computationthatleadstoclarificationof
singularity. Wehopethattheseconcretecomputationswillhelpthereaders
understand the problems we deal with. The book is written not only for
researchers with backgrounds in theories of algebraic curves and surfaces
but also graduate students who are interested in phenomena in positive
characteristic case. For the latter readers, we believe that the book will
become a source of many unsolved problems.
In reading chapters in Parts I and II the readers will encounter various
surface singularities which are mostly rational double points. Singularity
theory in positive characteristic is much more fascinating than in the case
of characteristic zero, in view of their complexity and richness.
Part III is devoted to the theory of rational double points. It begins
withArtin’sbasictheoryonrationalsingularitiesandLipman’slocalstudy
of these singularities. We then concentrate ourselves to a detailed study of
rational double points, their determinacy and algorithm to determine the
types of rational double points.
InPartIII, chapter2, we considerversaldeformationsandequisingular
loci. The results are worked out by Hiroyuki Ito and Natsuo Saito. The
authors are very grateful to N. Saito for approving with the inclusion of
these results into the present book.
Whenwerefertoaresultstatedinthesamepart,weindicateitwithout
mentioning the part number like Lemma 1.2.3. But a result in a different
part will be indicated with the part number like Lemma I-1.2.3 or Lemma
I.1.2.3.
Our sincere thanks go to Dr.Yuya Matsumoto of Tokyo University of
Science and Dr.Natsuo Saito of Hiroshima City University who read the
manuscriptwith greatcare andfound many errorsoftext as well as typos.
Last but not the least, we would like to express our indebtedness to the
editor Ms.Kwong Lai Fun of World Scientific Publ. Co. for giving us the
May27,2020 10:35 AlgebraicSurfaces—11690 9789811215209 pageviii
viii Algebraic surfaces inpositive characteristics
chance of writing a book on algebraic surfaces in positive characteristics
and constant encouragementduring the writing of this book.
M. Miyanishi and H. Ito
May27,2020 10:35 AlgebraicSurfaces—11690 9789811215209 pageix
Contents
Preface v
Part I Forms of the affine line 1
1. Picard scheme and Jacobian variety 3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Picard group and Picard group scheme . . . . . . . . . . . 4
0
1.3 Existence of Pic . . . . . . . . . . . . . . . . . . . . . . 7
X/k
1.4 Extendability of a rational map to Pic . . . . . . . . . 7
X/k
1.5 Lie algebra of Pic . . . . . . . . . . . . . . . . . . . . . 8
X/k
1.6 When is Pic an abelian variety? . . . . . . . . . . . . . 8
X/k
0
1.7 Pic for a complete normal curve C . . . . . . . . . . . 9
C/k
1.8 Generalized Jacobian variety. . . . . . . . . . . . . . . . . 10
1.9 Dualizing sheaf . . . . . . . . . . . . . . . . . . . . . . . . 15
2. Forms of the affine line 19
2.1 Forms of the rational function field . . . . . . . . . . . . . 19
2.2 Frobenius morphisms . . . . . . . . . . . . . . . . . . . . . 20
2.3 Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Forms of the affine line and one-place points at infinity . . 24
2.5 Explicit equations. . . . . . . . . . . . . . . . . . . . . . . 26
2.6 Smoothness criterion of points . . . . . . . . . . . . . . . . 29
2.7 Case of height ≤1 . . . . . . . . . . . . . . . . . . . . . . 31
2.8 Forms of A1 with arithmetic genus 0 or 1 . . . . . . . . . 34
ix
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x Algebraic surfaces in positive characteristics
3. Groups of Russell type 39
3.1 Forms of the additive group . . . . . . . . . . . . . . . . . 39
3.2 k-groups of Russell type as subgroups of G2 . . . . . . . . 41
a
3.3 G-torsors for k-groups of Russell type G . . . . . . . . . . 47
4. Hyperelliptic forms of the affine line 55
4.1 Birational forms of hyperelliptic curves . . . . . . . . . . . 55
4.2 k-normality of the birational form. . . . . . . . . . . . . . 56
4.3 Hyperelliptic k-forms of A1 . . . . . . . . . . . . . . . . . 58
4.4 Hyperelliptic k-forms of A1 in characteristic 2 . . . . . . . 61
4.5 Existence theorem of hyperelliptic forms of A1 . . . . . . . 69
5. Automorphisms 71
5.1 Finite Aut (X) for a k-form X of A1 . . . . . . . . . . . . 71
k
5.2 Quotients of k-forms of A1 by finite groups . . . . . . . . 73
5.3 Case of hyperelliptic forms of the affine line . . . . . . . . 74
5.4 Examples with concrete automorphism groups . . . . . . . 75
5.5 A remark . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6. Divisor class groups 83
6.1 Various invariants and their interrelations . . . . . . . . . 83
6.2 Divisor class groups. . . . . . . . . . . . . . . . . . . . . . 87
6.3 Picard varieties as unipotent groups . . . . . . . . . . . . 89
6.4 An example of unipotent Picard variety . . . . . . . . . . 94
Part II Purely inseparable and Artin-Schreier
coverings 95
1. Vector fields and infinitesimal group schemes 97
1.1 Differential 1-forms and derivations . . . . . . . . . . . . . 97
1.2 p-Lie algebra and Galois correspondence . . . . . . . . . . 100
1.3 Unramified extension of affine domains . . . . . . . . . . . 101
1.4 Sheaf of differential 1-forms and tangent sheaf . . . . . . . 106
1.5 Actions of affine group schemes . . . . . . . . . . . . . . . 109
1.6 Cartier dual of a commutative finite group scheme . . . . 115
1.7 Invariant subrings. . . . . . . . . . . . . . . . . . . . . . . 117
1.8 Frobenius sandwiches . . . . . . . . . . . . . . . . . . . . . 119