Algebraic Surfaces in Positive Characteristics

Contents
Preface
Part I Forms of the affine line
	1. Picard scheme and Jacobian variety
		1.1 Introduction
		1.2 Picard group and Picard group scheme
		1.3 Existence of Pic0X/k
		1.4 Extendability of a rational map to PicX/k
		1.5 Lie algebra of PicX/k
		1.6 When is PicX/k an abelian variety?
		1.7 Pic0C/k for a complete normal curve C
		1.8 Generalized Jacobian variety
		1.9 Dualizing sheaf
	2. Forms of the affine line
		2.1 Forms of the rational function field
		2.2 Frobenius morphisms
		2.3 Height
		2.4 Forms of the affine line and one-place points at infinity
		2.5 Explicit equations
		2.6 Smoothness criterion of points
		2.7 Case of height ≤ 1
		2.8 Forms of A1 with arithmetic genus 0 or 1
	3. Groups of Russell type
		3.1 Forms of the additive group
		3.2 k-groups of Russell type as subgroups of G2a
		3.3 G-torsors for k-groups of Russell type G
	4. Hyperelliptic forms of the affine line
		4.1 Birational forms of hyperelliptic curves
		4.2 k-normality of the birational form
		4.3 Hyperelliptic k-forms of A1
		4.4 Hyperelliptic k-forms of A1 in characteristic 2
		4.5 Existence theorem of hyperelliptic forms of A1
	5. Automorphisms
		5.1 Finite Aut k(X) for a k-form X of A1
		5.2 Quotients of k-forms of A1 by finite groups
		5.3 Case of hyperelliptic forms of the affine line
		5.4 Examples with concrete automorphism groups
		5.5 A remark
	6. Divisor class groups
		6.1 Various invariants and their interrelations
		6.2 Divisor class groups
		6.3 Picard varieties as unipotent groups
		6.4 An example of unipotent Picard variety
Part II Purely inseparable and Artin-Schreier coverings
	1. Vector fields and infinitesimal group schemes
		1.1 Differential 1-forms and derivations
		1.2 p-Lie algebra and Galois correspondence
		1.3 Unramified extension of affine domains
		1.4 Sheaf of differential 1-forms and tangent sheaf
		1.5 Actions of affine group schemes
		1.6 Cartier dual of a commutative finite group scheme
		1.7 Invariant subrings
		1.8 Frobenius sandwiches
			1.8.1 Some auxiliary results
			1.8.2 Singularity of Frobenius sandwiches
			1.8.3 Plane sandwiches
			1.8.4 Abelian sandwiches
	2. Zariski surfaces
		2.1 Affine Zariski surface
		2.2 A theorem of Kimura-Niitsuma
		2.3 A regularity criterion
		2.4 Generators of k[x, y]δ
		2.5 Divisor class group of affine Zariski surface
		2.6 Resolution of singularities
		2.7 Resolution data
		2.8 Nice ramification with given data
		2.9 μp-quotients
			2.9.1 A result of Rudakov-Shafarevich
			2.9.2 Case of algebraic surfaces
			2.9.3 μp-actions on the affine plane
	3. Quasi-elliptic or quasi-hyperelliptic fibrations
		3.0 Introduction
		3.1 General results
		3.2 Unirational case
		3.3 Main theorems on unirational quasi-elliptic surfaces
		3.4 Settings toward proofs
		3.5 σ−1(B ∪ L∞ ∪M∞) in the case p = 3
		3.6 σ−1(M∞ ∪ Lα ∪M0) in the case p = 3
		3.7 σ−1(B ∪ L∞ ∪M∞) in the case p = 2
		3.8 σ−1(M∞ ∪ Lα ∪M0) in the case p = 2
		3.9 Unirational surface with quasi-hyperelliptic fibration
		3.10 Canonical divisors for unirational surfaces with quasi hyperellipticfibrations
		3.11 Settings toward proving Theorem 3.9
		3.12 Expression of KV
		3.13 Explicit computation of KV in the case p = 5
		3.14 Final remark
	4. Mordell-Weil groups of quasi-elliptic or quasihyperellipticsurfaces
		4.0 Introduction
		4.1 Mordell-Weil groups and N´eron-Severi groups of quasiellipticsurfaces
			4.1.1 Group structure of E(K)
			4.1.2 Torsion rank of E(K)
			4.1.3 E(K)◦ = {0}
			4.1.4 Generators of NS(V )
		4.2 Reducible singular fibers and torsion rank
			4.2.1 Discriminant
			4.2.2 Determinant of the trivial lattice T
			4.2.3 Unboundedness of torsion rank
		4.3 Rational quasi-elliptic surfaces
			4.3.1 Defining equations
			4.3.2 Case p = 2
			4.3.3 Case p = 3
			4.3.4 Blowing-down of nine P1s
				4.3.4.1 Case p = 2
				4.3.4.2 Case p = 3
		4.4 Case of quasi-hyperelliptic surfaces
			4.4.1 Rational surfaces with quasi-hyperelliptic fibrations
			4.4.2 Examples of rational quasi-hyperelliptic surfaces
				4.4.2.1 Singularity of type E8(p)
				4.4.2.2 Singularity of type E6(p)
				4.4.2.3 Reducible fibers of zp = y2 + x3
	5. Artin-Schreier coverings
		5.1 Z/pZ-action and quotient morphism
		5.2 Z/pZ-action and fixed point
		5.3 Existence criterion of Z/pZ-fixed point
		5.4 Simple Artin-Schreier covering
		5.5 Geometry of Artin-Schreier coverings
		5.6 Local Z/pZ-actions near fixed points
			5.6.1 General observations
			5.6.2 Fogarty’s result on depthRG
			5.6.3 Peskin’s criterion for rational singularity
			5.6.4 Partial linearization
			5.6.5 Recent developments
	6. Higher derivations
		6.1 Basic properties
		6.2 Invariant subrings
		6.3 Miscellaneous problems
			6.3.1 When is fδ an lfihd if so is δ ?
			6.3.2 A1-fibrations and Ga-actions
			6.3.3 Ga-actions on A3
			6.3.4 A problem of Abhyankar-Moh
	7. Unified p-group scheme
		7.1 Group scheme Gλ and pseudo-derivation
			7.1.1 Gλ-action on an affine scheme
			7.1.2 Invariant subalgebra under Gλ ⊗R R[λ−1]
			7.1.3 Invariant subalgebra in terms of δ, σ − 1 and T
		7.2 Structure of A with Gλ-action
			7.2.1 Case where λ-depth is zero
			7.2.2 Case where λ-depth is positive
		7.3 Pseudo-derivation as a lift of p-closed derivation
			7.3.1 Characterization of a p-closed derivation in the case p = 2
			7.3.2 A recursive method of lifting a p-closed derivation
Part III Rational double points
	1. Basics on rational double points
		1.1 Artin theory of rational singularities
		1.2 Lipman’s local study of rational singularity
			1.2.1 The divisor class group of R
			1.2.2 The group H
			1.2.3 Case of geometric local rings
			1.2.4 Vanishing of the group H and factoriality of R
		1.3 Right and contact equivalence, Milnor number and Tjurina number
		1.4 Determinacy
		1.5 Simple singularities
			1.5.1 Determinacies for simple singularities
		1.6 Algorithm in characteristic 2 – detecting the type of rational double point
			1.6.1 Look at the quadratic part
			1.6.2 Quadratic Part - Case (1)
			1.6.3 Quadratic Part - Case (2)
			1.6.4 Quadratic Part - Case (3)
			1.6.5 Case (3)-(i)
			1.6.6 Case (3)-(ii)
			1.6.7 Case (3)-(iii)
			1.6.8 Case (3)-(iv)
			1.6.9 Non-rational double points
			1.6.10 Flowchart
	2. Deformation of rational double points
		2.1 Versal deformation
		2.2 Equisingular loci
			2.2.1 Type An in positive characteristic
			2.2.2 Characteristic p > 5
			2.2.3 Characteristic 5
			2.2.4 Characteristic 3
			2.2.5 Characteristic 2
			2.2.6 Proof of Theorem 2.2.5
			2.2.7 Odd dimension in characteristic 2
			2.2.8 Applications
				2.2.8.1 3-dimensional singularities
				2.2.8.2 Quasi-fibration
	3. Open problems on rational double points in positive characteristics
		3.1 Tables
			3.1.1 Complex rational double points
			3.1.2 Rational double points in characteristic p > 5
			3.1.3 Rational double points in characteristic p = 5
			3.1.4 Rational double points in characteristic p = 3
			3.1.5 Rational double points in characteristic p = 2
		3.2 Open problems
			3.2.1 Rational double points as quotients
			3.2.2 Miscellaneous questions
Bibliography
Index