Polynomial Rings and Affine Algebraic Geometry: PRAAG 2018, Tokyo, Japan, February 12-16

Preface
Contents
On Fano Schemes of Complete Intersections
	1 Introduction
	2 Hypersurfaces Containing Linear Subspaces
		2.1 Schubert Calculus
		2.2 Debarre–Manivel\'s Trick
		2.3 Bott\'s Residue Formula
	3 Fano Schemes of Complete Intersections
		3.1 Schubert Calculus
		3.2 Debarre–Manivel\'s Trick
	4 Numerical Invariants of Fano Schemes
	5 The Case of Fano Surfaces
	6 Irregular Fano Schemes
	7 Hypersurfaces Containing Conics
		7.1 The Codimension Count and Uniqueness
		7.2 The Degree Count
	References
Locally Nilpotent Sets of Derivations
	1 Preliminaries
	2 Locally Nilpotent Sets of Linear Maps
	3 Lie-Locally Nilpotent Sets of Linear Maps
	4 Nilpotency Conditions for Algebras
	5 Locally Nilpotent Sets of Derivations
	6 The Case of Derivation-Finite Algebras
	References
On the Theory of Gordan-Noether on Homogeneous Forms with Zero Hessian (Improved Version)
	1 Introduction
	2 Notation and Preliminaries
	3 Self-vanishing Systems of Polynomials
	4 Forms with Zero Hessian and Reduced Self-vanishing Systems
	5 Binary and Ternary Forms with Zero Hessian
	6 The Rational Map Defined by h
	7 Quaternary and Quinary Forms with Zero Hessian
	8 Proof of dim W leq1 in the Proof of Proposition 7.2
	9 Section6 of Gordan and Noether ch3GNzzz
	References
Rational Real Algebraic Models  of Compact Differential Surfaces  with Circle Actions
	1 Introduction
	2 Preliminaries
		2.1 Real and Complex Quasi-projective Algebraic Varieties
		2.2 Circle Actions as Real Forms of Hyperbolic mathbbGm-Actions
		2.3 Principal Homogeneous mathbbS1-Bundles
	3 Circle Actions on Smooth Real Affine Surfaces
		3.1 Real DPD-Presentations of Smooth Real Affine Surfaces with mathbbS1-Actions
		3.2 Real Fibers of the Quotient Morphism: Principal and Exceptional Orbits
	4 Rational Real Algebraic Models of Compact Differential Surfaces with Circle Actions
		4.1 Rational Affine Models with Compact Real Loci
		4.2 Uniqueness of Models up to Equivariant Birational Diffeomorphism
	References
The Super-Rank of a Locally Nilpotent Derivation of a Polynomial Ring
	1 Introduction
	2 The Affine Cone over a Determinantal Variety
	3 Derivations of Maximal Super-Rank
		3.1 The Derivation mathcalD(m,n)
		3.2 The Derivation D(m,n)
	References
Affine Space Fibrations
	1 Introduction
	2 Singular Fibers of mathbbA1- and mathbbP1-Fibrations
		2.1 Ga-Actions on Affine Surfaces
		2.2 Hidden Ga-Actions on Multiple Fiber Components
		2.3 Ga-Actions on Affine Threefolds
		2.4 Singular Fibers of mathbbP1-Fibrations on Smooth Projective Threefolds
		2.5 Freeness Conjecture of G. Freudenburg
	3 Equivariant Abhyankar-Sathaye Conjecture  in Dimension Three
		3.1 Arguments on Singular and Plinth Loci
		3.2 Statement of Theorem
	4 Forms of mathbbA3 with Unipotent Group Actions
		4.1 Preliminary Results
		4.2 Case of a Fixed-Point Free Ga-Action
		4.3 Case of an Effective GatimesGa-Action
		4.4 A k-Form of mathbbA4 with a Proper Action of a Unipotent Group of Dimension 2
		4.5 Forms of mathbbAntimesmathbbA1*
	5 Cancellation Problem in Dimension Three
	References
A Graded Domain Is Determined at Its Vertex. Applications to Invariant Theory
	1 Introduction
	2 Preliminaries
	3 Proof of Theorem 2
	4 Proof of Theorem 1
	References
Singularities of Normal Log Canonical del Pezzo Surfaces of Rank One
	1 Introduction
	2 Preliminaries
	3 Proof of Theorem 1
	References
O2(mathbbC)-Vector Bundles and Equivariant Real Circle Actions
	1 Introduction
	2 Real Structures
	3 O2(mathbbC)-Vector Bundles
	4 Real Vector Bundle Structures
	5 Families of Real Structures on Affine Four-Space
	6 Final Comments
	References
On Some Sufficient Conditions for Polynomials to Be Closed Polynomials over Domains
	1 Introduction
	2 Criteria of Closed Polynomials over Domains
	3 Closed Polynomials in Special Cases
	References
Variations on the Theme of Zariski\'s Cancellation Problem
	1 Introduction
		1.1 Notation, Conventions, and Terminology
	2 The Zariski Cancellation Problem
	3 Terminology
	4 Examples of Locally Flattenable Varieties
		4.1 Homogeneous Spaces
		4.2 Vector Bundles and Homogeneous Fiber Spaces
		4.3 Spherical Varieties
		4.4 Blow-Ups with Nonsingular Centers
		4.5 Curves and Surfaces
	5 Local Version of (ZCP)
	6 Flattenable Varieties Versus Locally Flattenable Varieties
	7 Answering (LZCP)
	8 Equivariantly Flattenable Varieties
	9 Equivariant Flattenability Versus Linear Equivariant Flattenability
	10 Equivariant Flattenability Versus Linearizability
	11 Equivariantly Flattenable Subgroups of the Cremona Groups
	12 Equivariantly Flattenable Groups and Special Groups in the Sense of Serre
	13 Classifying Equivariantly Flattenable Groups
	14 Locally Equivariantly Flattenable Varieties
	References
Tango Structures on Curves in Characteristic 2
	1 Introduction
	2 Tango Structures and Pre-Tango Structures on Curves
	3 The Induced Uniruled Surfaces
	4 Differential Forms
	5 Case of Genus >11
	6 Case of Small Genera
		6.1 Case of Genus 11
		6.2 Case of Genus 10
		6.3 Case of Genus 8
		6.4 Case of Genus 7
		6.5 Case of Genus 6
		6.6 Case of Genus 4
	7 Automorphisms of the Induced Uniruled Surfaces
	References
Exponential Matrices of Size Five-By-Five
	1 Introduction
		1.1 Matm, n(mathfrakR), Mat(n, mathfrakR), tA, A(i, j), i = 1r Ai, kn, ei
		1.2 GL(n, R), SL(n, R), Sym(n, R), GLS(n, R), GLC(n + 1, …, n + s)(n + s, k), Polynomial Matrices and Their Equivalence
		1.3 SE
		1.4 S S\'
		1.5 τ A, τmathfrakS
		1.6 σi, j
		1.7 νn, J(f0, …, fn - 1)
		1.8 mathfrakP
		1.9 Exp( A(T) ), Φn, Ψn - 1
		1.10 mathfrakUn, mathfrakU[d1, …, dt]
		1.11 mathfrakJ[n]
		1.12 mathfrakJ[n, 1], mathfrakJ[n, 1]0, mathfrakJ[n, 1]1
		1.13 mathfrakJ[1, n], mathfrakJ[1, n]0, mathfrakJ[1, n]1
		1.14 mathfrakA(i1, i2, i3)
		1.15 mathfrakHm + 2, mathfrakHm + 2°
	2 Preliminaries
		2.1 mathfrakUnE
		2.2 mathfrakU[n]E
		2.3 mathfrakU[n, 1]E
		2.4 mathfrakU[1, n]E
		2.5 mathfrakA(i1, i2, i3)E
		2.6 (mathfrakHm + 2°)E
	3 Polynomial Matrices of mathfrakU[n, 2]
		3.1 mathfrakJ[n, 2]0, 0
		3.2 mathfrakJ[n, 2]0, 1
		3.3 mathfrakJ[n, 2]1, 0
		3.4 mathfrakJ[n, 2]1, 1
		3.5 mathfrakJ[n, 2]0, 0, mathfrakJ[n, 2]0, 1, mathfrakJ[n, 2]1, 0, mathfrakJ[n, 2]1, 1 Are Mutually GL(n + 2, k)-Disjoint
		3.6 Exponential Matrices of mathfrakU[n , 2]
	4 Polynomial Matrices of mathfrakU[2, n]
	5 A Construction of Exponential Matrices
	6 Polynomial Matrices of mathfrakU5
		6.1 mathfrakH5
		6.2 mathfrakHmathfrakH5, mathfrakHmathfrakH5°
		6.3 mathfrakU[3, 1, 1], mathfrakU[1, 3, 1], mathfrakU[1, 1, 3]
		6.4 mathfrakU[2, 2, 1], mathfrakU[2, 1, 2], mathfrakU[1, 2, 2]
		6.5 mathfrakU[2, 1, 1, 1], mathfrakU[1, 2, 1, 1], mathfrakU[1, 1, 2, 1], mathfrakU[1, 1, 1, 2], mathfrakU[1, 1, 1, 1, 1]
		6.6 mathfrakU5
		6.7 Exponential Matrices of mathfrakU5
	7 Exponential Algebras
		7.1 Basics
		7.2 mathfrakj[n]
		7.3 mathfrakj[n, 2]1, 0(S)
		7.4 mathfrakj[n, 2]1, 1(λ)
		7.5 mathfrakj[2, n]0, 1(S)
		7.6 mathfrakj[2, n]1, 1(λ)
		7.7 mathfraka(i1, i2, i3)
		7.8 mathfrakhm + 2(S)
		7.9 mathfrakhhm + 3(S1, S2)
	8 Exponential Algebras and Modular Representations of Elementary Abelian p-Groups
	9 Five-Dimensional Modular Representations of Elementary Abelian p-Groups
	References
Mathieu-Zhao Spaces and the Jacobian Conjecture
	1 Introduction
	2 Mathieu-Zhao Spaces: Examples
	3 Background and History
	4 New Conjectures
	References