nach oben

2020 | Buch

Kapitel lesen Erstes Kapitel lesen

Polynomial Rings and Affine Algebraic Geometry

PRAAG 2018, Tokyo, Japan, February 12−16

herausgegeben von: Shigeru Kuroda, Nobuharu Onoda, Gene Freudenburg

Verlag: Springer International Publishing

Buchreihe : Springer Proceedings in Mathematics & Statistics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Über dieses Buch

This proceedings volume gathers selected, peer-reviewed works presented at the Polynomial Rings and Affine Algebraic Geometry Conference, which was held at Tokyo Metropolitan University on February 12-16, 2018. Readers will find some of the latest research conducted by an international group of experts on affine and projective algebraic geometry. The topics covered include group actions and linearization, automorphism groups and their structure as infinite-dimensional varieties, invariant theory, the Cancellation Problem, the Embedding Problem, Mathieu spaces and the Jacobian Conjecture, the Dolgachev-Weisfeiler Conjecture, classification of curves and surfaces, real forms of complex varieties, and questions of rationality, unirationality, and birationality. These papers will be of interest to all researchers and graduate students working in the fields of affine and projective algebraic geometry, as well as on certain aspects of commutative algebra, Lie theory, symplectic geometry and Stein manifolds.

Inhaltsverzeichnis

Frontmatter

On Fano Schemes of Complete Intersections

Abstract

We provide enumerative formulas for the degrees of varieties parameterizing hypersurfaces and complete intersections which contain projective subspaces and conics. Besides, we find all cases where the Fano scheme of the general complete intersection is irregular of dimension at least 2, and for the Fano surfaces we deduce formulas for their holomorphic Euler characteristic.

C. Ciliberto, M. Zaidenberg

Locally Nilpotent Sets of Derivations

Abstract

Let B be an algebra over a field \(\mathbf {k}\). We define what it means for a subset of \({{\,\mathrm{Der}\,}}_\mathbf {k}(B)\) to be a locally nilpotent set. We prove some basic results about that notion and explore the following questions. Let L be a Lie subalgebra of \({{\,\mathrm{Der}\,}}_\mathbf {k}(B)\); if \(L \subseteq {{\,\mathrm{LND}\,}}(B)\) then does it follow that L is a locally nilpotent set? Does it follow that L is a nilpotent Lie algebra?

Daniel Daigle

On the Theory of Gordan-Noether on Homogeneous Forms with Zero Hessian (Improved Version)

Abstract

We give a detailed proof for Gordan-Noether’s results in “Ueber die algebraischen Formen, deren Hesse’sche Determinante identisch verschwindet.” C. Lossen has written a paper in a similar direction as the present paper, but did not provide a proof for every result. In our paper, every result is proved. Furthermore, our paper is independent of Lossen’s paper and includes a considerable number of new observations.

Junzo Watanabe, Michiel de Bondt

Rational Real Algebraic Models of Compact Differential Surfaces with Circle Actions

Abstract

We give an algebro-geometric classification of smooth real affine algebraic surfaces endowed with an effective action of the real algebraic circle group \(\mathbb {S}^{1}\) up to equivariant isomorphisms. As an application, we show that every compact differentiable surface endowed with an action of the circle \(S^{1}\) admits a unique smooth rational real quasi-projective model up to \(\mathbb {S}^{1}\)-equivariant birational diffeomorphism.

Adrien Dubouloz, Charlie Petitjean

The Super-Rank of a Locally Nilpotent Derivation of a Polynomial Ring

Abstract

The super-rank of a k-derivation of a polynomial ring \(k^{[n]}\) over a field k of characteristic zero is introduced. Like rank, super-rank is invariant under conjugation, and thus gives a way to classify derivations of maximal rank n. For each \(m\ge 2\), we construct a locally nilpotent derivation of \(k^{[m(m+1)]}\) with maximal super-rank \(m(m+1)\).

Gene Freudenburg

Affine Space Fibrations

Abstract

We discuss various aspects of affine space fibrations \(f : X \rightarrow Y\) including the generic fiber, singular fibers and the case with a unipotent group action on X. The generic fiber \(X_\eta \) is a form of \({\mathbb A}^n\) defined over the function field k(Y) of the base variety. Singular fibers in the case where X is a smooth (or normal) surface or a smooth threefold have been studied, but we do not know what they look like even in the case where X is a singular surface. The propagation of properties of a given smooth fiber to nearby fibers will be studied in the equivariant case of Abhyankar-Sathaye Conjecture in dimension three. We also treat the triviality of a form of \({\mathbb A}^n\) if it has a unipotent group action. Treated subjects are classified into the following four themes

Singular fibers of \({\mathbb A}^1\)- and \({\mathbb P}^1\)-fibrations,

Equivariant Abhyankar-Sathaye Conjecture in dimension three,

Forms of \({\mathbb A}^3\) with unipotent group actions,

Cancellation problem in dimension three.

Rajendra V. Gurjar, Kayo Masuda, Masayoshi Miyanishi

A Graded Domain Is Determined at Its Vertex. Applications to Invariant Theory

Abstract

We will prove that a positively graded domain/\(\mathbf{C}\) is uniquely determined by its completion at the irrelevant maximal ideal. As an application we will prove that the logarithmic Kodaira dimension of the smooth locus of a quotient of an affine space modulo a reductive algebraic group is \(-\infty \).

R. V. Gurjar

Singularities of Normal Log Canonical del Pezzo Surfaces of Rank One

Abstract

Let X be a normal del Pezzo surface of rank one with only rational log canonical singular points. In this paper, we prove that X can have at most one non klt singular point.

Hideo Kojima

-Vector Bundles and Equivariant Real Circle Actions

Abstract

The main goal of this article is to give an expository overview of some new results on real circle actions on affine four-space and their relation to previous results on \(O_2(\mathbb {C})\)-equivariant vector bundles. In Moser-Jauslin (Infinite families of inequivalent real circle actions on affine four-space, 2019, [13]), we described infinite families of equivariant real circle actions on affine four-space. In the present note, we will describe how these examples were constructed, and some consequences of these results.

L. Moser-Jauslin

On Some Sufficient Conditions for Polynomials to Be Closed Polynomials over Domains

Abstract

In this paper, we study closed polynomials of the polynomial ring in n variables over an integral domain. By using the techniques on \(\mathbb {Z}\)-gradings on the polynomial ring, we give some sufficient conditions for a polynomial f to be a closed polynomial. We also give a correspondence between closed polynomials and derivations in the polynomial ring R[x, y] in two variables over a UFD R containing \(\mathbb {Q}\).

Takanori Nagamine

Variations on the Theme of Zariski’s Cancellation Problem

Abstract

This is an expanded version of the talk by the author at the conference Polynomial Rings and Affine Algebraic Geometry, February 12–16, 2018, Tokyo Metropolitan University, Tokyo, Japan. Considering a local version of the Zariski Cancellation Problem naturally leads to exploration of some classes of varieties of special kind and their equivariant versions. We discuss several topics inspired by this exploration, including the problem of classifying a class of affine algebraic groups that are naturally singled out in studying the conjugacy problem for algebraic subgroups of the Cremona groups.

Vladimir L. Popov

Tango Structures on Curves in Characteristic 2

Abstract

The (pre-)Tango structure is a certain ample invertible sheaf of exact differential 1-forms on a projective algebraic variety and it implies some typical pathological phenomena in positive characteristic. Moreover, by using the notion of (pre-)Tango structure, we can construct another variety accompanied by similar pathological phenomena. In this article, we explicitly show several interesting and mysterious phenomena on the induced uniruled surfaces from (pre-)Tango structures on curves in characteristic 2.

Yoshifumi Takeda

Exponential Matrices of Size Five-By-Five

Abstract

In the article, we supply examples of exponential matrices in positive characteristic, and then give an overlapping classification of exponential matrices of size five-by-five in positive characteristic. In characteristic zero, we can easily classify exponential matrices up to equivalence. But, in positive characteristic, we meet difficulties of classifying exponential matrices up to equivalence. At the conference Polynomial Rings and Affine Algebraic Geometry, February 12–16, 2018, Tokyo Metropolitan University, we gave a talk about classifying exponential matrices of size four-by-four in positive characteristic, up to equivalence. So, the article can be regarded as a continuation of the talk.

Ryuji Tanimoto

Mathieu-Zhao Spaces and the Jacobian Conjecture

Abstract

In this paper we define the notion of a Mathieu-Zhao space, give various examples of this concept and use the framework of these Mathieu-Zhao spaces to describe a chain of challenging conjectures, all implying the Jacobian Conjecture.

Arno van den Essen

Titel: Polynomial Rings and Affine Algebraic Geometry
herausgegeben von: Shigeru Kuroda
Nobuharu Onoda
Gene Freudenburg
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-42136-6
Print ISBN: 978-3-030-42135-9
DOI: https://doi.org/10.1007/978-3-030-42136-6