### Working paper

## Fano threefolds with infinite automorphism groups

In this article, the authors study the action of the additive group C on affine cones over projective varieties. They show that such actions always exist for the cones over del Pezzo surfaces of degree d≥4 which are canonically embedded, and give relations between the actions and existence of polar cylinders. The case of del Pezzo surfaces of degree 3 is still open; for example, it is not known if the variety of the equation w3+x3+y3+z3=0 in C4 admits an action of the additive group C.

We classify Fano threefolds with only Gorenstein terminal singularities and Picard number greater than 1, satisfying the additional assumption that the G-invariant part of the Weil divisor class group is of rank 1 with respect to an action of some group G.

We classify Fano threefolds with only terminal singularities whose canonical class is Cartier and divisible by 2 with the additional assumption that the G-invariant part of the Weil divisor class group is of rank 1 with respect to an action of some group G. In particular, we find a lot of examples of Fano 3-folds with “many” symmetries.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

We introduce a category of rigid geometries on smooth singular spaces of leaves of foliations.

A special category $\mathfrak F_0$ containing orbifolds is allocated. Unlike orbifolds, objects

of $\mathfrak F_0$ can have non-Hausdorff topology and can even not satisfy the separability axiom $T_0$.

It is shown that the rigid geometry $(N,\zeta)$, where $N\in (\mathfrak F_0)$, allows a desingularization. For each such geometry $( N,\zeta)$ we prove the existence and uniqueness of the structure of a finite-dimensional Lie group in the group of all automorphisms $Aut (N},\zeta)$.

The applications to the orbifolds are considered.

For the subgroups of the Cremona group $\mathrm{Cr}_3(\mathbb C)$ having the form $(\boldsymbol{\mu}_p)^s$, where $p$ is prime, we obtain an upper bound for $s$. Our bound is sharp if $p\ge 17$.

Preface

The workshop “Algebraic Varieties and Automorphism Groups” was held at the Research Institute of Mathematical Sciences (RIMS), Kyoto University during July 7-11, 2014. There were over eighty participants including twenty people from overseas Canada, France, Germany, India, Korea, Poland, Russia, Singapore, Switzerland, and USA.

Recently, there have been remarkable developments in algebraic geometry and related fields, especially, in the area of (birational) automorphism groups and algebraic group actions.

The purpose of this workshop was to provide the experts and young researchers with the opportunities to interact in the fields of affine and complete algebraic geometry, group actions and commutative algebra related to the topics listed below as well as to publicize the new results. We are confident that these purposes were achieved by the endeavors of the participants.

The main topics of the workshop were the following:

Algebraic varieties containing An-cylinders; Algebraic varieties with fibrations; Algebraic group actions and orbit stratifications on algebraic varieties; Automorphism groups and birational automorphism groups of algebraic varieties.There were 19 talks on the above and related topics by experts from the viewpoints of (affine) algebraic geometry, transformation groups, and commutative algebra. Inspired by the talks, there were active discussions and communication among participants during and after the workshop.

The present volume is the proceedings of the workshop and contains 15 articles on the workshop topics. We hope that this volume will contribute to the progress in the theories of algebraic varieties and their automorphism groups.

The workshop was financially supported by the RIMS and Grant- in-Aid for Scientific Research (B) 24340006, JSPS. We wish to thank all those who supported us in organizing the workshop and preparing this volume.

June, 2016

Kayo Masuda, Takashi Kishimoto, Hideo Kojima, Masayoshi Miyanishi, Mikhail Zaidenberg

All papers in this volume have been refereed and are in final form. No version of any of them will be submitted for publication elsewhere.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.