### Book

## Birational Geometry and Moduli Spaces

This volume collects contributions from speakers at the INdAM Workshop “Birational Geometry and Moduli Spaces”, which was held in Rome on 11–15 June 2018. The workshop was devoted to the interplay between birational geometry and moduli spaces and the contributions of the volume reflect the same idea, focusing on both these areas and their interaction. In particular, the book includes both surveys and original papers on irreducible holomorphic symplectic manifolds, Severi varieties, degenerations of Calabi-Yau varieties, uniruled threefolds, toric Fano threefolds, mirror symmetry, canonical bundle formula, the Lefschetz principle, birational transformations, and deformations of diagrams of algebras. The intention is to disseminate the knowledge of advanced results and key techniques used to solve open problems. The book is intended for all advanced graduate students and researchers interested in the new research frontiers of birational geometry and moduli spaces.

We survey some results about rational curves on hyperkähler manifolds, explaining how to prove a certain deformation-invariance statement for loci covered by rational curves with negative Beauville–Bogomolov square.

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of *GL _{n}*. We construct the action of the Yangian of

*sl*in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yangian (two-parametric deformation of the universal enveloping algebra of the universal central extension of

_{n}*sl*[

_{n}*s*

^{±1},

*t*]) in the cohomology of the affine version of Laumon spaces. We compute the matrix coefficients of the generators of the affine Yangian in the fixed point basis of cohomology. This basis is an affine analogue of the Gelfand-Tsetlin basis. The affine analogue of the Gelfand-Tsetlin algebra surjects onto the equivariant cohomology rings of the affine Laumon spaces. The cohomology ring of the moduli space

*M*of torsion free sheaves on the plane, of rank

_{n,d}*n*and second Chern class

*d*, trivialized at infinity, is naturally embedded into the cohomology ring of certain affine Laumon space. It is the image of the center

*Z*of the Yangian of

*gl*naturally embedded into the affine Yangian. In particular, the first Chern class of the determinant line bundle on

_{n}*M*is the image of a noncommutative power sum in

_{n,d}*Z*.

The *Handbook of Moduli,* comprising three volumes, offers a multi-faceted survey of a rapidly developing subject aimed not just at specialists but at a broad community of producers of algebraic geometry, and even at some consumers from cognate areas. The thirty-five articles in the *Handbook,* written by fifty leading experts, cover nearly the entire range of the field. They reveal the relations between these many threads and explore their connections to other areas of algebraic geometry, number theory, differential geometry, and topology. The goals of the *Handbook* are to introduce the techniques, examples, and results essential to each topic, and to say enough about recent developments to provide a gateway to the primary sources. Many articles are original treatments commissioned to bridge gaps in the literature and to make important problems accessible to a wide audience for the first time, and many others illustrate yogas and heuristics that experts use privately to guide intuition or simplify calculation, but that do not appear in published work aimed at other specialists.

This is the first of three volumes constituting the *Handbook of Moduli.*

Let M_{g;n} denote the moduli space of genus g stable algebraic curves with n marked points. It carries the Mumford cohomology classes k_{i}. A homology class in H_{*}(M_{g;n}) is said to be k-zero if the integral of any monomial in the k-classes vanishes on it. We show that any k-zero class implies a partial differential equation for generating series for certain intersection indices on the moduli spaces. The genus homogeneous components of the Witten–Kontsevich potential, as well as of the more general Hodge potential, which include, in addition to psi-classes, intersection indices for lambda-classes, are special cases of these generating series, and the well-known partial differential equations for them are instances of our general construction.

This book features recent developments in a rapidly growing area at the interface of higher-dimensional birational geometry and arithmetic geometry. It focuses on the geometry of spaces of rational curves, with an emphasis on applications to arithmetic questions. Classically, arithmetic is the study of rational or integral solutions of diophantine equations and geometry is the study of lines and conics. From the modern standpoint, arithmetic is the study of rational and integral points on algebraic varieties over nonclosed fields. A major insight of the 20th century was that arithmetic properties of an algebraic variety are tightly linked to the geometry of rational curves on the variety and how they vary in families. This collection of solicited survey and research papers is intended to serve as an introduction for graduate students and researchers interested in entering the field, and as a source of reference for experts working on related problems. Topics that will be addressed include: birational properties such as rationality, unirationality, and rational connectedness, existence of rational curves in prescribed homology classes, cones of rational curves on rationally connected and Calabi-Yau varieties, as well as related questions within the framework of the Minimal Model Program.

This book offers a concise yet thorough introduction to the notion of moduli spaces of complex algebraic curves. Over the last few decades, this notion has become central not only in algebraic geometry, but in mathematical physics, including string theory, as well.

The book begins by studying individual smooth algebraic curves, including the most beautiful ones, before addressing families of curves. Studying families of algebraic curves often proves to be more efficient than studying individual curves: these families and their total spaces can still be smooth, even if there are singular curves among their members. A major discovery of the 20th century, attributed to P. Deligne and D. Mumford, was that curves with only mild singularities form smooth compact moduli spaces. An unexpected byproduct of this discovery was the realization that the analysis of more complex curve singularities is not a necessary step in understanding the geometry of the moduli spaces.

The book does not use the sophisticated machinery of modern algebraic geometry, and most classical objects related to curves – such as Jacobian, space of holomorphic differentials, the Riemann-Roch theorem, and Weierstrass points – are treated at a basic level that does not require a profound command of algebraic geometry, but which is sufficient for extending them to vector bundles and other geometric objects associated to moduli spaces. Nevertheless, it offers clear information on the construction of the moduli spaces, and provides readers with tools for practical operations with this notion.

Based on several lecture courses given by the authors at the Independent University of Moscow and Higher School of Economics, the book also includes a wealth of problems, making it suitable not only for individual research, but also as a textbook for undergraduate and graduate coursework.

The name "K3 surfaces" was coined by A. Weil in 1957 when he formulated a research programme for these surfaces and theirmoduli. Since then, irreducible holomorphic symplectic manifolds have been introduced as a higher dimensional analogue of K3 surfaces. In this paper we present a review of this theory starting from the definition of K3 surfaces and going as far as the global Torelli theorem for irreducible holomorphic symplectic manifolds as recently proved by M. Verbitsky. For many years the last open question of Weil's programme was that of the geometric type of the moduli spaces of polarised K3 surfaces. We explain how this problem has been solved. Our method uses algebraic geometry, modular forms and Borcherds automorphic products. We collect and discuss the relevant facts from the theory of modular forms with respect to the orthogonal group O(2,n). We also give a detailed description of quasi pull-back of automorphic Borcherds products. This part contains previously unpublished results. We apply our geometric-automorphic method to study moduli spaces of both polarised K3 surfaces and irreducible symplectic varieties.

We consider (local) parameterizations of Teichmüller space Tg,n (of genus g hyperbolic surfaces with n boundary components) by lengths of 6 g- 6 + 3 n geodesics. We find a large family of suitable sets of 6 g- 6 + 3. n geodesics, each set forming a special structure called "admissible double pants decomposition". For admissible double pants decompositions containing no double curves we show that the lengths of curves contained in the decomposition determine the point of Tg,n up to finitely many choices. Moreover, these lengths provide a local coordinate in a neighborhood of all points of Tg,n{set minus}X where X is a union of 3 g- 3 + n hypersurfaces. Furthermore, there exists a groupoid acting transitively on admissible double pants decompositions and generated by transformations exchanging only one curve of the decomposition. The local charts arising from different double pants decompositions compose an atlas covering the Teichmüller space. The gluings of the adjacent charts are coming from the elementary transformations of the decompositions, the gluing functions are algebraic. The same charts provide an atlas for a large part of the boundary strata in Deligne-Mumford compactification of the moduli space Mg,n.

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.