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## Moduli of polarized Enriques surfaces

Cornell University
,
2015.
No. 02723.

Gritsenko V., Hulek K.

In press

In this paper we consider moduli spaces of polarized and numerically polarized Enriques surfaces. The moduli spaces of numerically polarized Enriques surfaces can be described as open subsets of orthogonal modular varieties of dimension 10. One of the consequences of our description is that there are only finitely many birational equivalence classes of moduli spaces of polarized and numerically polarized Enriques surfaces. We use modular forms to prove for a number of small degrees that the Kodaira dimension of the moduli space of numerically polarized Enriques surfaces is negative. Finally we prove that there are infinitely many polarizatons for which the moduli space of numerically polarized Enriques surfaces is birational to the moduli space of unpolarized Enriques surfaces with a level 2 structure.

Keywords: автоморфные формыпространства модулейmoduli spacesEnriques surficesautomorphic formsповерхности Энриквеса

Publication based on the results of:

Gritsenko V., Nikulin V., Proceedings of the London Mathematical Society 2018 Vol. 116 No. 3 P. 485–533

We describe a new large class of Lorentzian Kac--Moody algebras. For all ranks, we classify 2-reflective hyperbolic lattices $S$ with the group of 2-reflections of finite volume and with a lattice Weyl vector. They define the corresponding hyperbolic Kac--Moody algebras of restricted arithmetic type which are graded by S. For most of them, we construct ...

Added: October 23, 2017

Gritsenko V., Hulek K., Sankaran G., Compositio Mathematica 2010 Vol. 146 No. 2 P. 404–434

We study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of 2d polarised (split type) symplectic manifolds which are deformation equivalent to degree 2 Hilbert schemes of a K3 surface is of general type if d ...

Added: March 3, 2015

Gritsenko V., Hulek K., Sankaran G., , in : Handbook of Moduli. Vol. I. Vol. I.: Boston: International Press of Boston Inc, 2013. P. 469–525.

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 ...

Added: March 3, 2015

Poberezhny V. A., / ИТЭФ. Series "Препринты ИТЭФ". 2013. No. 52/13.

We give an w\explicit example of non-regular behaviour of fuchsian systems moduli space in the case of resonant singular points. Tha set of systems with same singularities, asymptotics and monodromy but still not globally equivalent is constructed. ...

Added: March 31, 2014

Finkelberg M. V., Rybnikov L. G., Algebraic Geometry 2014 Vol. 1 No. 2 P. 166–180

Drinfeld zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of an affine Lie algebra g^. In case g is the symplectic Lie algebra spN, we introduce an affine, reduced, irreducible, normal quiver variety Z which maps to the zastava space isomorphically in characteristic 0. The natural Poisson structure on ...

Added: October 25, 2013

Feigin B. L., Finkelberg M. V., Rybnikov L. G. et al., Selecta Mathematica, New Series 2011 Vol. 17 No. 3 P. 573–607

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GLn. We construct the action of the Yangian of sln in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yangian (two-parametric deformation of the universal ...

Added: October 9, 2012

Gritsenko V., Никулин В. В., TRANSACTIONS OF THE MOSCOW MATHEMATICAL SOCIETY 2017 Т. 78 № 1 С. 89–100

Using our results about Lorentzian Kac--Moody algebras and arithmetic mirror symmetry, we give six series of examples of lattice-polarized K3 surfaces with automorphic discriminant. ...

Added: October 11, 2017

Feigin B. L., Finkelberg M. V., Rybnikov L. G. et al., Selecta Mathematica, New Series 2011 Vol. 17 No. 2 P. 337–361

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GLn. We calculate the equivariant cohomology rings of the Laumon moduli spaces in terms of Gelfand-Tsetlin subalgebra of U(gln), and formulate a conjectural answer for the small quantum cohomology rings in terms of ...

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Natanzon S. M., Pratoussevitch A., Russian Mathematical Surveys 2016 Vol. 71 No. 2 P. 382–384

In this paper, we present all higher spinor structures on Klein surfaces. We present also topological invariants that describe the connected components of moduli of Klein surfaces with higher spinor structure. Each connected component is represented as a cell factorable by a discrete group . ...

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Verbitsky M., Duke Mathematical Journal 2013 Vol. 162 No. 15 (2013) P. 2929–2986

A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. We compute a mapping class group of a hyperkähler manifold $M$, showing that it is commensurable to an arithmetic lattice in SO(3, b_2-3). A Teichmüller space of $M$ is a space of complex structures on $M$ up ...

Added: December 10, 2013

Costa A., Gusein-Zade S., Natanzon S. M., Indiana University Mathematics Journal 2011 Vol. 60 No. 3 P. 985–995

Klein foams are analogues of Riemann and Klein surfaces with one-dimensional singularities. We prove that the field of dianalytic functions on a Klein foam Ω coincides with the field of dianalytic functions on a Klein surface K Ω We construct the moduli space of Klein foams, and we prove that the set of classes of ...

Added: November 24, 2012

Gorinov A., / Cornell University. Series math "arxiv.org". 2014. No. 1402.5946.

We present a modification of the method of conical resolutions \cite{quintics,tom}. We apply our construction to compute the rational cohomology of the spaces of equations of nodal cubics in CP2, nodal quartics in CP2 and nodal cubics in CP3. In the last two cases we also compute the cohomology of the corresponding moduli spaces. ...

Added: February 26, 2014

Kazaryan M., Lando S., Prasolov V., Switzerland: Springer, 2018

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, ...

Added: November 19, 2018

Kochetkov Y., / Cornell University Library. 2013. No. 1301.6059.

We consider the space $\mathcal{M}_{2,1}$ --- the open moduli space of complex curves of genus 2 with one marked point. Using language of chord diagrams we describe the cell structure of $\mathcal{M}_{2,1}$ and cell adjacency. This allows one to construct matrices of boundary operators and compute Betty numbers of $\mathcal{M}_{2,1}$ over $\mathbb{Q}$. ...

Added: February 24, 2013

Tikhomirov A. S., Matemática Contemporânea 2020 Vol. 47 P. 301–316

In this article, we will give a review of recent results on the geography and geometry of the Gieseker-Maruyama moduli scheme M = M (c1 , c2 ) of rank 2 semi-stable coherent sheaves with first Chern class c1 = 0 or −1, second Chern class c2 , and third Chern class 0 on the ...

Added: December 22, 2020

Kazaryan M., Lando S., Moscow Mathematical Journal 2012 Vol. 12 No. 2 P. 397–411

Let Mg;n denote the moduli space of genus g stable algebraic curves with n marked points. It carries the Mumford cohomology classes ki. A homology class in H*(Mg;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 ...

Added: May 24, 2012

Boston: International Press of Boston Inc, 2013

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 ...

Added: February 27, 2015

Buryak A., Tessler R., Communications in Mathematical Physics 2017 Vol. 353 No. 3 P. 1299–1328

In a recent work, R. Pandharipande, J. P. Solomon and the second author have initiated a study of the intersection theory on the moduli space of Riemann surfaces with boundary. They conjectured that the generating series of the intersection numbers satisfies the open KdV equations. In this paper we prove this conjecture. Our proof goes ...

Added: September 27, 2020

Fonarev A., Kuznetsov A., Journal of London Mathematical Society 2018 Vol. 97 No. 2 P. 24–46

We prove that the derived category D(C) of a generic curve of genus greater than one embeds into the derived category D(M) of the moduli space M of rank two stable bundles on C with fixed determinant of odd degree. ...

Added: November 7, 2017

Kochetkov Y., Фундаментальная и прикладная математика 2014 Т. 19 № 1 С. 45–63

Мы рассматриваем открытое пространство модулей $\mathcal{M}_{2,1}$ комплексных кривых рода 2 с одной отмеченной точкой. На языке хордовых диаграмм мы описываем клеточную структуру пространства $\mathcal{M}_{2,1}$ и структуру примыкания клеток. Это позволяет нам построить матрицы граничных операторов и найти числа Бетти пространства $\mathcal{M}_{2,1}$ над Q. ...

Added: November 11, 2014

Gritsenko V., / Cornell University. Series math "arxiv.org". 2010. No. 3753.

We prove that the existence of a strongly reflective modular form of a large weight implies that the Kodaira dimension of the corresponding modular variety is negative or, in some special case, it is equal to zero. Using the Jacobi lifting we construct three towers of strongly reflective modular forms with the simplest possible divisor. ...

Added: March 3, 2015

Springer Publishing Company, 2020

This book offers an introduction to the research in several recently discovered and actively developing mathematical and mathematical physics areas. It focuses on: 1) Feynman integrals and modular functions, 2) hyperbolic and Lorentzian Kac-Moody algebras, related automorphic forms and applications to quantum gravity, 3) superconformal indices and elliptic hypergeometric integrals, related instanton partition functions, 4) ...

Added: September 9, 2020

Gorsky E., Advances in Mathematics 2014 Vol. 250 P. 588–595

We derive a formula for the Sn-equivariant Euler characteristic of the moduli space Mg,n of genus g curves with n marked points. ...

Added: December 9, 2014

Springer, 2020

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. ...

Added: August 13, 2020