Uniruledness of orthogonal modular varieties

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Uniruledness of orthogonal modular varieties

Journal of Algebraic Geometry. 2014. Vol. 23. No. 4. P. 711–725.

A strongly reflective modular form with respect to an orthogonal group of signature (2,n) determines a Lorentzian Kac--Moody algebra. We find a new geometric application of such modular forms: we prove that if the weight is larger than n then the corresponding modular variety is uniruled. We also construct new reflective modular forms and thus provide new examples of uniruled moduli spaces of lattice polarised K3 surfaces. Finally we prove that the moduli space of Kummer surfaces associated to (1,21)-polarised abelian surfaces is uniruled.

Priority areas: mathematics

Bifurcations and Structural Stability of Generic PC-HC Families

Dorovskiy A., / Series arXiv "math". 2026.

In this paper the structural stability of generic families of vector fields of the PC-HC class on the two-dimensional sphere is proved. A classification of these families up to moderate equivalence in neighborhoods of their large bifurcation supports is presented, based on such invariants as the configuration and the characteristic set. The realization lemma is proved. ...

Added: May 14, 2026

On the minimum number of maximal distance-k independent sets in trees

Taletskii D., / Series arXiv "math". 2026.

A vertex subset of a graph is called a \textit{distance-$k$ independent set} if the distance between any two of its distinct vertices is at least $k + 1$. For all $n,k \geq 1$, we determine the minimum possible number of inclusion-wise maximal distance-$k$ independent sets among all $n$-vertex trees. It equals~$n$ if $n \leq k ...

Added: May 1, 2026

On Arithmetic Mirror Symmetry for smooth Fano fourfolds

Ovcharenko M., / Series arXiv "math". 2026.

We introduce an explicit class of tempered Laurent polynomials in the sense of Villegas and Doran--Kerr in n⩽4 variables including all Landau--Ginzburg models for smooth Fano threefolds with very ample anticanonical class. We check that it contains Landau--Ginzburg models for various Fano fourfolds which are complete intersections in smooth toric varieties and Grassmannians of planes, ...

Added: April 30, 2026

On weak solutions to the 1d compressible Navier-Stokes equations: a Lipschitz continuous dependence on data in weaker norms and an error of their homogenization

Zlotnik Alexander, / Series arXiv "math". 2026. No. 2602.03481v1.

We deal with the global in time weak solutions to the 1D compressible Navier-Stokes system of equations for large discontinuous initial data and nonhomogeneous boundary conditions of three standard types. We prove the Lipschitz-type continuous dependence of the solution $(\eta,u,\theta)$, in a norm slightly stronger than $L^{2,\infty}(Q)\times L^2(Q)\times L^2(Q)$, on the initial data $(\eta^0,u^0,e^0)$ in a ...

Added: April 18, 2026

On the dimension of the space of static potentials on three-manifolds

Medvedev V., / Series arXiv "math". 2026.

We investigate the interplay between the dimension of the space of static potentials and the geometric and topological structure of the underlying static three-manifold. A partial classification of boundaryless static manifolds is obtained in terms of this dimension. We also treat the case of static manifolds with boundary. In particular, we prove that if a ...

Added: April 3, 2026

Using predefined vector systems to speed up neural network multimillion class classification

Gabdullin N., Androsov I., / Series Computer Science "arxiv.org". 2026.

Label prediction in neural networks (NNs) has O(n) complexity proportional to the number of classes. This holds true for classification using fully connected layers and cosine similarity with some set of class prototypes. In this paper we show that if NN latent space (LS) geometry is known and possesses specific properties, label prediction complexity can ...

Added: April 2, 2026

Homogeneous maximizers of the Blaschke-Santalo-type functionals

Kolesnikov A., / Series arXiv "math". 2025.

We study Blaschke--Santal{ó}-type inequalities for N>=2 sets (functions) and a special class of cost functions. In particular, we prove new results about reduction of the maximization problem for the Blaschke--Santal{ó}-type functional to homogeneous case (functional inequalities on the sphere) and extend the symmetrization argument to the case of N>2 sets. We also discuss links to the ...

Added: February 13, 2026

Iterative Ricci-Foster Curvature Flow with GMM-Based Edge Pruning: A Novel Approach to Community Detection

Sorokin K., Beketov M., Онучин А. et al., / arxiv.org. Серия cs.SI "Social and Information Networks ". 2025.

Community detection in complex networks is a fundamental problem, open to new approaches in various scientific settings. We introduce a novel community detection method, based on Ricci flow on graphs. Our technique iteratively updates edge weights (their metric lengths) according to their (combinatorial) Foster version of Ricci curvature computed from effective resistance distance between the ...

Added: January 15, 2026

On finding formal power-logarithmic expansions of solutions to q-difference equations

Gaianov N., Parusnikova A., / Cornell University. Серия math "arxiv.org". 2025.

An algebraic q-difference equation is considered. A sufficient condition for the existence of a formal power-logarithmic expansion of a solution to such an equation in the neighborhood of zero is proposed. An example of applying this sufficient condition for constructing a formal expansion of a solution to a certain q-difference analogue of the fifth Painlevé equation ...

Added: December 25, 2025

Ideal of the variety of flexes of plane cubics

Popov V., / Series arXiv "math". 2025. No. 2502.01539.

We prove that the variety of flexes of algebraic curves of degree 3 in the projective plane is an ideal theoretic complete intersection in the product of a two-dimensional and a nine-dimensional projective spaces. ...

Added: December 16, 2025

Random walks on rank one symmetric spaces of noncompact type

Gnetov F., Konakov V., / Series arXiv "math". 2025. No. 2512.04667.

We establish a central limit theorem, a local limit theorem, and a law of large numbers for a natural random walk on a symmetric space M of non-compact type and rank one. This class of spaces, which includes the complex and quaternionic hyperbolic spaces and the Cayley hyperbolic plane, generalizes the real hyperbolic space Hn. Our approach introduces ...

Added: December 5, 2025

Cascades of Lorenz attractors in the Shimizu-Morioka model

Kazakov A., Koryakin V., Safonov K. et al., / Series arXiv "math". 2025.

The Lorenz attractor is the first example of a robustly chaotic non-hyperbolic attractor. Each orbit of such an attractor has a positive top Lyapunov exponent, and this property persists under small perturbations despite possible bifurcations of the attractor. In this paper, we study the boundary of the Lorenz attractor existence region in the Shimizu-Morioka model. ...

Added: December 4, 2025

Асимптотический вариант метода параметрикс для цепей Маркова, сходящихся к диффузиям

Bitter I., Konakov V., / Cornell University. Серия arXiv "math". 2025. № 2505.24548.

В работе приводится обобщение локальной предельной теоремы о сходимости неоднородных цепей Маркова к диффузионному пределу на случай, когда соответ- ствующие коэффициенты процессов удовлетворяют слабым условиям регулярности и совпадают лишь асимптотически. В частности, рассматриваемые нами коэффици- енты сноса могут быть неограниченными с не более чем линейным ростом, а оценки отражают перенос терминального состояния неограниченным трендом через ...

Added: December 3, 2025

Stabilization of direct images for curves

Bogomolov F. A., Schrandt S., / Series arXiv "math". 2025.

We discuss phenomena of stabilization for direct images of line bundles over projective curves mapping onto the projective line, for maps of sufficiently big degree. ...

Added: December 1, 2025

Upper Bounds on the Torsion Index of Half-Spin Groups

Deviatov R., Baek S., / Series arXiv "math". 2025.

The torsion index of split simple groups has been extensively studied, notably by Totaro, who calculated the torsion indexes of the spin groups and $E_{8}$ in [5] and [6], respectively. The aim of this paper is to provide upper bounds for the torsion index of half-spin groups, the only remaining case in the calculation of ...

Added: December 1, 2025

Hessian-based lightweight neural network for brain vessel segmentation on a minimal training dataset

Меньшиков И. А., Бернадотт А. К., Elvimov N. S., / Series arXie "Statistical mechanics". 2025.

Accurate segmentation of blood vessels in brain magnetic resonance angiography (MRA) is essential for successful surgical procedures, such as aneurysm repair or bypass surgery. Currently, annotation is primarily performed through manual segmentation or classical methods, such as the Frangi filter, which often lack sufficient accuracy. Neural networks have emerged as powerful tools for medical image ...

Added: December 1, 2025

Birational transformations of threefold Q-conic bundles

Prokhorov Y., / Series arXiv "math". 2025.

A $\mathbf{Q}$-conic bundle is a contraction $f: X\to Z$ of a three-dimensional algebraic variety $X$ to a surface~$Z$ such that the variety~$X$ has only terminal $\mathbf{Q}$-factorial singularities, the anticanonical divisor $-K_X$ is~$f$-ample, and $\uprho(X/Z)=1$. We provide an algorithm to transform a $\mathbf{Q}$-conic bundle to its standard form. ...

Added: December 1, 2025

Birational geometry of hyperkahler manifolds and the Hu-Yau conjecture

Amerik E., Verbitsky M., Soldatenkov A., / Series arXiv "math". 2025.

Wierzba and Wisniewski proved that in dimension 4, every bimeromorphic map of hyperkahler manifolds is represented as a composition of Mukai flops. Hu and Yau conjectured that this result can be generalized to arbitrary dimension. They defined ``Mukai's elementary transformation'' as the blow-up of a subvariety ruled by complex projective spaces, composed with the contraction ...

Added: December 1, 2025

Reflective modular forms and applications

Gritsenko V., Russian Mathematical Surveys 2018 Vol. 73 No. 5 P. 797–864

The reflective modular forms of orthogonal type are fundamental automorphic objects generalizing the classical Dedekind eta-function. This article describes two methods for constructing such modular forms in terms of Jacobi forms: automorphic products and Jacobi lifting. In particular, it is proved that the first non-zero Fourier–Jacobi coefficient of the Borcherds modular form Φ12 (the generating ...

Added: October 29, 2019

Рефлективные модулярные формы и их приложения

Gritsenko V., Успехи математических наук 2018 Т. 73 № 5 С. 53–122

Рефлективные модулярные формы ортогонального типа — это фундаментальные автоморфные объекты, обобщающие классическую эта-функцию Дедекинда. В этой статье мы опишем в терминах форм Якоби две конструкции для построения таких модулярных форм: автоморфные произведения и подъем Якоби. В частности, мы докажем, что первый коэффициент Фурье--Якоби модулярной формы Борчердса $\Phi_{12}$ (производящая функция для ``Fake Monster Lie Algebra’’) в любом из 23 ...

Added: October 2, 2018

Examples of lattice-polarized K3 surfaces with automorphic discriminant, and Lorentzian Kac–Moody algebras

Gritsenko V., Nikulin V. V., Transactions of the Moscow Mathematical Society 2017 Vol. 78 P. 75–83

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: January 29, 2018

K3 Surfaces and Their Moduli

Basel: Birkhäuser, 2016.

This book provides an overview of the latest developments concerning the moduli of K3 surfaces. It is aimed at algebraic geometers, but is also of interest to number theorists and theoretical physicists, and continues the tradition of related volumes like “The Moduli Space of Curves” and “Moduli of Abelian Varieties,” which originated from conferences on ...

Added: April 25, 2016

d=46, 50, 52, 54, 57. Appendix to the paper of A. Peterson, G.K.Sankaran. On some lattice computations related to moduli problems

Gritsenko V., Rendiconti del Seminario Matematico Università e Politecnico di Torino 2010 Vol. 68 No. 3 P. 298–299

We show how to analyse the cusp forms of small weights for the moduli spaces of polarised K3 surfaces with these degrees 2d. ...

Added: March 3, 2015

Reflective modular forms in algebraic geometry

Gritsenko V., / 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