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Birational geometry via moduli spaces
Ch. 5. P. 93–132.
In this paper we connect degenerations of Fano threefolds by projections. Using mirror symmetry we transfer these connections to the side of Landau–Ginzburg models. Based on that we suggest a generalization of Kawamata’s categorical approach to birational geometry enhancing it via geometry of moduli spaces of Landau–Ginzburg models. We suggest a conjectural application to the Hassett–Kuznetsov–Tschinkel program, based on new nonrationality “invariants”—gaps and phantom categories. We formulate several conjectures about these invariants in the case of surfaces of general type and quadric bundles.
Alexandrov A., Bychkov B., Dunin-Barkowski P. et al., International Mathematics Research Notices 2026 Vol. 2026 No. 14 Article rnag146
We prove a recent conjecture of the fourth named author with P. Norbury that states a system of universal polynomial relations among the kappa classes on the moduli spaces of algebraic curves. The proof involves localization and materialization analysis of the spin Gromov–Witten theory of the projective line and is dictated by Z 2 -equivariant ...
Added: August 31, 2026
Жакупов О. Б., European Journal of Mathematics 2025 Vol. 11 Article 84
We provide examples of smooth three-dimensional Fano complete intersections of degree 2, 4, 6, and 8 that have absolute coregularity 0. Considering the main theorem of Avilov, Loginov, and Przyjalkowski (CNTP 18:506–577, 2024) on the remaining 101 families of smooth Fano threefolds, our result implies that each family of smooth Fano threefolds has an element of absolute ...
Added: May 18, 2026
Merkulov S., Letters in Mathematical Physics 2023 Vol. 113 No. 3 Article 62
Added: December 19, 2025
Yu. Prokhorov, Russian Journal of Mathematical Physics 2025 Vol. 32 P. 160–184
We study quotients of projective and affine spaces by various actions of the icosahedral group. Basically we concentrate on the rationality questions. ...
Added: November 28, 2025
Buryak A., Rossi P., International Mathematics Research Notices 2025 Vol. 2025 No. 20 Article rnaf315
The Riemann hierarchy is the simplest example of rank one, (+)-dimensional integrable system of nonlinear evolutionary PDEs. It corresponds to the dispersionless limit of the Korteweg–de Vries hierarchy. In the language of formal variational calculus, we address the classification problem for deformations of the Riemann hierarchy satisfiying different extra requirements (general deformations, deformations as systems ...
Added: November 27, 2025
A V Zaitsev, International Mathematics Research Notices 2025 Vol. 2025 No. 6 Article rnaf051
In this paper we compute the Jordan constants of the Cremona group of rank two over all fields of characteristic zero. ...
Added: March 25, 2025
Nenasheva M., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2023 Т. 514 № 1 С. 74–78
The moduli space of holomorphic differentials on curves of genus g admits a natural action of the group GL2(R). The study of orbits of this action and their closures has attracted the interest of a wide range of researchers in the last few decades. In the 2000s, C. McMullen described an infinite family of orbifolds that are closures ...
Added: February 17, 2025
Nenasheva M., Алгебра и анализ 2024 Т. 36 № 2 С. 93–107
Мероморфный дифференциал на римановой поверхности называется вещественно-нормированным, если его периоды вещественны. Пространства модулей вещественно-нормированных дифференциалов были впервые рассмотрены в работах И. М. Кричевера; с их помощью ряд известных теорем о геометрии пространств модулей алгебраических кривых получил более простые доказательства. Пространства модулей вещественно-нормированных дифференциалов с данным набором порядков полюсов стратифицируются в соответствии с порядками нулей дифференциала. В недавней ...
Added: February 17, 2025
Kazaryan M., Norbury P., International Mathematics Research Notices 2024 Vol. 2024 No. 3 P. 1825–1867
We construct an infinite collection of universal—independent of (g,n)—polynomials in the Miller–Morita–Mumford classes κm ∈ H2m(Mg,n, Q), defined over the moduli space of genus g stable curves with n labeled points. We conjecture vanishing of these polynomials in a range depending on g and n. ...
Added: February 19, 2024
Aleksei Golota, Mathematische Nachrichten 2023 Vol. 296 No. 11 P. 5012–5029
The aim of this paper is to classify codimension 1 foliations ℱ with canonical
singularities and 𝜈(𝐾 ℱ ) < 3 on threefolds of general type. I prove a classification
result for foliations satisfying these conditions and having nontrivial algebraic
part. We also describe purely transcendental foliations ℱ with the canonical class
𝐾 ℱ being not big on manifolds ...
Added: September 4, 2023
Bolbachan V., Geometriae Dedicata 2021 No. 215 P. 443–455
To any cubic surface, one can associate a cubic threefold given by a triple cover of P3P3 branched in this cubic surface. D. Allcock, J. Carlson, and D. Toledo used this construction to define the period map for cubic surfaces. It is interesting to calculate this map for some specific cubic surfaces. In this paper, we have ...
Added: May 28, 2022
Buryak A., Clader E., Tessler R., International Mathematics Research Notices 2022 Vol. 2022 No. 14 P. 10458–10532
We lay the foundation for a version of r-spin theory in genus zero for Riemann surfaces with boundary. In particular, we define the notion of r-spin disks, their moduli space, and the Witten bundle; we show that the moduli space is a compact smooth orientable orbifold with corners, and we prove that the Witten bundle is canonically ...
Added: October 29, 2021
Иванов А. Н., Математический сборник 2020 Т. 211 № 7 С. 72–92
We construct a new infinite series of irreducible components of the Gieseker-Maruyama moduli scheme M(k), k≥3, of semistable rank-2 sheaves on P3 with Chern classes c1=0, c2=k and c3=0, whose general points are sheaves with singularities of mixed dimension. These sheaves are constructed by elementary transformations of stable and properly μ-semistable reflexive sheaves along disjoint unions of collections of points and smooth irreducible curves which ...
Added: October 11, 2021
Katzarkov L. V., Lupercio E., Meersseman L. et al., Advances in Mathematics 2021 Vol. 391 Article 107945
n this paper, we will introduce Quantum Toric Varieties which are (non-commutative) generalizations of ordinary toric varieties where all the tori of the classical theory are replaced by quantum tori. Quantum toric geometry is the non-commutative version of the classical theory; it generalizes non-trivially most of the theorems and properties of toric geometry. By considering ...
Added: September 24, 2021
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