Related papers: Maximally Mutable Laurent Polynomials

If an $m+2$-manifold $M$ is locally modeled on $\RR^{m+2}$ with coordinate changes lying in the subgroup $G=\RR^{m+2}\rtimes ({\rO}(m+1,1)\times \RR^+)$ of the affine group ${\rA}(m+2)$, then $M$ is said to be a \emph{Lorentzian similarity…

We argue that algebraic and combinatorial polytope mutations of Fano 3-folds can be identified with mass deformations of associated 2d (0,2) supersymmetric gauge theories realized by brane brick models. These are Type IIA brane…

We prove the LeBrun-Salamon Conjecture in low dimensions. More precisely, we show that a contact Fano manifold X of dimension 2n+1 that has reductive automorphism group of rank at least n-2 is necessarily homogeneous. This implies that any…

Polynomial machine learning potentials (MLPs) based on polynomial rotational invariants have been systematically developed for various systems and applied to efficiently predict crystal structures. In this study, we propose a robust…

We study prime Fano threefolds of genus 12 ($V_{22}$-varieties) with positive-dimensional automorphism groups in positive and mixed characteristic. We classify such varieties over any perfect field. In particular, we prove that…

This paper extends the seven-dimensional Fano plane to a 15-dimensional Fano volume, which is related to sedenions. The Fano plane visualises the octonions and their structure as seven quaternions and is derived from a calibration in…

The Manin-Peyre conjecture is established for a class of smooth spherical Fano varieties of semisimple rank one. This includes all smooth spherical Fano threefolds of type T as well as some higher-dimensional smooth spherical Fano…

In the past two decades, Sorin Popa's breakthrough deformation/rigidity theory has produced remarkable rigidity results for von Neumann algebras $M$ which can be deformed inside a larger algebra $\widetilde M \supseteq M$ by an action…

For a complex connected semisimple linear algebraic group $G$ of adjoint type and of rank $n$, De Concini and Procesi constructed its wonderful compactification $\bar{G}$, which is a smooth Fano $G \times G$-variety of Picard number $n$…

We study weighted Fano fourfolds of K3 type realized as hypersurfaces in weighted projective spaces. Under the additional assumption that the singular locus has dimension at most one, we prove that only finitely many such families exist. We…

The cohomological rigidity problem for toric manifolds asks whether toric manifolds are diffeomorphic (or homeomorphic) if their integral cohomology rings are isomorphic. Many affirmative partial solutions to the problem have been obtained…

This is an outline of work in progress concerning an algebro-geometric form of the Strominger-Yau-Zaslow conjecture. We introduce a limited type of degeneration of Calabi-Yau manifolds, which we call toric degenerations. For these, the…

We study locally conformally homogeneous Lorentzian manifolds of dimension at least $3$, admitting an essential pseudo-group of local conformal transformations. Generalizing a recent result of Alekseevsky and Galaev, we show that any such…

We classify all 1-nodal degenerations of smooth Fano threefolds with Picard number 1 (both nonfactorial and factorial) and describe their geometry. In particular, we describe a relation between such degenerations and smooth Fano threefolds…

In this paper we investigate the moduli spaces of semistable coherent sheaves of rank two on the projective space $\mathbb{P}^3$ and the following rational Fano manifolds of the main series - the three-dimensional quadric $X_2$, the…

The aim of this work is to provide the first examples of $n$-dimensional varieties of wild representation type, for arbitrary $n\geq 2$. More precisely, we prove that all Fano blow-ups of $\PP^n$ at a finite number of points are of wild…

We say a mirror pair of Calabi-Yau varieties exhibits strong arithmetic mirror symmetry if the number of points on each variety over a finite field is equivalent, modulo the order of that field. We search for strong mirror symmetry in…

We give an approximate algorithm of computing holonomic systems of linear differential equations for definite integrals with parameters. We show that this algorithm gives a correct answer in finite steps, but we have no general stopping…

Let $(X,L)$ be any Fano manifold polarized by a positive multiple of its fundamental divisor $H$. The polynomial defining the Hilbert curve of $(X,L)$ boils down to being the Hilbert polynomial of $(X,H)$, hence it is totally reducible over…

We compute global log canonical thresholds, or equivalently alpha invariants, of birationally rigid orbifold Fano threefolds embedded in weighted projective spaces as codimension two or three. As an important application, we prove that most…