Toric Surfaces Research Articles

Towers of Looijenga pairs and asymptotics of ECH capacities

ECH capacities are rich obstructions to symplectic embeddings in 4-dimensions that have also been seen to arise in the context of algebraic positivity for (possibly singular) projective surfaces. We extend this connection to relate general convex toric domains on the symplectic side with towers of polarised toric surfaces on the algebraic side, and then use this perspective to show that the sub-leading asymptotics of ECH capacities for all convex and concave toric domains are $O(1)$. We obtain sufficient criteria for when the sub-leading asymptotics converge in this context, generalising results of Hutchings and of the author, and derive new obstructions to embeddings between toric domains of the same volume. We also propose two invariants to more precisely describe when convergence occurs in the toric case. Our methods are largely non-toric in nature, and apply more widely to towers of polarised Looijenga pairs.

Toric Surfaces with Reflection Symmetries

Toric Surfaces with Reflection Symmetries

Toric Surfaces with Reflection Symmetries

Пусть $W$ - группа, порожденная отражениями на плоскости, и $P$ - рациональный многоугольник, инвариантный относительно действия группы $W$. Действие группы $W$ на $P$ индуцирует ее действие на торическом многообразии $X_P$, ассоциированном с $P$. В работе изучается представление группы $W$ на кольце когомологий $H^*(X_P)$. Показано, что инвариантное подкольцо $H^*(X_P)^W$ изоморфно кольцу когомологий торического многообразия, ассоциированного с фундаментальной областью $P/W$. В качестве примера дается явное описание основного результата в случае торического многообразия, ассоциированного с веером камер Вейля типа $G_2$.

K-theoretic balancing conditions and the Grothendieck group of a toric variety

We introduce a ring of Z-valued functions on a complete fan Δ called Grothendieck weights to describe the ordinary operational K-theory of the associated toric variety X. These functions satisfy a K-theoretic analogue of the balancing condition for Minkowski weights, which is induced by a presentation of the Grothendieck group of X. We explicitly give a combinatorial presentation in low dimensions, and relate Grothendieck weights to other fan-based invariants such as piecewise exponential functions and Minkowski weights. As an application, we give an example of a projective toric surface X such that the forgetful map KT∘(X)→K∘(X) is not surjective.

On the number of vertices of Newton–Okounkov polygons

The Newton–Okounkov body of a big divisor D on a smooth surface is a numerical invariant in the form of a convex polygon. We study the geometric significance of the shape of Newton–Okounkov polygons of ample divisors, showing that they share several important properties of Newton polygons on toric surfaces. In concrete terms, sides of the polygon are associated to some particular irreducible curves, and their lengths are determined by the intersection numbers of these curves with D . As a consequence of our description, we determine the numbers k such that D admits some k -gon as a Newton–Okounkov body, elucidating the relationship of these numbers with the Picard number of the surface, which was first hinted at by work of Küronya, Lozovanu and Maclean.

An imaginary refined count for some real rational curves

In 2015, Mikhalkin introduced a refined count for the real rational curves in a toric surface which pass through a set consisting of real points and pairs of complex conjugated points chosen generically on the toric boundary of the surface. He then proved that the result of this refined count depends only on the number of pairs of complex conjugated points on each toric divisor. Using the tropical geometry approach and the correspondence theorem, we address the computation of the refined count when the pairs of complex conjugated points are chosen purely imaginary and belonging to the same component of the toric boundary. Despite the non-genericity, we relate this refined count for purely imaginary values to the refined invariant of Mikhalkin for generic values. That allows us to extend the relation between these classical refined invariants and the tropical refined invariants from Block–Göttsche.

Classification of noncommutative monoid structures on normal affine surfaces

In 2021, Dzhunusov and Zaitseva classified two-dimensional normal affine commutative algebraic monoids. In this work, we extend this classification to noncommutative monoid structures on normal affine surfaces. We prove that two-dimensional algebraic monoids are toric. We also show how to find all monoid structures on a normal toric surface. Every such structure is induced by a comultiplication formula involving Demazure roots. We also give descriptions of opposite monoids, quotient monoids, and boundary divisors.

Flow field characteristics and experimental research on inner-jet electrochemical face grinding of SUS420J2 stainless steel

Electrochemical grinding (ECG) is processed by the combination of dissolution and grinding. It is very suitable for the processing of difficult-to-cut stainless steel, but its processing performance is restricted by the matching effect of dissolution and grinding. In this work, the processing of the torus surfaces of the stainless steel shaver cap was taken as the research object. A flow field model including the through-hole structure and the rotation of the grinding head was proposed to optimize the flow field distribution and promote the uniform dissolution of materials. The flow field simulation results showed that the rotational flow formed by the high-speed rotation prolonged the electrolyte flow path and was not conducive to the discharge of electrolytic products, and the reasonable selection of the diameter and distribution of the through-hole could reduce the velocity difference. The effects of rotational speed, feed rate, and inlet pressure on the flatness and surface roughness of the torus surfaces were experimentally investigated, and a better matching effect of dissolution and grinding was obtained. Moreover, the experimental results showed that the inner-jet ECG had a good prospect in the batch processing of high-hardness stainless steel parts.

Multipoint Okounkov bodies

Starting from the data of a big line bundle $L$ on a projective manifold $X$ with a choice of $N\geq 1$ different points on $X$ we provide a new construction of $N$ Okounkov bodies which encodes important geometric features of ($L\to X,p_{1},\dots,p_{N}$) such as the volume of $L$, the (moving) multipoint Seshadri constant of $L$ at $p_{1},\dots,p_{N}$, and the possibility to construct Kahler packings centered at $p_{1},\dots,p_{N}$. Toric manifolds and surfaces are examined in detail.

Polynomiality properties of tropical refined invariants

Tropical refined invariants of toric surfaces constitute a fascinating interpolation between real and complex enumerative geometries via tropical geometry. They were originally introduced by Block and Göttsche, and further extended by Göttsche and Schroeter in the case of rational curves. In this paper, we study the polynomial behavior of coefficients of these tropical refined invariants. We prove that coefficients of small codegree are polynomials in the Newton polygon of the curves under enumeration, when one fixes the genus of the latter. This provides a surprising reappearance, in a dual setting, of the so-called node polynomials and the Göttsche conjecture. Our methods, based on floor diagrams introduced by Mikhalkin and the first author, are entirely combinatorial. Although the combinatorial treatment needed here is different, we follow the overall strategy designed by Fomin and Mikhalkin and further developed by Ardila and Block. Hence our results may suggest phenomena in complex enumerative geometry that have not been studied yet. In the particular case of rational curves, we extend our polynomiality results by including the extra parameter \(s\) recording the number of \(\psi\) classes. Contrary to the polynomiality with respect to \( \Delta\), the one with respect to \(s\) may be expected from considerations on Welschinger invariants in real enumerative geometry. This pleads in particular in favor of a geometric definition of Göttsche-Schroeter invariants.Mathematics Subject Classifications: Primary 14T15, 14T90, 05A15; Secondary 14N10, 52B20Keywords: Tropical refined invariants, enumerative geometry, Welschinger invariants, Gromov-Witten invariants, floor diagrams

Quantum mechanics of a fermion confined to a curved surface in Foldy-Wouthuysen representation

In Foldy-Wouthuysen representation, this paper deduces the effective quantum mechanics for a relativistic particle confined to a curved surface in an external field by using the thin-layer quantization scheme. It is found that the quantum effects caused by introducing the confining potential can be taken as the results of relativistic correction in the nonrelativistic limit. Meanwhile, the spin connection of the curved surface can produce a Zeeman-like gap through the relativistic correction term. In addition, the confining potential can provide a curvature-independent energy shift resulting from the Zitterbewegung effect. As an example, this paper applies the effective Hamiltonian to a torus surface, in which the spin effects related to the introduced confining potential are obtained. These results demonstrate the scaling of the noncommutation of the nonrelativistic limit and the thin-layer quantization procedure.

Fermions on a torus knot

In this work, we investigate the effects of a nontrivial topology (and geometry) of a system considering \textit{interacting} and \textit{noninteracting} particle modes, which are restricted to follow a closed path over the torus surface. In order to present a prominent thermodynamical investigation of this system configuration, we carry out a detailed analysis using statistical mechanics within the grand canonical ensemble approach to deal with \textit{noninteracting} fermions. In an analytical manner, we study the following thermodynamic functions in such context: the Helmholtz free energy, the mean energy, the magnetization and the susceptibility. Further, we take into account the behavior of Fermi energy of the thermodynamic system. Finally, we briefly outline how to proceed in case of \textit{interacting} fermions.

On secant defectiveness and identifiability of Segre–Veronese varieties

We give an almost asymptotically sharp bound for the non-secant defectiveness and identifiability of Segre–Veronese varieties. We also provide new examples of defective Segre–Veronese varieties, and implement our methods in Magma. Finally, we give two applications of our techniques: we classify possibly singular 2-secant defective toric surfaces and we study secant defectiveness of Losev–Manin spaces.

Late-onset toxic anterior segment syndrome after possible aluminum-contaminated and silicon-contaminated intraocular lens implantation.

To describe an outbreak of late-onset toxic anterior segment syndrome (TASS) after the implantation of a specific hydrophilic acrylic intraocular lens (IOL). University Hospitals of Leuven, Belgium. Retrospective, single-center, observational study. All eyes undergoing cataract surgery with a monofocal, toric, or enhanced depth-of-focus (EDoF) Synthesis (Cutting Edge) IOL between August 2019 and March 2020 were reviewed. Data were collected on the surgical procedure, postoperative course, time until onset of symptoms, clinical features, and additional treatments. A laboratory surface analysis of all 3 IOL subtypes was performed in the Intermountain Ocular Research Center at the University of Utah, USA. Furthermore, other possible causes of prolonged postoperative inflammation rather than the IOL itself were investigated. Among the 203 eyes included, 28 TASS cases were identified (13.8%), among which 25 received a monofocal IOL, and 3 received an EDoF IOL. The mean time until onset was 28.9 (±19.9) days. Patients presented with anterior chamber cells (92.9%), deposits on the IOL (57.1%), or fibrinous inflammation (35.7%). 4 eyes (14.3%) underwent a surgical intervention, whereas 24 eyes showed a resolution of inflammation with topical therapy alone. Laboratory analysis showed the presence of both aluminum and silicon particles on the monofocal IOL, silicon particles only on the EDoF IOL, and no particles on the toric IOL surface. This report describes an outbreak of atypical, late-onset TASS after cataract surgery, possibly correlated with aluminum and silicon contamination of the IOL surfaces.

Degeneration of curves on some polarized toric surfaces

Abstract We address the following question: Given a polarized toric surface ( S , L ) {(S,{\mathcal{L}})} , and a general integral curve C of geometric genus g in the linear system | L | {|{\mathcal{L}}|} , do there exist degenerations of C in | L | {|{\mathcal{L}}|} to general integral curves of smaller geometric genera? We give an affirmative answer to this question for surfaces associated to h-transverse polygons, provided that the characteristic of the ground field is large enough. We give examples of surfaces in small characteristic, for which the answer to the question is negative. In case the answer is affirmative, we deduce that a general curve C as above is nodal. In characteristic 0, we use the result to show the irreducibility of Severi varieties of a large class of polarized toric surfaces with h-transverse polygon.

A note on the Severi problem for toric surfaces

In this note, we make a step towards the classification of toric surfaces admitting reducible Severi varieties. We provide two families of toric surfaces admitting reducible Severi varieties. The first family is general, and is obtained by a quotient construction. The second family is exceptional, and corresponds to certain narrow polygons, which we call kites. We introduce two types of invariants that distinguish between the components of the Severi varieties, and allow us to provide lower bounds on the numbers of the components. The sharpness of the bounds is verified in some cases, and is expected to hold in general for ample enough linear systems. In the appendix, we establish a connection between the Severi problem and the topological classification of univariate polynomials.

The quantitative behavior of asymptotic syzygies for Hirzebruch surfaces

We study the quantitative behavior of asymptotic syzygies for certain toric surfaces, including Hirzebruch surfaces. In particular, we show that the asymptotic linear syzygies of Hirzebruch surfaces embedded by 𝒪(d,2) conform to Ein, Erman, and Lazarsfeld’s normality heuristic. We also show that the higher degree asymptotic syzygies are not asymptotically normally distributed.

Relativistic superluminal quantum-vortex electron model internally crosses lightspeed

This article continues the development of the relativistic quantum-vortex electron and positron models composed of a helically-circulating electrically-charged superluminal energy quantum (SEQ). Parametric equations similar to that of a coiled slinky toy are proposed and analyzed for the relativistic electron and positron models, which can each have either spin-up or spin-down. The point-like electric charge of the SEQ moves along a positively or negatively-turning helical trajectory, depending on the sign of the electric charge, to form the electron/positron model. At low electron-model speeds, the SEQ moves along the surface of a mathematical horn torus that moves forward at the electron/positron model’s speed. At highly relativistic electron-positron speeds this horn-torus surface, along which the SEQ moves, transforms into a spherical surface whose radius decreases with increasing electron/positron speed. There is a range of speeds of the superluminal energy quantum for the relativistic electron/positron model. The maximum and minimum speeds range from superluminal to subluminal, passing easily through the speed of light. These speed ranges depend on the electron/positron speed and the combination of electric charge and spin direction of the electron/positron model.

H-Refinement method for toric parameterization of planar multi-sided computational domain in isogeometric analysis

Toric surface patches are a class of multi-sided surface patches that can represent multi-sided domains without mesh degeneration. In this paper, we propose an improved subdivision algorithm for toric surface patches, which subdivides an N-sided toric surface patch into N rational tensor product Bézier surface patches. By the proposed subdivision algorithm, a Ck-continuous spline surface composed of piecewise toric surface patches is subdivided into a set of rational tensor product Bézier surface patches with Gk-continuity. Additionally, after subdivision, toric surface patches are compatible with CAD systems. Combining the subdivision algorithm with the classical knot insertion algorithm of non-uniform rational B-splines, we develop a novel h-refinement scheme for isogeometric analysis with planar toric parameterizations. Several numerical examples are given to demonstrate the effectiveness and numerical stability of the presented method.

Analytical regularities of inversionless superradiance

The article reports the results of a theoretical study of superradiance of three-level optical systems with a doublet in the ground state (Λ-scheme) placed in a high-quality cavity. Hyperbolic chaos and unpredictable dynamic movements of the system appear on the surface of a multidimensional torus without dissipative losses in the conditions of superradiance without population inversion. The study resulted in the development of conservation laws to reduce the dimension of the phase space. An analytical result is obtained for the special case of a degenerate doublet.