Rational Curves Research Articles

Mirrors to Del Pezzo Surfaces and the Classification of $T$-Polygons

We give a new geometric proof of the classification of $T$-polygons, a theorem originally due to Kasprzyk, Nill and Prince, using ideas from mirror symmetry. In particular, this gives a completely geometric proof that any two toric $\mathbb{Q}$-Gorenstein degenerations of a smooth del Pezzo $X$ surface are connected via trees of rational curves in the moduli space of $X$.

Rational Curves on Moduli Spaces of Vector Bundles

Rational Curves on Moduli Spaces of Vector Bundles

Derived deformation theory of crepant curves

AbstractThis paper determines the full, derived deformation theory of certain smooth rational curves in Calabi–Yau 3‐folds, by determining all higher ‐products in its controlling DG‐algebra. This geometric setup includes very general cases where does not contract, cases where the curve neighbourhood is not rational, all known simple smooth 3‐fold flops, and all known divisorial contractions to curves. As a corollary, it is shown that the non‐commutative deformation theory of is described via a superpotential algebra derived from what we call free necklace polynomials, which are elements in the free algebra obtained via a closed formula from combinatorial gluing data. The description of these polynomials, together with the above results, establishes a suitably interpreted string theory prediction due to Ferrari (Adv. Theor. Math. Phys. 7 (2003) 619–665), Aspinwall–Katz (Comm. Math. Phys.. 264 (2006) 227–253) and Curto–Morrison (J. Algebraic Geom. 22 (2013) 599–627). Perhaps most significantly, the main results give both the language and evidence to finally formulate new contractibility conjectures for rational curves in CY 3‐folds, which lift Artin's (Amer. J. Math. 84 (1962) 485–496) celebrated results from surfaces.

Exploring topological indices and entropy measure via rational curve fitting models for calcium hydroxide network

This work uses rational curve fitting models to investigate the complex link between topological indices and entropy metrics in calcium hydroxide (Ca(OH)2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(Ca(OH)_{2})$$\\end{document} networks. Entropy measurements shed light on the complexity and disorder of these networks, whereas topological indices are essential instruments for comprehending the structural characteristics of chemical substances. Our goal is to find new patterns and connections by merging these two fields, improving materials science’s prediction capacity. We calculate many topological indices, such as the Randic, Balaban, and Zagreb indices, and examine the relationship between them and entropy measurements obtained from (Ca(OH)2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(Ca(OH)_{2})$$\\end{document} structural arrangement.

Castelnuovo–Mumford regularity and splitting criteria for logarithmic bundles over rational normal scroll surfaces

We introduce and study a notion of Castelnuovo–Mumford regularity suitable for rational normal scroll surfaces. In this setting we prove analogs of some classical properties. We prove splitting criteria for coherent sheaves and a characterization of Ulrich bundles. Finally we study logarithmic bundles associated to arrangements of lines and rational curves.

The non-degeneracy invariant of Brandhorst and Shimada’s families of Enriques surfaces

Brandhorst and Shimada described a large class of Enriques surfaces, called (\tau,\bar{\tau}) -generic, for which they gave generators for the automorphism groups and calculated the elliptic fibrations and the smooth rational curves up to automorphisms. In the present paper, we give lower bounds for the non-degeneracy invariant of such Enriques surfaces, we show that in most cases the invariant has generic value 10 , and we present the first known example of complex Enriques surface with infinite automorphism group and non-degeneracy invariant not equal to 10 .

Tropical Refined Curve Counting with Descendants

We prove a q-refined tropical correspondence theorem for higher genus descendant logarithmic Gromov–Witten invariants with a λg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _g$$\\end{document} class in toric surfaces. Specifically, a generating series of such logarithmic Gromov–Witten invariants agrees with a q-refined count of rational tropical curves satisfying higher valency conditions. As a corollary, we obtain a geometric proof of the deformation invariance of this tropical count. In particular, our results give an algebro-geometric meaning to the tropical count defined by Blechman and Shustin. Our strategy is to use the logarithmic degeneration formula, and the key new technique is to reduce to computing integrals against double ramification cycles and connect these integrals to the non-commutative KdV hierarchy.

Inverse Relations for Blossoms and Parametrisations of Rational Curves and Surfaces

Inverse Relations for Blossoms and Parametrisations of Rational Curves and Surfaces

The method for comparing different blade shapes of flat knives of a rotary cultivator

BACKGROUND: In the process of tillage, flat knives of rotary cultivators are to carry out the process of cutting soil and weeds with minimal energy consumption and not to become clogged with weeds, which is due to the compliance of the parameters of their blades with operating conditions. However, despite the abundance of works devoted to the study of the working process of flat knives of rotary cultivators, there is not any method of a comparative assessment of the various shapes of their blades. AIM: Development of the method for comparing different blade shapes of flat knives of rotary cultivators. METHODS: The research design included studying the following issues: clarifying the conditions for weeds to fall off the curved blade of a flat knife performing a plane-parallel movement; determining the equation of the curve of a knife blade that satisfies the condition of weeds to fall off the blade; development of methods for comparative analysis of the energy intensity of the process of cutting soil with flat knives with blades of various shapes; experimental verification of theoretical conceptions. The study objects were the processes of interaction of weeds and soil with the blades of flat knives of various shapes. The studies were conducted during 2022 and 2023. RESULTS: The condition for weeds to fall off the curved blade of a flat knife performing a plane-parallel movement has been clarified (a more informative form of notation), based on which an equation for the rational curve of the blade has been obtained. The comprehensive method has been developed for comparing different shapes of blades of flat knives of rotary cultivators, according to the criteria of compliance with the condition of weeds to fall off the blade and the energy intensity of soil cutting. Examples of the application of the method for the analysis of various shapes of blades of flat knives of rotary cultivators and the development of one of the effective shapes, as well as the results of experimental studies are given. CONCLUSION: The proposed method helps to give specific practical recommendations on the advisability of using knives of one shape or another in the operating conditions under consideration, taking into account the operating parameters of the machines, as well as to create effective combined blades.

Rational conic fibrations containing special curves transverse to the fibers

Rational conic fibrations embedded in P N and containing an irreducible curve C transverse to the fibers are classified when C is either a plane curve or a rational normal curve of any degree d. This extends our previous results concerning lines, conics and twisted cubics. To emphasize the role of linear normality, the case in which C is a non-normal rational quartic is also discussed. Moreover, the rank of the d-th jet map of the surface at all points lying on such curves is evaluated.

From computed tomography to 3D printed heterogeneous scaffolds: A novel approach to simulating bone characteristics using octree adaptive subdivision

Bone microstructures or scaffolds mimicking natural bone characteristics find widespread applications in bone tissue engineering (BTE). To accurately replicate a personalized bone structure, we developed a heterogeneous scaffold by combining medical image data with the octree adaptive subdivision technique. The transformations between Hounsfield units (HU)-strength and strength-lattice features were implemented for mapping bone information to the scaffold. Initially, HU values from computed tomography (CT) images were smoothly converted into corresponding strengths through a rational Bezier curve. Then, these strengths were further translated into the features of lattice structures, specifically the strut diameter and subdivision level. The novel concept of adaptive octree subdivision, executed on the lattices, allows for controlling bone porosity and mechanical property even at local levels. As a result, a personalized heterogeneous scaffold accurately simulating natural bone structure was designed and manufactured for bone grafting application on a 3D-printed alveolar model. The scaffold has an outer cortical bone layer capable of withstanding pressures of up to 208.2 MPa, while the inner cancellous bone exhibits a high porosity ranging from 77.6 % to 93.6 %. The research provides an elegant yet effective approach to the generation of bone microstructures, opening up practical BTE applications.

Enumeration of non-nodal real plane rational curves

Welschinger invariants enumerate real nodal rational curves in the plane or in another real rational surface. We analyze the existence of similar enumerative invariants that count real rational plane curves having prescribed non-nodal singularities and passing through a generic conjugation-invariant configuration of appropriately many points in the plane. We show that an invariant like this is unique: it enumerates real rational three-cuspidal quartics that pass through generically chosen four pairs of complex conjugate points. As a consequence, we show that through any generic configuration of four pairs of complex conjugate points, one can always trace a pair of real rational three-cuspidal quartics.

Robust Containment Queries over Collections of Rational Parametric Curves via Generalized Winding Numbers

Point containment queries for regions bound by watertight geometric surfaces, i.e., closed and without self-intersections, can be evaluated straightforwardly with a number of well-studied algorithms. When this assumption on domain geometry is not met, such methods are either unusable, or prone to misclassifications that can lead to cascading errors in downstream applications. More robust point classification schemes based on generalized winding numbers have been proposed, as they are indifferent to these imperfections. However, existing algorithms are limited to point clouds and collections of linear elements. We extend this methodology to encompass more general curved shapes with an algorithm that evaluates the winding number scalar field over unstructured collections of rational parametric curves. In particular, we evaluate the winding number for each curve independently, making the derived containment query robust to how the curves are arranged. We ensure geometric fidelity in our queries by treating each curve as equivalent to an adaptively constructed polyline that provably has the same generalized winding number at the point of interest. Our algorithm is numerically stable for points that are arbitrarily close to the model, and explicitly treats points that are coincident with curves. We demonstrate the improvements in computational performance granted by this method over conventional techniques as well as the robustness induced by its application.

Rationality and parametrizations of algebraic curves under specializations

Rational algebraic curves have been intensively studied in the last decades, both from the theoretical and applied point of view. In applications (e.g. level curves, linear homotopy deformation, geometric constructions in computer aided design, image detection, algebraic differential equations, etc.), there often appear unknown parameters. It is possible to adjoin these parameters to the coefficient field as transcendental elements. In some particular cases, however, the curve has a different behavior than in the generic situation treated in this way. In this paper, we show when the singularities and thus the (geometric) genus of the curves might change. More precisely, we give a partition of the affine space, where the parameters take values, so that in each subset of the partition the specialized curve is either reducible or its genus is invariant. In particular, we give a Zariski-closed set in the space of parameter values where the genus of the curve under specialization might decrease or the specialized curve gets reducible. For the genus zero case, and for a given rational parametrization, a finer partition is possible such that the specialization of the parametrization parametrizes the specialized curve. Moreover, in this case, the set of parameters where Hilbert's irreducibility theorem does not hold can be identified. We conclude the paper by illustrating these results by some concrete applications.

Inverse Geometric Reconstruction Based on MW-NURBS Curves

Currently, rational curves such as the Non-Uniform Rational B-Spline (NURBS) play a significant role in both shape representation and shape reconstruction. NURBS weights are often real in nature and are referred to as challenging to assign, with the exception of conics. ‘Matrix Weighted Rational Curves’ are the expanded form of rational curves that result from replacing these real weights with matrices, or matrix weights. The only difference between these curves and conventional curves is the geometric definition of the matrix weights. In this paper, MW-NURBS curves are used to reconstruct space curves from their stereo perspectives. In particular, MW-NURBS fitting is carried out in stereo views, and the weight matrices for the MW-NURBS curves are produced using the normal vectors provided at the control points. Instead of needing to solve a complicated system, the MW-NURBS model can reconstruct curves by choosing control points and control normals from the input data. The efficacy of the proposed strategy is verified by using many examples based on both synthetic and real images. The various error types are compared to those of conventional methods like point-based and NURBS-based approaches. The results demonstrate that the errors acquired from the proposed approach are much fewer than those obtained from the point-based method and the NURBS-based method.

Rational curves and Seshadri constants on Enriques surfaces

We prove that classes of rational curves on very general Enriques surfaces are always 2 2 -divisible. As a consequence, we compute the Seshadri constant of any big and nef line bundle on a very general Enriques surface, proving that it coincides with the value of the ϕ \phi -function introduced by Cossec [Math. Ann. 271 (1985), pp. 577–600].

Motivic Geometry of two-Loop Feynman Integrals

Abstract We study the geometry and Hodge theory of the cubic hypersurfaces attached to two-loop Feynman integrals for generic physical parameters. We show that the Hodge structure attached to planar two-loop Feynman graphs decomposes into mixed Tate pieces and the Hodge structures of families of hyperelliptic, elliptic or rational curves depending on the space-time dimension. For two-loop graphs with a small number of edges, we give more precise results. In particular, we recover a result of Bloch (Double box motive. SIGMA 2021;17,048) that in the well-known double-box example, there is an underlying family of elliptic curves, and we give a concrete description of these elliptic curves. We show that the motive for the non-planar two-loop tardigrade graph is that of a K3 surface. In an appendix by Eric Pichon-Pharabod, we argue via high-precision numerical computations that the Picard number of this K3 surface is generically 11 and we compute the expected lattice polarization. Lastly, we show that generic members of the ice cream cone family of graph hypersurfaces correspond to the pairs of sunset Calabi–Yau varieties.

Application of X-ray principles to quantify sulfur oxidation states in concrete

This study introduces a novel mathematical approach rooted in X-ray spectroscopy principles to evaluate the oxidation state of sulfur, which is then applied in concrete and cementitious materials. Unlike previous studies that relied on an empirical quadratic relationship based on the ratios of characteristic X-ray lines (SKα, SKβ, and SKβ') for sulfur oxidation state determination, unique linear relationships between the intensities of the three X-ray lines for each oxidation state and total sulfur were utilized. Rational equations are then derived by combining these linear relationships for each species. Two independent rational calibration curves are constructed using the SKβ/SKα and SKβ'/SKβ ratios for samples containing sulfide-sulfate mixtures, irrespective of total sulfur concentration within the range of 0.3% to 2.0% by weight of concrete. These equations demonstrated interchangeability across three simulated matrices (Portland cement, concrete, and silica) and two sulfide species (pyrrhotite and pyrite). Absolute quantification of sulfide and sulfate species in concrete is achieved by independently measuring total sulfur. Compared to previous quadratic and linear equations, this approach is shown to minimize the difference between the experimental and calculated ratios of the calibration samples. Agreement was also observed between the sulfate ratio calculated through the rational approach and gravimetric analysis for concrete samples.

Optimization of machining path for integral impeller side milling based on SA-PSO fusion algorithm in CNC machine tools

The five axis linkage Computer Numerical Control machine tool for integral impeller can achieve blade machining through side milling, which is of great significance for improving the machining accuracy, production efficiency, and long-term stability of integral impeller blades. This study is based on non-uniform rational B-spline curves and aims to reduce the surface over cutting or under cutting of integral turbine blades. The path planning of non deployable ruled surfaces was analyzed in depth through side milling, and the path planning model of the side milling cutter axis was solved through a fusion algorithm of simulated annealing algorithm and particle swarm optimization algorithm, in order to find the optimal path through iterative process. As the number of iterations increased, the error values of particle swarm optimization algorithm and simulated annealing particle swarm optimization fusion algorithm gradually decreased, with convergence times of about 7 and 6, respectively. The stable error value of the fusion algorithm was 0.253, which is 30.45% lower than that of the particle swarm optimization algorithm. The optimal number of iterations for solving the model using particle swarm optimization algorithm and fusion algorithm was the 7th, with range values of 0.0213 and 0.0165 mm, respectively. The tool axis trajectory surface optimized by the fusion algorithm was closer to the tool axis motion state compared to the initial tool axis trajectory surface. The range of the sum of mean squared deviations for single and global cutting was 0.0011–0.0198 and 0.046–0.0341, but the overall error value was relatively small. This study effectively reduces the envelope error of machining tools and improves machining accuracy, thereby solving the principle error of non expandable ruled surfaces in the motion trajectory of the blade axis of the integral turbine. This provides new research ideas for the intelligent development of Computer Numerical Control machining technology.

Three paths to rational curves with rational arc length

We solve the so far open problem of constructing all spatial rational curves with rational arc length functions. More precisely, we present three different methods for this construction. The first method adapts a recent approach of (Kalkan et al. 2022) to rational PH curves and requires solving a modestly sized system of linear equations. The second constructs the curve by imposing zero-residue conditions, thus extending ideas of previous papers by (Farouki and Sakkalis 2019) and the authors themselves (Schröcker and Šír 2023). The third method generalizes the dual approach of (Pottmann 1995) from planar to spatial curves. The three methods share the same quaternion based representation in which not only the PH curve but also its arc length function are compactly expressed. We also present a new proof based on the quaternion polynomial factorization theory of the well known characterization of the Pythagorean quadruples.