Algebraic Surfaces Research Articles

CMC surfaces of revolution, elliptic curves, Weierstrass-$$\wp $$ functions, and algebraicity

AbstractThis paper establishes an interesting connection between the family of CMC surfaces of revolution in $$\mathbb {E}_1^3$$ E 1 3 and some specific families of elliptic curves. As a consequence of this connection, we show in the class of spacelike CMC surfaces of revolution in $$\mathbb {E}_1^3$$ E 1 3 , only spacelike cylinders and standard hyperboloids are algebraic. We also show that a similar connection exists between CMC surfaces of revolution in $$\mathbb E^3$$ E 3 and elliptic curves. Further, we use this to reestablish the fact that the only CMC algebraic surfaces of revolution in $$\mathbb E^3$$ E 3 are spheres and right circular cylinders.

Correction to: H-stability of syzygy bundles on some regular algebraic surfaces

Correction to: H-stability of syzygy bundles on some regular algebraic surfaces

Rational surfaces with a non‐arithmetic automorphism group

AbstractIn [Algebraic surfaces and hyperbolic geometry, Cambridge University Press, Cambridge, 2012], Totaro gave examples of a K3 surface such that its automorphism group is not commensurable with an arithmetic group, answering a question of Mazur in Section 7 of [Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 14–50]. We give examples of rational surfaces with the same property. Our examples are Looijenga pairs, that is, there is a connected singular nodal curve such that .

Investigation of torsional and fatigue resistance properties based on triply periodic minimal surfaces (TPMS)

Abstract Triply periodic minimal surfaces (TPMS) are algebraic surfaces found in nature, defined by implicit functions, and exhibit exceptional properties in various fields such as mechanics, computer graphics, and tissue engineering. This study focuses on four common minimal surface structures: Gyroid, Diamond, I-WP, and Schwarz P structures, and presents two methods for converting these surfaces into TPMS cells. These four surface structures are converted into TPMS cells, which are then organized into TPMS cylindrical units for finite element analysis. Through a comparison of the torsional mechanical properties of different minimal surfaces using the finite element method, the stress-strain curves of TPMS were generated. The findings suggest that the Schwarz P cylindrical structure remains within the elastic deformation stage under the same load, with a stress-load curve exhibiting a smaller slope, indicating superior torsional resistance. Additionally, a fatigue strength analysis of various minimal surface structures revealed that, under equivalent load conditions, the fatigue life of the Schwarz P cylindrical structure tends towards infinity.

Convex hulls of surfaces in fourspace

AbstractThis is a case study of the algebraic boundary of convex hulls of varieties. We focus on surfaces in fourspace to showcase new geometric phenomena that neither curves nor hypersurfaces exhibit. Our method is a detailed analysis of a general purpose formula by Ranestad and Sturmfels in the case of smooth real algebraic surfaces of low degree (that are rational over the complex numbers). We study both the complex and the real features of the algebraic boundary of Veronese and Del Pezzo surfaces. The main difficulties and the possible approaches to the case of general surfaces are discussed for and complemented by the example of Bordiga surfaces.

Connected algebraic groups acting on algebraic surfaces

Connected algebraic groups acting on algebraic surfaces

Topics in group schemes and surfaces in positive characteristic

This paper is based on a series of talks that the author gave at the University of Edinburgh in March 2023 on surfaces in positive characteristic. In particular, infinitesimal group schemes and their actions on algebraic surfaces are discussed, the structure of the automorphism group scheme of a surface of general type and the failure of the Kodaira vanishing theorem.

Geometric Investigation of Three Thin Shells with Ruled Middle Surfaces with the Same Main Frame

It is proved and illustrated that by taking the main frame of the surface, consisting of three plane curves placed in three coordinate planes, three different algebraic surfaces with the same rigid frame can be designed. For the first time, one three of new ruled surfaces in a family of five threes of ruled surfaces, formed on the basis of some shapes of hulls of river and see ships, which, in turn, are projected in the form of algebraic surfaces with a main frame of three superellipses or of three other plane curves, is under consideration in detail with a standpoint of differential geometry. The geometrical properties of the ruled surfaces taken as the middle surfaces of thin shells for industrial and civil engineering are presented. Analytical formulas for determination of force resultants with using the approximate momentless theory of shells of zero Gaussian curvature given by non-orthogonal conjugate curvilinear coordinates are offered for the first time. The results derived using these formulae will help to correct the results obtained by numerical methods.

Dupin Cyclides Passing through a Fixed Circle

Dupin cyclides are classical algebraic surfaces of low degree. Recently, they have gained popularity in computer-aided geometric design (CAGD) and architecture owing to the fact that they contain many circles. We derive algebraic conditions that fully characterize the Dupin cyclides passing through a fixed circle. The results are applied to the basic problem in CAGD of the blending of Dupin cyclides along circles.

Revealing More Hidden Attractors from a New Sub-Quadratic Lorenz-Like System of Degree 6 5

In the sense that the descending powers of some certain variables may widen the range of parameters of self-excited and hidden attractors, this technical note proposes a new three-dimensional Lorenz-like system of degree [Formula: see text]. In contrast to the previously studied one of degree [Formula: see text], the newly reported one creates more hidden Lorenz-like attractors coexisting with the unstable origin and a pair of stable node-foci in a broader range of parameters, which confirms the generalization of the second part of the celebrated Hilbert’s 16th problem once more. In addition, some other dynamics, i.e. Hopf bifurcation, the generic and degenerate pitchfork bifurcation, invariant algebraic surface, first integral, singularly degenerate heteroclinic cycle with nearby chaotic attractor, ultimate bounded set and existence of a pair of heteroclinic orbits, are discussed.

Surfaces of General Type with Maximal Picard Number Near the Noether Line

Abstract The first published non-trivial examples of algebraic surfaces of general type with maximal Picard number are due to Persson, who constructed surfaces with maximal Picard number on the Noether line $K^{2}=2\chi -6$ for every admissible pair $(K^{2},\chi )$ such that $\chi \not \equiv 0 \ \text {mod}\ 6$. In this note, given a non-negative integer $k$, algebraic surfaces of general type with maximal Picard number lying on the line $K^{2}=2\chi -6+k$ are constructed for every admissible pair $(K^{2},\chi )$ such that $\chi \geq 2k+10$. These constructions, obtained as bidouble covers of rational surfaces, not only allow to fill in Persson’s gap on the Noether line, but also provide infinitely many new examples of algebraic surfaces of general type with maximal Picard number above the Noether line.

The Euler Characteristic of Parabolic Sheaves

Objectives: The primary aim of this study is to explicitly determine the Euler characteristic of the parabolic sheaves with rank 2 on a smooth projective algebraic surface defined over complex numbers with the smooth irreducible parabolic divisor . Methods: The computation of the parabolic Hilbert polynomial involves the use of -filtered sheaves on a smooth projective surface , with weights corresponding to the points where the filtration jumps. The Riemann-Roch theorem and Chern class computation have also been used. Findings: The study provides explicit computations of the parabolic Hilbert polynomial as well as the parabolic Chern classes for parabolic rank 2 bundles. Novelty: This work contributes to the understanding of parabolic sheaves on smooth projective surfaces, bridging the gap between different constructions of stable bundles. The explicit computation of the parabolic Hilbert polynomial for rank 2 bundles adds valuable insights to the study of moduli spaces of parabolic bundles. Keywords: Euler characteristic, Hilbert polynomial, Chern class, Parabolic sheaves, Smooth projective algebraic surface

A Numerical Method for Solving Elliptic Equations on Real Closed Algebraic Curves and Surfaces

A Numerical Method for Solving Elliptic Equations on Real Closed Algebraic Curves and Surfaces

K3 surfaces with two involutions and low Picard number

Let X be a complex algebraic K3 surface of degree 2d and with Picard number ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document}. Assume that X admits two commuting involutions: one holomorphic and one anti-holomorphic. In that case, ρ≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho \\ge 1$$\\end{document} when d=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=1$$\\end{document} and ρ≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho \\ge 2$$\\end{document} when d≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 2$$\\end{document}. For d=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=1$$\\end{document}, the first example defined over Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}$$\\end{document} with ρ=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho =1$$\\end{document} was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by Kondō, also defined over Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}$$\\end{document}, can be used to realise the minimum ρ=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho =2$$\\end{document} for all d≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 2$$\\end{document}. In these notes we construct new explicit examples of K3 surfaces over the rational numbers realising the minimum ρ=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho =2$$\\end{document} for d=2,3,4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2,3,4$$\\end{document}. We also show that a nodal quartic surface can be used to realise the minimum ρ=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho =2$$\\end{document} for infinitely many different values of d. Finally, we strengthen a result of Morrison by showing that for any even lattice N of rank 1≤r≤10\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le r \\le 10$$\\end{document} and signature (1,r-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1,r-1)$$\\end{document} there exists a K3 surface Y defined over R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}$$\\end{document} such that PicYC=PicY≅N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Pic}\\,}}Y_{\\mathbb {C}}={{\\,\ extrm{Pic}\\,}}Y \\cong N$$\\end{document}.

Algebraic Surfaces with Three Super Ellipses for Formation of Submarine Hull Surfaces

_ Definite stereotype for the outward look of submarines was worked out in the process of evolution. It is very difficult to take into account all inner and outside factors having an influence on the choice of submarine hull shape. Choosing either factor, designers’ model a great variety of outside lines. A light hull of a submarine is the outer nonwatertight hull which provides a hydrodynamically efficient shape. The pressure hull is the inner hull of a submarine; this holds the difference between outside and inside pressure. Closed algebraic surfaces with the main cross-sections in the form of three superellipses lying in three coordinate planes can help to select a submarine form during early-stage design. Having used a parallel middle body, one can extend considerably the choice of suitable shapes. In this paper, a method of modeling smooth compound closed algebraic surfaces that approximates the outside lines of a submarine is presented. The authors offer to form an outside submarine hull from six fragments of algebraic surfaces. The whole hull surface and its fragments are given by the same parametric equations. A method is illustrated in three examples, and it is realized easily in the form of a computer program. The initial data contain large quantities of constants, and it gives an opportunity to consider an infinitely large quantity of variants of submarine hull surfaces. Keywords submarine hull surface; superellipse; compound algebraic surface; buttock line; waterline; midship section

Dynamics of Fricke–Painlevé VI Surfaces

The symmetries of a Riemann surface Σ∖{ai} with n punctures ai are encoded in its fundamental group π1(Σ). Further structure may be described through representations (homomorphisms) of π1 over a Lie group G as globalized by the character variety C=Hom(π1,G)/G. Guided by our previous work in the context of topological quantum computing (TQC) and genetics, we specialize on the four-punctured Riemann sphere Σ=S2(4) and the ‘space-time-spin’ group G=SL2(C). In such a situation, C possesses remarkable properties: (i) a representation is described by a three-dimensional cubic surface Va,b,c,d(x,y,z) with three variables and four parameters; (ii) the automorphisms of the surface satisfy the dynamical (non-linear and transcendental) Painlevé VI equation (or PVI); and (iii) there exists a finite set of 1 (Cayley–Picard)+3 (continuous platonic)+45 (icosahedral) solutions of PVI. In this paper, we feature the parametric representation of some solutions of PVI: (a) solutions corresponding to algebraic surfaces such as the Klein quartic and (b) icosahedral solutions. Applications to the character variety of finitely generated groups fp encountered in TQC or DNA/RNA sequences are proposed.

Detecting isometries and symmetries of implicit algebraic surfaces

<abstract><p>We presented a new and complete algorithm for detecting isometries and symmetries of implicit algebraic surfaces. First, our method reduced the problem to the case of isometries fixing the origin. Second, using tools from elimination theory and polynomial factoring, we determined the desired isometries between the surfaces. We have implemented the algorithm in Maple to provide evidences of the efficiency of the method.</p></abstract>

Minimal compactifications of the affine plane with nef canonical divisors

We shall consider minimal analytic compactifications of the affine plane with singularities. In previous work, Kojima and Takahashi proved that any minimal analytic compactification of the affine plane, which has at worse log canonical singularities, is a numerical del Pezzo surface (i.e., a normal complete algebraic surface with the numerically ample anti-canonical divisor) and has only rational singularities. In this article, we show that any minimal analytic compactification of the affine plane with the nef canonical divisor has an irrational singularity.

Detecting and Parametrizing Polynomial Surfaces without Base Points

Given an algebraic surface implicitly defined by an irreducible polynomial, we present a method that decides whether or not this surface can be parametrized by a polynomial parametrization without base points and, in the affirmative case, we show how to compute this parametrization.

Existence of Ulrich bundle on some surfaces of general type

Let X be a smooth projective algebraic surface of Picard rank one with very ample canonical bundle KX. We further assume that q+1≤χ(OX). In this article, we will study the existence of an Ulrich bundle and its stability property of it with respect to KX.