Algebraic Surfaces Research Articles

Asymptotic Behavior of a Parametric Algebraic Surface

Starting from the concept of infinite branches and approximation surfaces, we present a method to compute infinite branches and surfaces having the same asymptotic behavior as an input parametric surface. The results obtained in this work represent a breakthrough for the study of surfaces and their applications.

Limit Cycles from Perturbed Center on the Invariant Algebraic Surface of Unified Lorenz-Type System

For a three-dimensional chaotic system, little seems to be known about the perturbation of invariant algebraic surface and the center on this surface. This question is very interesting and worth investigating. This paper is devoted to analyzing the limit cycles from perturbed center (trivial and nontrivial equilibria) on the invariant algebraic surface of the unified Lorenz-type system (ULTS), which contains some common chaotic systems as its particular cases. First, based on the parameter-dependent center manifold, we obtain the approximate two-dimensional center manifold from the perturbation of invariant algebraic surface, as well as the two-dimensional system on this center manifold. Second, by applying the averaging method of third order to the above two-dimensional system, we show that under suitable perturbation of parameters of the ULTS, there is one limit cycle bifurcating from the perturbed center on the invariant algebraic surface of the ULTS, and the stability of this limit cycle is determined as well. By using the averaging method of fourth order, we show the same results with the averaging method of third order. Finally, numerical simulation is used to verify the theoretical analyses.

Stability of Syzygy bundles corresponding to stable vector bundles on algebraic surfaces

Let (X,H) be a polarized smooth projective algebraic surface and E is globally generated, stable vector bundle on X. Then the Syzygy bundle ME associated to it is defined as the kernel bundle corresponding to the evaluation map. In this article we will study the stability property of ME with respect to H.

On K3 surfaces of Picard rank 14

AbstractWe study complex algebraic K3 surfaces with finite automorphism groups and polarized by rank 14, 2‐elementary lattices. Three such lattices exist, namely, , , and . As part of our study, we provide birational models for these surfaces as quartic projective hypersurfaces and describe the associated coarse moduli spaces in terms of suitable modular invariants. Additionally, we explore the connection between these families and dual K3 families related via the Nikulin construction.

Unimodal singularities and boundary divisors in the KSBA moduli of a class of Horikawa surfaces

AbstractSmooth minimal surfaces of general type with , , and constitute a fundamental example in the geography of algebraic surfaces, and the 28‐dimensional moduli space of their canonical models admits a modular compactification via the minimal model program. We describe eight new irreducible boundary divisors in such compactification parameterizing reducible stable surfaces. Additionally, we study the relation with the GIT compactification of and the Hodge theory of the degenerate surfaces that the eight divisors parameterize.

Improved Elekes-Szabó type estimates using proximity

We prove a new Elekes-Szabó type estimate on the size of the intersection of a Cartesian product A×B×C with an algebraic surface {f=0} over the reals. In particular, if A,B,C are sets of N real numbers and f is a trivariate polynomial, then either f has a special form that encodes additive group structure (for example, f(x,y,x)=x+y−z), or A×B×C∩{f=0} has cardinality O(N12/7). This is an improvement over the previous bound O(N11/6). We also prove an asymmetric version of our main result, which yields an Elekes-Ronyai type expanding polynomial estimate with exponent 3/2. This has applications to questions in combinatorial geometry related to the Erdős distinct distances problem.Like previous approaches to the problem, we rephrase the question as an L2 estimate, which can be analyzed by counting additive quadruples. The latter problem can be recast as an incidence problem involving points and curves in the plane. The new idea in our proof is that we use the order structure of the reals to restrict attention to a smaller collection of proximate additive quadruples.

On the Monodromy Manifold of q-Painlevé VI and Its Riemann–Hilbert Problem

We study the q-difference sixth Painlevé equation (qtext {P}_{text {VI}}) through its associated Riemann–Hilbert problem (RHP) and show that the RHP is always solvable for irreducible monodromy data. This enables us to identify the solution space of qtext {P}_{text {VI}} with a monodromy manifold for generic parameter values. We deduce this manifold explicitly and show it is a smooth and affine algebraic surface when it does not contain reducible monodromy. Furthermore, we describe the RHP for reducible monodromy data and show that, when solvable, its solution is given explicitly in terms of certain orthogonal polynomials yielding special function solutions of qtext {P}_{text {VI}}.

On the holomorphic convexity of reductive Galois coverings over compact Kähler surfaces

This article generalizes the result of Katzarkov and Ramachandran from algebraic surfaces to Kähler surfaces. We follow their argument to prove the holomorphic convexity of a reductive Galois covering over a compact Kähler surface which does not have two ends, except that we replace the p-adic factorization theorem by an analysis of the singularities of the continuous subanalytic plurisubharmonic exhaustion function.

A Riemann–Hilbert method to algebro-geometric solutions of the Korteweg–de Vries equation

A proof without using Dubrovin’s equations is given for the Its–Matveev formula for algebro-geometric solutions of the Korteweg–de Vries (KdV) equation. It is shown that the Baker–Akhiezer (BA) function of the KdV equation can be described in terms of two types of holomorphic vector Riemann–Hilbert problems with σ1-symmetry condition in the complex k-plane. Further, a sufficient condition on determining whether the algebro-geometric KdV solution has mKdV-relevant property is presented. This provides an explanation for the classical Miura transformation in terms of the deformations of associated algebro-geometric datum. Our main tools include algebraic curve and Riemann surface, Riemann theta function, Riemann–Hilbert problem and associated deformation procedure.

Real tight contact structures on lens spaces and surface singularities

We give a partial classification for the real tight contact structures on solid tori up to equivariant contact isotopy and apply the results to the classification of real tight structures on [Formula: see text] and real lens spaces [Formula: see text]. We prove that there is a unique real tight [Formula: see text] and [Formula: see text]. We show that there is at most one real tight [Formula: see text] with respect to one of its two possible real structures. With respect to the other we give lower and upper bounds for the count. To establish lower bounds we explicitly construct real tight manifolds through equivariant contact surgery, real open book decompositions and isolated real algebraic surface singularities. As a by-product we observe the existence of an invariant torus in an [Formula: see text] which cannot be made convex equivariantly.

Dynamics of Chebyshev endomorphisms on some affine algebraic varieties

The Chebyshev polynomials Td in one variable are typical chaotic maps on C. Chebyshev endomorphisms PAnd: Cn→Cn are also chaotic. We consider the action of the dihedral group Dn+1 on Cn. The endomorphism PAnd maps any Dn+1-orbit of z∈Cn to a Dn+1-orbit of PAnd(z). The endomorphism PAnd induces a mapping on Cn∕Dn+1. Using invariant theory, we embed Cn∕Dn+1 as an affine subvariety X in Cm. Then we have morphisms gd on X. We study the cases n=2 and 3. In these cases, the morphisms gd are defined over Z. We find a class of affine subvarieties V of X which are invariant under gd. These varieties are concerned with branch loci or critical loci. The class contains C2, a cuspidal cubic, a parabola, a quadric hypersurface in C4, an affine algebraic surface in C4 which is birationally equivalent to an affine quadric cone in C3, and others. For each affine variety V in the class, there exists a polynomial parameterization PV satisfying gd|V(PV(y1,…,yk))=PV(Td(y1),…,Td(yk)), where Td(z) is a Chebyshev polynomial in one variable. Then we determine the set of bounded orbits of gd|V in each invariant set V and give relations between them.

Examples of surfaces with canonical maps of degree 12, 13, 15, 16 and 18

In this note, we present examples of complex algebraic surfaces with canonical maps of degree 12, 13, 15, 16 and 18. They are constructed as quotients of a product of two curves of genus 10 and 19 using certain non-free actions of the group S3×Z32\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_3\ imes {\\mathbb {Z}}_3^2$$\\end{document}. To our knowledge, there are no other examples in the literature of surfaces with canonical map of degree 13, 15 and 18.

Smooth Q$\mathbb {Q}$‐homology planes satisfying the negativity conjecture

AbstractA complex algebraic surface is a ‐homology plane if for . The Negativity Conjecture of Palka asserts that , where is a log smooth completion of . We give a complete description of smooth ‐homology planes satisfying the Negativity Conjecture. We restrict our attention to those of log general type, as otherwise their geometry is well‐understood. We show that, as conjectured by tom Dieck and Petrie, they can be arranged in finitely many discrete series, each obtained in a uniform way from an arrangement of lines and conics on . We infer that these surfaces satisfy the Rigidity Conjecture of Flenner and Zaidenberg; and a conjecture of Koras, which asserts that .

Some algebraic surfaces with canonical map of degree 10, 12, 14

Surfaces of general type with canonical map of degree d bigger than 8 have bounded geometric genus and irregularity. In particular the irregularity is at most 2 if . In the present paper, the existence of surfaces with d = 10 and all possible irregularities, surfaces with d = 12 and irregularity 1 and 2, and surfaces with d = 14 and irregularity 0 and 1 is proven, by constructing these surfaces as -covers of certain rational surfaces. These results together with the construction by C. Rito of a surface with d = 12 and irregularity 0 show that all the possibilities for the irregularity in the cases d = 10, d = 12 can occur, whilst the existence of a surface with d = 14 and irregularity 2 is still an open problem.

Complex dynamics of a sub-quadratic Lorenz-like system

Abstract Motivated by the generic dynamical property of most quadratic Lorenz-type systems that the unstable manifolds of the origin tending to the stable manifold of nontrivial symmetrical equilibria forms a pair of heteroclinic orbits, this technical note reports a new 3D sub-quadratic Lorenz-like system: x ˙ = a ( y − x ) \dot{x}=a(y-x) , y ˙ = c x 3 + d y − x 3 z \dot{y}=c\sqrt[3]{x}+{\rm{d}}y-\sqrt[3]{x}z and z ˙ = − b z + x 3 y \dot{z}=-bz+\sqrt[3]{x}y . Instead, the unstable manifolds of nontrivial symmetrical equilibria tending to the stable manifold of the origin creates a pair of heteroclinic orbits. This drives one to further investigate it and reveal its other hidden dynamics: Hopf bifurcation, invariant algebraic surfaces, ultimate bound sets, globally exponentially attractive sets, existence of homoclinic and heteroclinic orbits, singularly degenerate heteroclinic cycles, and so on. The main contributions of this work are summarized as follows: First, the ultimate boundedness of that system yields the globally exponentially attractive sets of it. Second, the existence of another heteroclinic orbits is also proved by utilizing two different Lyapunov functions. Finally, on the invariant algebraic surface z = 3 4 a x 4 3 z=\frac{3}{4a}\sqrt[3]{{x}^{4}} , the existence of a pair of homoclinic orbits to the origin, and two pairs of heteroclinic orbits to two pairs of nontrivial symmetrical equilibria is also proved by utilizing a Hamiltonian function. In addition, the correctness of the theoretical results is illustrated via numerical examples.

Some surfaces with canonical maps of degrees 10, 11, and 14

AbstractIn this note, we present examples of complex algebraic surfaces of general type with canonical maps of degrees 10, 11, and 14. They are constructed as quotients of a product of two Fermat septics using certain free actions of the group .

Computing symmetries of implicit algebraic surfaces

We present a complete algorithm to compute the rotational, axial, reflectional and central symmetries of an algebraic surface defined by means of its implicit equation. The algorithms rely on the fact that the symmetries of the surface satisfy an algebraic condition verified by the polynomial defining the surface, and use a variety of ideas, some of them extensions to space of ideas already used to compute the symmetries of planar implicit curves, and some of them new. We flesh out our algorithms with several examples.

Chow dilogarithm and strong Suslin reciprocity law

We prove a conjecture of A. Goncharov concerning strong Suslin reciprocity law. The main idea of the proof is the construction of the norm map on so-called lifted reciprocity maps. This construction is similar to the construction of the norm map on Milnor K K -theory. As an application, we express Chow dilogarithm in terms of Bloch-Wigner dilogarithm. Also, we obtain a new reciprocity law for four rational functions on an arbitrary algebraic surface with values in the pre-Bloch group.

Erratum: Topological characterization of ℂ2 via open algebraic surface theory

International Journal of MathematicsOnline Ready Free AccessErratum: Topological characterization of ℂ2 via open algebraic surface theoryis erratum ofTopological characterization of C2 via open algebraic surface theoryR. V. Gurjar and S. GurjarR. V. GurjarDepartment of Mathematics, Indian Institute of Technology, Bombay, Mumbai 400076, IndiaCorresponding author. and S. GurjarDepartment of Mathematics, Indian Institute of Technology, Bombay, Mumbai 400076, Indiahttps://doi.org/10.1142/S0129167X23920015Cited by:0 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail [Int. J. Math. Vol. 31, No. 7 (2020) Article ID: 2050055, DOI: 10.1142/S0129167X2050055X] FiguresReferencesRelatedDetailsRelated articlesTopological characterization of C2 via open algebraic surface theory2 Jun 2020International Journal of Mathematics Recommended Online Ready Metrics History Published: 28 April 2023 PDF download

A K3 surface related to Leonardo Pisano’s work on congruent numbers

This note recalls an early 13th century result on congruent numbers by Leonardo Pisano (“Fibonacci”), and shows how it relates to a specific much studied K3 surface and to an elliptic fibration on this surface. As an aside, the discussion reveals how, via explicit maps of degree two, the surface is covered by the Fermat quartic surface and also covers one of the two famous ‘most algebraic K3 surfaces’.