Algebraic Surfaces Research Articles

Automorphism groups of rational elliptic and quasi-elliptic surfaces in all characteristics

We study the groups of automorphisms of rational algebraic surfaces that admit a relatively minimal pencil of curves of arithmetic genus one over an algebraically closed field of arbitrary characteristic. In particular, we classify such surfaces that admit non-trivial automorphisms that act trivially on the Picard group. As an application, we classify classical Enriques surfaces in characteristic 2 that admit non-trivial numerically trivial automorphisms.

Quasi-regular Sasakian and K-contact structures on Smale–Barden manifolds

Smale–Barden manifolds are simply-connected closed $5$-manifolds. It is an important and difficult question to decide when a Smale–Barden manifold admits a Sasakian or a K-contact structure. The known constructions of Sasakian and K-contact structures are obtained mainly by two techniques. These are either links (Boyer and Galicki), or semi-regular Seifert fibrations over smooth orbifolds (Kollár). Recently, the second named author of this article started the systematic development of quasi-regular Seifert fibrations, that is, over orbifolds which are not necessarily smooth. The present work is devoted to several applications of this theory. First, we develop constructions of a Smale–Barden manifold admitting a quasi-regular Sasakian structure but not a semi-regular K-contact structure. Second, we determine all Smale–Barden manifolds that admit a null Sasakian structure. Finally, we show a counterexample in the realm of cyclic Kähler orbifolds to the algebro-geometric conjecture by Muñoz, Rojo and Tralle that claims that for an algebraic surface with $b_1=0$ and $b_2>1$ there cannot be $b_2$ smooth disjoint complex curves of genus $g>0$ spanning the (rational) homology.

Reconstructing curves from their Hodge classes

Let S be a smooth algebraic surface in {mathbb {P}}^3({mathbb {C}}). Movasati and Sertöz (Rend. Circ. Mat. Palermo 2:1–17, 2020) associate an ideal I_{alpha (C)} to the primitive cohomology class alpha (C) of C in S. We show that the equations of C can be determined by I_{alpha (C)} under numerical conditions. We apply this result to reconstruct rational curves and arithmetically Cohen-Macaulay curves from their cohomology classes. On the other hand, we show that the class alpha (C) of a rational quartic curve C on a smooth quartic surface S is not even perfect, that is, that I_{alpha (C)} is bigger than the sum of the Jacobian ideal of S and of the homogeneous ideals of curves D in S for which I_{alpha (D)}=I_{alpha (C)}.

On birational transformations of Hilbert schemes of points on K3 surfaces

We classify the group of birational automorphisms of Hilbert schemes of points on algebraic K3 surfaces of Picard rank one. We study whether these automorphisms are symplectic or non-symplectic and if there exists a hyperk\"ahler birational model on which they become biregular. We also present new geometrical constructions of these automorphisms.

On the equation of a parametric algebraic surface

On the equation of a parametric algebraic surface

Algebraic k-Sets and Generally Neighborly Embeddings

Given a set S of n points in $${{\mathbb {R}}}^d$$ , a k-set is a subset of k points of S that can be strictly separated by a hyperplane from the remaining $$n-k$$ points. Similarly, one may consider k-facets, which are hyperplanes that pass through d points of S and have k points on one side. A notorious open problem is to determine the asymptotics of the maximum number of k-sets. In this paper we study a variation on the k-set/k-facet problem with hyperplanes replaced by algebraic surfaces. In stark contrast to the original k-set/k-facet problem, there are some natural families of algebraic curves for which the number of k-facets can be counted exactly. For example, we show that the number of halving conic sections for any set of $$2n+5$$ points in general position in the plane is $$2\left( {\begin{array}{c}n+2\\ 2\end{array}}\right) ^2$$ . To understand the limits of our argument we study a class of maps we call generally neighborly embeddings, which map generic point sets into neighborly position. Additionally, we give a simple argument which improves the best known bound on the number of k-sets/k-facets for point sets in convex position.

(ζ−m, ζm)-Type Algebraic Minimal Surfaces in Three-Dimensional Euclidean Space

We introduce the real minimal surfaces family by using the Weierstrass data (ζ−m,ζm) for ζ∈C, m∈Z≥2, then compute the irreducible algebraic surfaces of the surfaces family in three-dimensional Euclidean space E3. In addition, we propose that family has a degree number (resp., class number) 2m(m+1) in the cartesian coordinates x,y,z (resp., in the inhomogeneous tangential coordinates a,b,c).

Central extensions and the Riemann-Roch theorem on algebraic surfaces

We study canonical central extensions of the general linear group over the ring of adeles on a smooth projective algebraic surface $X$ by means of the group of integers. Via these central extensions and the adelic transition matrices of a rank $n$ locally free sheaf of $\mathcal{O}_X$-modules we obtain a local (adelic) decomposition for the difference of Euler characteristics of this sheaf and the sheaf $\mathcal{O}_X^n$. Two distinct calculations of this difference lead to the Riemann-Roch theorem on $X$ (without Noether's formula). Bibliography: 21 titles.

Self-oscillations in a certain Belousov–Zhabotinsky model

We consider the dynamic properties of a system of three differential equations known as the oreganator model. This model depends on four external parameters and describes one of the periodic Belousov–Zhabotinsky reactions. We obtain broad conditions for the parameters that ensure the existence of nonstationary steady-state regimes in oregonator model. With classical values of the parameters, the localization of the limit (at a long time) dynamics in the phase space has been improved. In fact, using numerical analysis, we significantly narrow the bounded region of the phase space containing the trajectories of the system. An iterative procedure is proposed for the approximate localization of closed trajectories (cycles) of the system on algebraic surfaces inR3. A promising problem of theoretical substantiation of the numerical convergence of this procedure is posed.

Invariant Algebraic Surfaces of the Vallis System

In this paper, we characterize the Darboux polynomials and invariant algebraic surfaces of the Vallis system. In the proofs, we use the weight homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations. Finally, we have obtained three types of Darboux polynomials of the Vallis system under suitable conditions of parameters.

Methodology for a reverse engineering process chain with focus on customized segmentation and iterative closest point algorithms

One-off construction is characterized by a multiplicity of manual manufacturing processes whereby it is based on consistent use of digital models. Since the actual state of construction does not match the digital models without manually updating them, the authors propose a method to automatically detect deviations and reposition the model data according to reality. The first essential method is based on the “Segmentation of Unorganized Points and Recognition of Simple Algebraic Surfaces” presented by Marek Vanco (2003). The second method is the customization of the iterative closest point (ICP) algorithm. The authors present the overall structure of the implemented software, based on open source and relate it to the general reverse engineering (RE) framework by Buonamici et al. (2017).A highlight will be given on.•The general architecture of the software prototype.•A customized segmentation and clustering of unorganized points and recognition of simple algebraic surfaces.•The deviation analysis with a customized iterative closest point (CICP) algorithm.Especially in the field of one-off construction, characterized by small and medium companies, automated assessment of 3D scan data during the design process is still in its infancy. By using an open source environment progress for consistent use of digital models could be accelerated.

Objectivity and Rigor in Classical Italian Algebraic Geometry

The classification of algebraic surfaces by the Italian School of algebraic geometry is universally recognized as a breakthrough in 20th century mathematics. The methods by which it was achieved do not, however, meet the modern standard of rigor and therefore appear dubious from a contemporary viewpoint. In this article, we offer a glimpse into the mathematical practice of the three leading exponents of the Italian School of algebraic geometry: Castelnuovo, Enriques, and Severi. We then bring into focus their distinctive conception of rigor and intuition. Unlike what is often assumed today, from their perspective, rigor is neither opposed to intuition nor understood as a unitary phenomenon –Enriques distinguishes between small-scale rigor and large-scale rigor and Severi between formal rigor and substantial rigor. Finally, we turn to the notion of mathematical objectivity. We draw from our case study in order to advance a multi-dimensional analysis of objectivity. Specifically, we suggest that various types of rigor may be associated with different conceptions of objectivity: namely objectivity as faithfulness to facts and objectivity as intersubjectivity.

Special triple covers of algebraic surfaces

We study special triple covers f:T\to S of algebraic surfaces, where the Tschirnhausen bundle \mathcal{E}=\left(f_*\mathcal{O}_T/\mathcal{O}_S\right)^\vee is a quotient of a split rank three vector bundle, and we provide several necessary and sufficient criteria for the existence. As an application, we give a complete classification of special triple planes, finding among others two nice families of K3 surfaces.

Hydrodynamic surfaces with midsection in the form of Lame curve

General representation of ship geometry is given by the method of slicing the ship hull by three mutually perpendicular planes: vertical symmetry plane which runs along the middle of hull width, horizontal plane which divides the hull into underwater and abovewater parts, and vertical plane perpendicular to the other two which coincides with midsection. By taking the same three predefined sections of the theoretical hull shape, it is possible to obtain three algebraic surfaces of different order, which are called hydrodynamic in this article. By introducing alphabetic parameters to signify orders of ship skeleton main curves and then by giving them various numerical values, it is possible to consider a large number of hull shapes, having only three explicit surface equations. Method of deriving the equations, obtained by other authors, using only three explicit algebraic equations is demonstrated. The proposed technique is illustrated on six new ship hull shapes.

The Algebraic Surfaces of the Enneper Family of Maximal Surfaces in Three Dimensional Minkowski Space

We consider the Enneper family of real maximal surfaces via Weierstrass data (1,ζm) for ζ∈C, m∈Z≥1. We obtain the irreducible surfaces of the family in the three dimensional Minkowski space E2,1. Moreover, we propose that the family has degree (2m+1)2 (resp., class 2m(2m+1)) in the cartesian coordinates x,y,z (resp., in the inhomogeneous tangential coordinates a,b,c).

Turbulent Premixed Flame Modeling Using the Algebraic Flame Surface Wrinkling Model: A Comparative Study between OpenFOAM and Ansys Fluent

Hereafter, we used the Algebraic Flame Surface Wrinkling (AFSW) model to conduct numerical simulations of the Paul Scherrer Institute (PSI) high-pressure, turbulent premixed Bunsen flame experiments. We implemented the AFSW model in OpenFOAM and in Ansys Fluent, and we compared the outcome of both solvers against the experimental results. We also highlight the differences between both solvers. All the simulations were performed using a two-dimensional axisymmetric model with the standard k−ϵ turbulence model with wall functions. Two different fuel/air mixtures were studied, namely, a 100%CH4 volumetric ratio and a 60%CH4+ 40%H2 volumetric ratio. The thermophysical and transport properties of the mixture were calculated as a function of temperature using the library Cantera (open-source suite of tools for problems involving chemical kinetics, thermodynamics, and transport processes), together with the GRI-Mech 3.0 chemical mechanism. It was found that the outcome of the AFSW model implemented in both solvers was in good agreement with the experimental results, quantitatively and qualitatively speaking. Further assessment of the results showed that, as much as the chemistry, the turbulence model and turbulent boundary/initial conditions significantly impact the flame shape and height.

Analytical surfaces for ship hulls

The choice of optimal shape of ship hull surface is one of the main problems of ship architects and designs. A choice of the form is based on empirical formulae or on intuition of designers. In the article a method of determination of explicit algebraic equations of theoretical shape of ship hull with three main cross-sections given in advance and coinciding with the design waterline, the midship section, and with the main buttock line is given. The forms of the lines in the main cross-sections are chosen from conditions taken in advance. These surfaces are called hydrodynamic. A method is illustrated for three threes of main cross-sections of the ship hulls, i.e. nine hydrodynamic surfaces were constructed. All algebraic equations were converted to parametrical form for comfort of computer modelling. With their help, all nine ship surfaces proposed for the introduction were visualized. Having changed constants containing in the surface equations, i.e. correcting the forms of three main geometric parameters of ship hull, one can select the most rational shape of hull surface for the first approach. Further, it is possible to begin planning parallel middle bodies or to combine bow and stern extremities of a ship from different fragments of algebraic surfaces but with the same midship sections. In a paper, only geometrical problems of design of theoretical hull shape are described.

Algebraic surfaces determine analyticity of functions

Let f :X rightarrow mathbb {R} be a function defined on a nonsingular real algebraic set X of dimension at least 3. We prove that f is an analytic (resp. a Nash) function whenever the restriction f|_{S} is an analytic (resp. a Nash) function for every nonsingular algebraic surface S subset X whose each connected component is homeomorphic to the unit 2-sphere. Furthermore, the surfaces S can be replaced by compact nonsingular algebraic curves in X, provided that dimX ge 2 and f is of class mathcal {C}^{infty }.

Algebraic capacities

We study invariants coming from certain optimisation problems for nef divisors on surfaces. These optimisation problems arise in work of the author and collaborators tying obstructions to embeddings between symplectic 4-manifolds to questions of positivity for (possibly singular) algebraic surfaces. We develop the general framework for these invariants and prove foundational results on their structure and asymptotics. We describe the connections these invariants have to embedded contact homology (ECH) and the Ruelle invariant in symplectic geometry, and to min–max widths in the study of minimal hypersurfaces. We use the first of these connections to obtain optimal bounds for the sub-leading asymptotics of ECH capacities for many toric domains.

Some examples of algebraic surfaces with canonical map of degree 20

In this note, we construct two minimal surfaces of general type with geometric genus p_g= 3, irregularity q = 0, self-intersection of the canonical divisor K^22 =20,24 such that their canonical map is of degree 20. In one of these surfaces, the canonical linear system has a non-trivial fixed part. These surfaces, to our knowledge, are the first examples of minimal surfaces of general type with canonical map of degree 20.