Algebraic Surfaces Research Articles

Ramification in the cohomology of algebraic surfaces arising from ordinary double point singularities

Let K be a field complete with respect to a discrete valuation whose residue field is algebraically closed of an odd positive characteristic. We study the ramification in the cohomology of a smooth proper surface X defined over K, under the assumption that X admits an integral model X whose special fibre has at worst ordinary double points. We will introduce a numerical invariant of X, in terms of which the ramification in the cohomology of X is determined.

An Algebraic Surface Projecting onto Squares

An Algebraic Surface Projecting onto Squares

Computing projective equivalences of special algebraic varieties

This paper is devoted to the investigation of selected situations when computing projective (and other) equivalences of algebraic varieties can be efficiently solved via finding projective equivalences of finite sets of points on the projective line. In particular, we design a method that finds for two algebraic varieties X,Y from special classes an associated set of automorphisms of the projective line (the so called good candidate set) consisting of suitable candidates for the subsequent construction of possible mappings X→Y. The functionality of the designed approach is presented for computing projective equivalences of rational curves, determining projective equivalences of rational ruled surfaces, detecting affine transformations between planar algebraic curves, and computing similarities between two implicitly given algebraic surfaces. When possible, symmetries of given shapes are also discussed as special cases.

Algebraic affine rotation surfaces of parabolic type

Affine rotation surfaces, which appear in the context of affine differential geometry, are generalizations of surfaces of revolution. These affine rotation surfaces can be classified into three different families: elliptic, hyperbolic and parabolic. In this paper we investigate some properties of algebraic parabolic affine rotation surfaces, i.e. parabolic affine rotation surfaces that are algebraic, generalizing some previous results on algebraic affine rotation surfaces of elliptic type (classical surfaces of revolution) and hyperbolic type (hyperbolic surfaces of revolution). In particular, we characterize these surfaces in terms of the structure of their implicit equation, we describe the structure of the form of highest degree defining an algebraic parabolic affine rotation surface, and we prove that these surfaces can have either one, or two, or infinitely many axes of affine rotation. Additionally, we characterize the surfaces with more than one parabolic axis.

Algebraic surfaces with pg = q = 1,K2 = 4 and nonhyperelliptic Albanese fibrations of genus 4

In this article, we study minimal algebraic surfaces with and nonhyperelliptic Albanese fibrations of genus 4. We construct for the first time a family of such surfaces as complete intersections of type (2, 3) in a -bundle over an elliptic curve. For the surfaces we construct here, the direct image of the canonical sheaf under the Albanese map is decomposable (which is a topological invariant property). Moreover, we prove that all minimal surfaces with and nonhyperelliptic Albanese fibrations of genus 4 such that the direct image of the canonical sheaf under the Albanese map is decomposable are contained in our family. As a consequence, we show that these surfaces constitute a four-dimensional irreducible subset of , the Gieseker moduli space of minimal surfaces with . Moreover, the closure of is an irreducible component of .

Enriques involutions on singular K3 surfaces of small discriminants

We classify Enriques involutions on a K3 surface, up to conjugation in the automorphism group, in terms of lattice theory. We enumerate such involutions on singular K3 surfaces with transcendental lattice of discriminant smaller than or equal to 36. For 11 of these K3 surfaces, we apply Borcherds method to compute the automorphism group of the Enriques surfaces covered by them. In particular, we investigate the structure of the two most algebraic Enriques surfaces.

Q_l-cohomology projective planes and singular Enriques surfaces in characteristic two

We classify singular Enriques surfaces in characteristic two supporting a rank nine configuration of smooth rational curves. They come in one-dimensional families defined over the prime field, paralleling the situation in other characteristics, but featuring novel aspects. Contracting the given rational curves, one can derive algebraic surfaces with isolated ADE-singularities and trivial canonical bundle whose Q_l-cohomology equals that of a projective plane. Similar existence results are developed for classical Enriques surfaces. We also work out an application to integral models of Enriques surfaces (and K3 surfaces).

On the Orbits of One Non-Solvable 5-Dimensional Lie Algebra

This paper studies holomorphic homogeneous real hypersurfaces in C3 associated with the unique non-solvable indecomposable 5-dimensional Lie algebra 𝑔5 (in accordance with Mubarakzyanov’s notation). Unlike many other 5-dimensional Lie algebras with “highly symmetric” orbits, non-degenerate orbits of 𝑔5 are “simply homogeneous”, i.e. their symmetry algebras are exactly 5-dimensional. All those orbits are equivalent (up to holomorphic equivalence) to the specific indefinite algebraic surface of the fourth order. The proofs of those statements involve the method of holomorphic realizations of abstract Lie algebras. We use the approach proposed by Beloshapka and Kossovskiy, which is based on the simultaneous simplification of several basis vector fields. Three auxiliary lemmas formulated in the text let us straighten two basis vector fields of 𝑔5 and significantly simplify the third field. There is a very important assumption which is used in our considerations: we suppose that all orbits of 𝑔5 are Levi non-degenerate. Using the method of holomorphic realizations, it is easy to show that one need only consider two sets of holomorphic vector fields associated with 𝑔5. We prove that only one of these sets leads to Levi non-degenerate orbits. Considering the commutation relations of 𝑔5, we obtain a simplified basis of vector fields and a corresponding integrable system of partial differential equations. Finally, we get the equation of the orbit (unique up to holomorphic transformations) (𝑣 − 𝑥2𝑦1)2 + 𝑦2 1𝑦2 2 = 𝑦1, which is the equation of the algebraic surface of the fourth order with the indefinite Levi form. Then we analyze the obtained equation using the method of Moser normal forms. Considering the holomorphic invariant polynomial of the fourth order corresponding to our equation, we can prove (using a number of results obtained by A.V. Loboda) that the upper bound of the dimension of maximal symmetry algebra associated with the obtained orbit is equal to 6. The holomorphic invariant polynomial mentioned above differs from the known invariant polynomials of Cartan’s and Winkelmann’s types corresponding to other hypersurfaces with 6- dimensional symmetry algebras.

The period-index problem for real surfaces

We study when the period and the index of a class in the Brauer group of the function field of a real algebraic surface coincide. We prove that it is always the case if the surface has no real points (more generally, if the class vanishes in restriction to the real points of the locus where it is well-defined), and give a necessary and sufficient condition for unramified classes. As an application, we show that the u-invariant of the function field of a real algebraic surface is equal to 4, answering questions of Lang and Pfister. Our strategy relies on a new Hodge-theoretic approach to de Jong's period-index theorem on complex surfaces.

ALGEBRAIC SURFACES WITH INFINITELY MANY TWISTOR LINES

We prove that a reduced and irreducible algebraic surface in $\mathbb{CP}^{3}$ containing infinitely many twistor lines cannot have odd degree. Then, exploiting the theory of quaternionic slice regularity and the normalisation map of a surface, we give constructive existence results for even degrees.

A new example of an algebraic surface with canonical map of degree 16

In this note, we construct a minimal surface of general type with geometric genus $$ p_g =4 $$ , self-intersection of the canonical divisor $$ K^2 = 32$$ , and irregularity $$ q = 1 $$ such that its canonical map is an Abelian cover of degree 16 of $$\mathbb P^1\times \mathbb P^1$$ .

Global dynamics of a virus model with invariant algebraic surfaces

In this paper by using the Poincare compactification in $${{\mathbb {R}}}^3$$ we make a global analysis for the virus system $$\begin{aligned} {\dot{x}}=\lambda -dx-\beta xz, \quad {\dot{y}} = -ay+\beta xz \quad {\dot{z}} = ky-\mu z \end{aligned}$$with $$(x, y, z) \in {{\mathbb {R}}}^3$$, $$\beta >0$$, $$\lambda , a, d, k$$ and $$\mu $$ are nonnegative parameters due to their biological meaning. We give the complete description of its dynamics on the sphere at infinity. For two sets of the parameter values the system has invariant algebraic surfaces. For these two sets we provide the global phase portraits of the virus system in the Poincare ball (i.e. in the compactification of $${{\mathbb {R}}}^3$$ with the sphere $${\mathbb {S}}^2$$ of the infinity).

Invariant algebraic surfaces of the FitzHugh-Nagumo system

In this paper, we characterize all the irreducible Darboux polynomials and polynomial first integrals of the FitzHugh-Nagumo system by the called characteristic curves method which involves the theory of weight homogeneous polynomials. It has been ever widely used to give a complete classification of Darboux polynomials of some systems. However, it does not work for the FitzHugh-Nagumo system. Thus, we introduce an “assistant” system and classify its invariant algebraic surfaces finally.

High U(1) charges in type IIB models and their F-theory lift

We construct models with U(1) gauge group and matter with charges up to 6, in the context of type IIB compactifications. We show explicitly that models with charges up to 4 can be derived from corresponding models in F-theory by applying the Sen weak coupling limit. We derive which type IIB models should be the limit of charge 5 and 6 F-theory models. Explicit six dimensional type IIB models with maximal charge 5 and 6 are constructed on an algebraic K3 surface that is the double cover of ℂℙ2. By using type IIB results we are also able to rediscover the F-theory charge 4 model in a straightforward way.

First Integrals of the May–Leonard Asymmetric System

For the May–Leonard asymmetric system, which is a quadratic system of the Lotka–Volterra type depending on six parameters, we first look for subfamilies admitting invariant algebraic surfaces of degree two. Then for some such subfamilies we construct first integrals of the Darboux type, identifying the systems with one first integral or with two independent first integrals.

Singularities of the isospectral Hilbert scheme

We study the singularities of the isospectral Hilbert scheme Bn of n points over a smooth algebraic surface and we prove that they are canonical if , log-canonical if and not log-canonical if . We describe as well two explicit log-resolutions of B3, one crepant and the other -equivariant.

Real algebraic curves with large finite number of real points

We address the problem of the maximal finite number of real points of a real algebraic curve (of a given degree and, sometimes, genus) in the projective plane. We improve the known upper and lower bounds and construct close to optimal curves of small degree. Our upper bound is sharp if the genus is small as compared to the degree. Some of the results are extended to other real algebraic surfaces, most notably ruled.

Three Topological Results on the Twistor Discriminant Locus in the 4-Sphere

We exploit techniques from classical (real and complex) algebraic geometry for the study of the standard twistor fibration $\pi:\mathbb{CP}^{3}\to S^{4}$. We prove three results about the topology of the twistor discriminant locus of an algebraic surface in $\mathbb{CP}^{3}$. First of all we prove that, with the exception of two exceptional cases, the real dimension of the twistor discriminant locus of an algebraic surface is always equal to 2. Secondly we describe the possible intersections of a general surface with the family of twistor lines: we find that only 4 configurations are possible and for each of them we compute the dimension. Lastly we give a decomposition of the twistor discriminant locus of a given cone in terms of its singular locus and its dual variety.

Algebraic surfaces, four-folds and moonshine

The aim of this note is to point out an interesting fact related to the elliptic genus of complex algebraic surfaces in the context of Mathieu moonshine. We also discuss the case of 4-folds.

Hirzebruch–Kummer covers of algebraic surfaces

The aim of this paper is to show that using some natural curve arrangements in algebraic surfaces and Hirzebruch-Kummer covers one cannot construct new examples of ball-quotients, i.e., minimal smooth complex projective surfaces of general type satisfying equality in the Bogomolov-Miyaoka-Yau inequality.