Real Algebraic Surfaces Research Articles

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.

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.

Exponential rarefaction of maximal real algebraic hypersurfaces

Given an ample real Hermitian holomorphic line bundle L over a real algebraic variety X , the space of real holomorphic sections of L^{\otimes d} inherits a natural Gaussian probability measure. We prove that the probability that the zero locus of a real holomorphic section s of L^{\otimes d} defines a maximal hypersurface tends to 0 exponentially fast as d goes to infinity. This extends to any dimension a result of Gayet and Welschinger (2011) valid for maximal real algebraic curves inside a real algebraic surface. The starting point is a low degree approximation property which relates the topology of the real vanishing locus of a real holomorphic section of L^{\otimes d} with the topology of the real vanishing locus a real holomorphic section of L^{\otimes d'} for a sufficiently smaller d'<d . Such a statement is inspired by the recent work of Diatta and Lerario (2022).

Real-Fibered Morphisms of del Pezzo Surfaces and Conic Bundles

It goes back to Ahlfors that a real algebraic curve admits a real-fibered morphism to the projective line if and only if the real part of the curve disconnects its complex part. Inspired by this result, we are interested in characterising real algebraic varieties of dimension n admitting real-fibered morphisms to the n-dimensional projective space. We present a criterion to classify real-fibered morphisms that arise as finite surjective linear projections from an embedded variety which relies on topological linking numbers. We address special attention to real algebraic surfaces. We classify all real-fibered morphisms from real del Pezzo surfaces to the projective plane and determine which such morphisms arise as the composition of a projective embedding with a linear projection. Furthermore, we give some insights in the case of real conic bundles.

Viro–Zvonilov Inequalities for Flexible Curves on an Almost Complex Four-Dimensional Manifold

The restrictions on the topology of nonsingular plane projective real algebraic curves of odd degree, obtained by O. Viro and the author in the paper published in the early 90s, are extended to flexible curves lying on an almost complex four-dimensional manifold. Some examples of real algebraic surfaces and real curves on them prove the sharpness of the obtained inequalities. In addition, it is proved that a compact Lie group smooth action can be lifted to a cyclic branched covering space $ \tilde{X} $ over a closed four-dimensional manifold, and a sufficient condition for $ H_1(\tilde{X})=0 $ was found.

On the invariance of Welschinger invariants

We collect in this note some observations about original Welschinger invariants of real symplectic fourfolds. None of their proofs is difficult, nevertheless these remarks do not seem to have been made before. Our main result is that when $X$ is a real rational algebraic surface, Welschinger invariants only depend on the number of real interpolated points, and some homological data associated to $X$. This strengthened the invariance statement initially proved by Welschinger. This main result follows easily from a formula relating Welschinger invariants of two real symplectic manifolds differing by a surgery along a real Lagrangian sphere. In its turn, once one believes that such formula may hold, its proof is a mild adaptation of the proof of analogous formulas previously obtained by the author on the one hand, and by Itenberg, Kharlamov and Shustin on the other hand. We apply the two aforementioned results to complete the computation of Welschinger invariants of real rational algebraic surfaces, and to obtain vanishing, sign, and sharpness results for these invariants that generalize previously known statements. We also discuss some hypothetical relations of our work with tropical refined invariants defined by Block-G\ottsche and G\ottsche-Schroeter.

Numerical roadmap of smooth bounded real algebraic surface

For a smooth bounded real algebraic surface in Rn, a roadmap of it is a one-dimensional semi-algebraic subset of the surface whose intersection with each connected component of the surface is nonempty and semi-algebraically connected. In this paper, we introduce the notion of a numerical roadmap of a surface, which is a set of disjoint polygonal chains such that there is a bijective map between the chains and the connected components of a given roadmap of the surface. Moreover, the chains are ϵ-close to the connected components. We present an algorithm to compute such a numerical roadmap through constructing a topological graph. The topological graph also enables us to compute an approximate graph and a more intrinsic connectivity graph to represent the roadmap and its connectivity property, which is important for applications such as determining if two points on the surface belong to the same connected component and if so, finding a connected path between them.

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.

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.

Algebraic models of the Euclidean plane

We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that allows to distinguish smooth real algebraic surfaces up to birational diffeomorphism. As an application, we construct infinite families of smooth rational real algebraic surfaces with trivial homology groups, whose real loci are diffeomorphic to $\mathbb{R}^2$, but which are pairwise not birationally diffeomorphic. There are thus infinitely many non-trivial models of the euclidean plane, contrary to the compact case.

On the moduli space of equilateral plane pentagons

We give two new proofs of the well-known result that the moduli space M_5 of equilateral plane pentagons is a closed surface of genus four. Moreover, we construct a new algebraic description of this space, also in the non-equilateral case, as a real affine algebraic surface F defined by a polynomial p(x, y, z) of degree 12. This allows a visualization using the Surfer software.

On the geometric structure of certain real algebraic surfaces

In this paper we study the affine geometric structure of the graph of a polynomial $$f \in \mathbb {R}[x,y]$$ . We provide certain criteria to determine when the parabolic curve is compact and when the unbounded component of its complement is hyperbolic or elliptic. We analyse the extension to the real projective plane of both fields of asymptotic lines and the Poincare index at its singular points at infinity. We exhibit an index formula for the field of asymptotic lines involving the number of connected components of the projective Hessian curve of f and the number of godrons. As an application of this investigation, we obtain upper bounds, respectively, for the number of godrons having an interior tangency and when they have an exterior tangency.

Linear equations on real algebraic surfaces

We prove that if a linear equation, whose coefficients are continuous rational functions on a nonsingular real algebraic surface, has a continuous solution, then it also has a continuous rational solution. This is known to fail in higher dimensions.

Real Algebraic Surfaces with Many Handles in $(\mathbb{CP}^1)^3$

In this text, we study Viro’s conjecture and related problems for real algebraic surfaces in |$(\mathbb{CP}^1)^3$|⁠. We construct a counterexample to Viro’s conjecture in tridegree |$(4,4,2)$| and a family of real algebraic surfaces of tridegree |$(d_1,d_2,2)$| in |$(\mathbb{CP}^1)^3$| with asymptotically maximal first Betti number of the real part. To perform such constructions, we consider double covers of blow-ups of |$(\mathbb{CP}^1)^2$| and we glue singular curves with special position of the singularities adapting the proof of Shustin’s theorem for gluing singular hypersurfaces.

The monodromy of a general algebraic function

We consider a general reduced algebraic equation of degree n with complex coefficients. The solution to this equation, a multifunction, is called a general algebraic function. In the coefficient space we consider the discriminant set ∇ of the equation and choose in its complement the maximal polydisk domain D containing the origin. We describe the monodromy of the general algebraic function in a neighborhood of D. In particular, we prove that ∇ intersects the boundary ∂D along n real algebraic surfaces \(S^{(j)} \) of dimension n − 2. Furthermore, every branch yj(x) of the general algebraic function ramifies in D only along the pair of surfaces \(S^{(j)} \) and \(S^{(j - 1)} \).

Investigation of a real algebraic surface

A description of a real algebraic variety in ?3 is given. This variety plays an important role in the investigation of the Einstein metrics whose evolution is studied using the normalized Ricci flow. To reveal the internal structure of this variety, a description of all its singular points is given. Due to the internal symmetry of this variety, a part of the investigation uses elementary symmetric polynomials. All the computations are performed using computer algebra algorithms (in particular, Grobner bases) and algorithms for dealing with polynomial ideals. As an auxiliary result, a proposition about the structure of the discriminant surface of a cubic polynomial is proved.

On Higher Genus Welschinger Invariants of del Pezzo Surfaces

The Welschinger invariants of real rational algebraic surfaces count real rational curves which represent a given divisor class and pass through a generic conjugation-invariant configuration of points. No invariants counting real curves of positive genera are known in general. We indicate particular situations, when Welschinger-type invariants counting real curves of positive genera can be defined. We also prove the positivity and give asymptotic estimates for such Weischinger-type invariants for several del Pezzo surfaces of degree $\ge2$ and suitable real nef and big divisor classes. In particular, this yields the existence of real curves of given genus and of given divisor class passing through any appropriate configuration of real points on the given surface.

Computation of topological invariants for real projective surfaces with isolated singularities

Given a real algebraic surface S in RP3, we propose a procedure to determine the topology of S and to compute non-trivial topological invariants for the pair (RP3,S) under the hypothesis that the real singularities of S are isolated. In particular, starting from an implicit equation of the surface, we compute the number of connected components of S, their Euler characteristics and the labeled 2-adjacency graph of the surface.

Cubic and quartic cyclic homogeneous inequalities of three variables

We determine the geometric structures of the families of three variables cubic and quartic cyclic homogeneous inequalities of certain classes. These structures are determined by studying some real algebraic surfaces. Mathematics subject classification (2010): Primary 26D05.

Encomplexed Brown invariant of real algebraic surfaces in ℝP3

We construct an invariant of parametrized generic real algebraic surfaces in RP^3 which generalizes the Brown invariant of immersed surfaces from smooth topology. The invariant is constructed using the self intersection, which is a real algebraic curve with points of three local characters: the intersection of two real sheets, the intersection of two complex conjugate sheets or a Whitney umbrella. The Brown invariant was expressed through a self linking number of the self intersection by Kirby and Melvin. We extend the definition of this self linking number to the case of parametrized generic real algebraic surfaces.