The Viro method for construction of Bernstein-Bézier algebraic hypersurface piece

This paper proposes a new method for the construction of Bernstein-Bézier algebraic hypersurface on a simplex with prescribed topology.The method is based on the combinatorial patchworking of Viro method.The topology of the Viro Bernstein-Bézier algebraic hypersurface piece is also described.     

Viro theorem and topology of real and complex combinatorial hypersurfaces

We introduce a class of combinatorial hypersurfaces in the complex projective space. They are submanifolds of codimension 2 inℂP ⁿ and are topologically “glued” out of algebraic hypersurfaces in (ℂ^*) ⁿ . Our construction can be viewed as a version of the Viro gluing theorem relating topology of algebraic hypersurfaces to the combinatorics of subdivisions of convex lattice polytopes. If a subdivision is convex, then according to the Viro theorem a combinatorial hypersurface is isotopic to an algebraic one. We study combinatorial hypersurfaces resulting from non-convex subdivisions of convex polytopes, show that they are almost complex varieties, and in the real case, they satisfy the same topological restrictions (congruences, inequalities etc.) as real algebraic hypersurfaces. A part of the present work was done during the stay of the second author at the Fields Institute, Toronto, and at the NSF Science and Technology Research Center for the Computation and Visualization of Geometric Structures, funded by NSF/DMS89-20161. The work was completed during the stay of both authors at Max-Planck-Institu für Mathematik. The authors thank these funds and institutions for hospitality and financial support.     

Symmetric Orientations of Dividing T-curves

In 1974, Rokhlim introduced complex orientations for nonsingular real algebraic plane projective curves of type I. Here we give a definition of symmetric orientations and of "type" for T-curves which are PL-curves constructed using a combinatorial method called T-construction. An important aspect of T-construction is that, under particular conditions, the constructed T-curve has the isotopy type of a nonsingular real algebraic plane projective curve. T-construction is in fact a particular case of the method of construction of real algebraic projective varieties due to O. Ya. Viro. We prove that if an algebraic curve is associated to a T-curve by the Viro process, then the type of the T-curve coincides with the type of the algebraic curve and its symmetric orientations are complex orientations as defined by Rokhlin. The main result of this paper is the classification theorem for T-curves of type I.     

Patchworking singular algebraic curves I

In this paper we present a general patchworking procedure for the construction of reduced singular curves having prescribed singularities and belonging to a given linear system on algebraic surfaces. It originates in the Viro “gluing” method for the construction of real non-singular algebraic hypersurfaces. The general procedure includes almost all known particular modifications, and goes far beyond. Some applications and examples illustrate the construction. Both authors were partially supported by the Herman Minkowsky-Minerva Center for Geometry at Tel Aviv University, and by grant no. G-616-15.6/99 from the German-Israeli Foundation for Research and Development. The first author was also supported by the Bessel Research Award from the Alexander von Humboldt Foundation. The second author was also partially supported by the EC-network ‘Algebraic Lie Representations” contract no. ERB-FMRX-CT97-0100.     

Rational versus real cohomology algebras of low-dimensional toric varieties

We show that the real cohomology algebra of a compact toric variety of complex dimension  is determined, up to isomorphism, by the combinatorial data of its defining fan. Surprisingly enough, this is no longer the case when taking rational coefficients. Moreover, we show that neither the rational nor the real or complex cohomology algebras of compact quasi-smooth toric varieties are combinatorial invariants in general.

     

M-curves of degree 10

The known restrictions on the disposition of ovals of a nonsingular projective real algebraic curve leave possible for M-curves of degree 10 about 70,000 real schemes. These schemes divide naturally into 18 families. Basic result: in each of these families there exist representatives realizing M-curves of degree 10. Using the method of O. Ya. Viro, based on perturbations of curves with involved singularities, there are constructed more than 500 M-curves of degree 10. Incidentally there are obtained a large number of perturbations of points of tangency of five branches.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 122, pp. 146–161, 1982.In conclusion, I want to thank O. Ya. Viro for constant interest in my work and a series of important comments.     

Patchworking Arrangements of a cubic and a Quartic

B. Sturmfels modified Viro's "patchworking" method and applied it for construction of complete intersections. In the paper, this modification is used for construction of decomposable curves. 11 new arrangements of an M-cubic and an M-quartic with 12 common points lying on the odd branch of the cubic and an oval of the quartic are realized. Bibliography: 16 titles.     

A nonalgebraic patchwork

Itenberg and Shustin’s pseudoholomorphic curve patchworking is in principle more flexible than Viro’s original algebraic one. It was natural to wonder if the former method allows one to construct nonalgebraic objects. In this paper we construct the first examples of patchworked real pseudoholomorphic curves in Σ _n whose position with respect to the pencil of lines cannot be realized by any real algebraic curve of the same bidegree. Both authors are very grateful to the Max Planck Institute für Mathematik in Bonn for its financial support and excellent working conditions.     

Finite type invariants of nanowords and nanophrases

Homotopy classes of nanowords and nanophrases are combinatorial generalizations of virtual knots and links. Goussarov, Polyak and Viro defined finite type invariants for virtual knots and links via semi-virtual crossings. We extend their definition to nanowords and nanophrases. We study finite type invariants of low degrees. In particular, we show that the linking matrix and T invariant defined by Fukunaga are finite type of degree 1 and degree 2 respectively. We also give a finite type invariant of degree 4 for open homotopy of Gauss words.     

A Counterexample for Subadditivity of Multiplier Ideals on Toric Varieties

We construct a 3-dimensional complete intersection toric variety on which the subadditivity formula doesn't hold, answering negatively a question by Takagi and Watanabe. A combinatorial proof of the subadditivity formula on 2-dimensional normal toric varieties is also provided.     

Formal specification and proofs for the topology and classification of combinatorial surfaces

We describe one of the first attempts at using modern specification techniques in the field of geometric modeling and computational geometry. Using the Coq system, we developed a formal multi-level specification of combinatorial maps, used to represent subdivisions of geometric manifolds, and then exploited it to formally prove fundamental theorems. In particular, we outline here an original and constructive proof of a combinatorial part of the famous Surface Classification Theorem, based on a set of so-called “conservative” elementary operations on subdivisions.     

Algebras and triangle relations

In this paper the new concept of an n-algebra is introduced, which embodies the combinatorial properties of an n-tensor, in an analogous manner to the way ordinary algebras embody the properties of compositions of maps. The work of Turaev and Viro on 3-manifold invariants is seen to fit naturally into the context of 3-algebras. A new higher dimensional version of Yang-Baxter's equation, distinct from Zamolodchikov's equation, which resides naturally in these structures, is proposed. A higher dimensional analogue of the relationship between the Yang-Baxter equation and braid groups is then seen to exhibit a similar relationship with Manin and Schechtman's higher braid groups.     

Critical Points of Real Polynomials and Topology of Real Algebraic T-Surfaces

The paper is devoted to a special class of real polynomials, so-called T-polynomials, which arise in the combinatorial version of the Viro theorem. We study the relation between the numbers of real critical points of a given index of a T-polynomial and the combinatorics of lattice triangulations of Newton polytopes. We obtain upper bounds for the numbers of extrema and saddles of generic T-polynomials of a given degree in three variables, and derive from them upper bounds for Betti numbers of real algebraic surfaces in defined by T-polynomials. The latter upper bounds are stronger than the known upper bounds for arbitrary real algebraic surfaces in . Another result is the existence of an asymptotically maximal family of real polynomials of degree min three variables with 31m ³/36 + O(m ²) saddle points.     

Totally real embeddings of the torus intoC 2

In this short note we combine a construction of Viro and a result of Eliashberg and Harlamov to prove that there exist smooth totally real embeddings of the torus intoC ² which are isotopic but not so within the class of totally real surfaces. We also show how Viro's construction can be used to define an isotopy invariant for a certain class of complex curves inC P ².     

A TQFT of Turaev–Viro Type on Shaped Triangulations

A shaped triangulation is a finite triangulation of an oriented pseudo-three-manifold where each tetrahedron carries dihedral angles of an ideal hyperbolic tetrahedron. To each shaped triangulation, we associate a quantum partition function in the form of an absolutely convergent state integral which is invariant under shaped 3–2 Pachner moves and invariant with respect to shape gauge transformations generated by total dihedral angles around internal edges through the Neumann–Zagier Poisson bracket. Similarly to Turaev–Viro theory, the state variables live on edges of the triangulation but take their values on the whole real axis. The tetrahedral weight functions are composed of three hyperbolic gamma functions in a way that they enjoy a manifest tetrahedral symmetry. We conjecture that for shaped triangulations of closed three-manifolds, our partition function is twice the absolute value squared of the partition function of Techmüller TQFT defined by Andersen and Kashaev. This is similar to the known relationship between the Turaev–Viro and the Witten–Reshetikhin–Turaev invariants of three-manifolds. We also discuss interpretations of our construction in terms of three-dimensional supersymmetric field theories related to triangulated three-dimensional manifolds.     

Topology of Maximally Writhed Real Algebraic Knots

Oleg Viro introduced an invariant of rigid isotopy for real algebraic knots in ??³ which can be viewed as a first order Vassiliev invariant. In this paper we look at real algebraic knots of degree d with the maximal possible value of this invariant. We show that for a given d all such knots are topologically isotopic and explicitly identify their knot type.     

Infinite combinatorics and the foundations of regular variation

The theory of regular variation is largely complete in one dimension, but is developed under regularity or smoothness assumptions. For functions of a real variable, Lebesgue measurability suffices, and so does having the property of Baire. We find here that the preceding two properties have common combinatorial generalizations, exemplified by ‘containment up to translation of subsequences’. All of our combinatorial regularity properties are equivalent to the uniform convergence property.     

The Cayley Trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings

In 1994, Sturmfels gave a polyhedral version of the Cayley Trick of elimination theory: he established an order-preserving bijection between the posets of coherent mixed subdivisions of a Minkowski sum ?₁+...+? _r of point configurations and of coherent polyhedral subdivisions of the associated Cayley embedding ?(?₁,...,? _r ). In this paper we extend this correspondence in a natural way to cover also non-coherent subdivisions. As an application, we show that the Cayley Trick combined with results of Santos on subdivisions of Lawrence polytopes provides a new independent proof of the Bohne-Dress theorem on zonotopal tilings. This application uses a combinatorial characterization of lifting subdivisions, also originally proved by Santos. Received February 18, 1999 / final version received January 25, 2000?Published online May 22, 2000     

Tarski's Fixpoint Lemma and combinatorial games

Using Tarski's Fixpoint Lemma for order preserving maps of a complete lattice into itself, a new, lattice theoretic proof is given for the existence of persistent strategies for combinatorial games as well as for games with a topological tolerance and games on lattices. Further, the existence of winning strategies is obtained for games on superalgebraic lattices, which includes the case of ordinary combinatorial games. Finally, a basic representation theorem is presented for those lattices.