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The discriminant set of a real polynomial is studied. It is shown that this set has a complex hierarchical structure and consists of algebraic varieties of various dimensions. A constructive algorithm for a polynomial parameterization of the discriminant set in the space of the coefficients of the polynomial is proposed. Each variety of a greter dimension can be geometrically considered as a tangent developable surface formed by one-dimensional linear varieties. The role of the directrix is played by the component of the discriminant set with the dimension by one less on which the original polynomial has a single multiple root and the other roots are simple. The relationship between the structure of the discriminant set and the partitioning of natural numbers is revealed. Various algorithms for the calculation of subdiscriminants of polynomials are also discussed. The basic algorithms described in this paper are implemented as a library for Maple. [ABSTRACT FROM AUTHOR]

Abstract: The intersection between a nilpotent orbit and the Lie algebra of a Borel subgroup is an equidimensional, quasi-affine algebraic variety. Its irreducible components are called orbital varieties. In this Note, we provide criteria to guarantee that an orbital variety is smooth or has a dense orbit for the adjoint action of B. In addition, we point out a possible relation between these two properties. [Copyright &y& Elsevier]

We present a decomposition of the space of twisted arcs of a toric stack. As a consequence, we give a combinatorial description of the motivic integral associated to a torus-invariant divisor of a toric stack. [ABSTRACT FROM AUTHOR]

Abstract: Starting from the well-known transformation law for the Klein functions, we give a proof of a fairly general multiplicative distribution formula for the Siegel functions associated to isogenous complex lattices. This formula has as an immediate consequence the remarkable distribution formula proved by Jarvis in 2000 on the occasion of Rolshausen''s thesis on the second K-group of an elliptic curve. [Copyright &y& Elsevier]

We consider an obstacle-type problem Δu = ƒ(x)χΩ in D, u = ∣∇u∣ = 0 on D \ Ω where D is a given open set in ℝn and Ω is an unknown open subset of D. The problem originates in potential theory, in connection with harmonic continuation of potentials. The qualitative difference between this problem and the classical obstacle problem is that the solutions here are allowed to change sign. Using geometric and energetic criteria in delicate combination we show the C1,1 regularity of the solutions, and the regularity of the free boundary, below the Lipschitz threshold for the right-hand side. [ABSTRACT FROM AUTHOR]

We show that any normal algebraic monoid is an extension of an abelian variety by a normal affine algebraic monoid. This extends (and builds on) Chevalley's structure theorem for algebraic groups. [ABSTRACT FROM AUTHOR]

Let W → 2 be the universal Weierstrass family of cubic curves over . For each N ≥ 2, we construct surfaces parameterizing the three standard kinds of level N structures on the smooth fibers of W. We then complete these surfaces to finite covers of 2. Since W → 2 is the versal deformation space of a cusp singularity, these surfaces convey information about the level structure on any family of curves of genus g degenerating to a cuspidal curve. Our goal in this note is to determine for which values of N these surfaces are smooth over (0, 0). From a topological perspective, the results determine the homeomorphism type of certain branched covers of S3 with monodromy in SL2 (/N). [ABSTRACT FROM AUTHOR]

Abstract: We provide a geometric study of the problem of finding and describing the double point locus of a bicubic surface. Our motivation is to determine whether a real bicubic patch over the unit square will have self intersections. And if so, to identify useful points and curves in order to determine basic features and to help graph the surface accurately. Here, we consider special interesting cases with additional structures, which are among the surfaces commonly used in CAGD (Computer Aided Geometric Design). [Copyright &y& Elsevier]

The article discusses the mathematical principles and applications concerning the families of plane algebraic curves that contain a special set of points. The case of curves in cages is associated with the elementary methods that presume the familiarity with algebraic geometry and geometrical applications. These applications can be considered as natural generalizations of classical theorems in projective geometry and are associated with familiarity with Pascal's theorem and its comparison with the classical Pascal diagram.

Abstract: In [K. Kato, Toric singularities, Amer. J. Math. 116 (5) (1994) 1073–1099], Kato defined his notion of a log regular scheme and studied the local behavior of such schemes. A toric variety equipped with its canonical logarithmic structure is log regular. And, these schemes allow one to generalize toric geometry to a theory that does not require a base field. This paper will extend this theory by removing normality requirements. [Copyright &y& Elsevier]

Type III unprojection plays a very important role in the birational geometry of Fano threefolds (cf. Corti et al., 2000; Reid, 2000). According to Reid (2000, p. 43), it was first introduced by A. Corti on his calculations of Fano threefolds of genus 6 and 7. It seems that at present a general definition of Type III unprojection is still missing. After proving some general facts about residual ideals in Section 2, we propose a definition for the generic Type III unprojection (Definition 3.3), and prove in Theorem 3.5 that it gives a Gorenstein ring. Communicated by W. Bruns [ABSTRACT FROM AUTHOR]

The Hilbert scheme of 3-folds in ℙ n , n ≥ 6 , that are scrolls over ℙ 2 or over a smooth quadric surface Q ⊂⃒ ℙ 3 or that are quadric or cubic fibrations over ℙ 1 is studied. All known such threefolds of degree 7 ≤ d ≤ 11 are shown to correspond to smooth points of an irreducible component of their Hilbert scheme, whose dimension is computed. [ABSTRACT FROM AUTHOR]

LetGbe a connected reductive linear algebraic group. We use geometric methods to investigateG-completely reducible subgroups ofG, giving new criteria forG-complete reducibility. We show that a subgroup ofGisG-completely reducible if and only if it is strongly reductive inG; this allows us to use ideas of R.W. Richardson and Hilbert-Mumford-Kempf from geometric invariant theory. We deduce that a normal subgroup of aG-completely reducible subgroup ofGis againG-completely reducible, thereby providing an affirmative answer to a question posed by J.-P. Serre, and conversely we prove that the normalizer of aG-completely reducible subgroup ofGis againG-completely reducible. Some rationality questions and applications to the spherical building ofGare considered. Many of our results extend to the case of non-connectedG. [ABSTRACT FROM AUTHOR]

We apply the Murasugi–Tristram inequality to real algebraic curves of odd degree in RP2 with a deep nest, i.e. a nest of the depth k - 1 where 2k + 1 is the degree. For such curves, the ingredients of the Murasugi–Tristram inequality can be computed (or estimated) inductively using the computations for iterated torus links due to Eisenbud and Neumann as the base case of the induction and Conway's skein relation as the induction step. As an example of applications, we prove that some isotopy types are not realizable by M-curves of degree 9. In Appendix B, we give some generalization of the skein relation for Conway polynomial. [ABSTRACT FROM AUTHOR]

We study some geometric configurations related to projections of an irreducible algebraic curve embedded in CP³ onto embedded projective planes. These configurations are motivated by applications to static and dynamic computational vision. More precisely, we study how an irreducible closed algebraic curve X embedded in CP³, of degree d and genus g, can be recovered using its projections from points onto embedded projective planes. The embeddings are unknown. The only input is the defining equation of each projected curve. We show how both the embeddings and the curve in CP³ can be recovered modulo some action of the group of projective transformations of CP³. In particular in the case of two projections, we show how in a generic situation, a characteristic matrix of the pair of embeddings can be recovered. In the process we address dimensional issues and as a result find the minimal number of irreducible algebraic curves required to compute this characteristic matrix up to a finite-fold ambiguity, as a function of their degrees and genus. Then we use this matrix to recover the class of the couple of maps and as a consequence to recover the curve. In a generic situation, two projections define a curve with two irreducible components. One component has degree d(d - 1) and the other has degree d, being the original curve. Then we consider another problem. N projections, with known projection operators and N ≫ 1, are considered as an input and we want to recover the curve. The recovery can be done by linear computations in the dual space and in the Grassmannian of lines in CP³. Those computations are respectively based on the dual variety and on the variety of intersecting lines. In both cases a simple lower bound for the number of necessary projections is given as a function of the degree and the genus. A closely related question is also considered. Each point of a finite closed subset of an irreducible algebraic curve is projected onto a plane from a point: For each point the projection center is different. The projection operators are known. We show when and how the recovery of the algebraic curve is possible, in terms of the degree of the curve, and of the degree of the curve of minimal degree generated by the projection centers. [ABSTRACT FROM AUTHOR]

In Keilen (2003) we gave sufficient conditions for the irreducibility of the family Virr([This symbol cannot be presented in ASCII format]l,...,[This symbol cannot be presented in ASCII format]r) of irreducible curves in the linear system \D\t with precisely r singular points of topological, respectively, analytical, types [This symbol cannot be presented in ASCII format]l,... ,[This symbol cannot be presented in ASCII format]r on several classes of smooth projective surfaces Σ. The conditions were of the form [This symbol cannot be presented in ASCII format] where τ* is some invariant of singularity types, KΣ is the canonical divisor of Σ, and γ is some constant. In the present paper we improve this condition, that is, the constant γ, by a factor of 9. [ABSTRACT FROM AUTHOR]

The paper presents examples of complete singular toric varieties associated to the Niemeier lattices. The singularities and automorphisms of these varieties are seen to be closely related to the Golay codes. [ABSTRACT FROM AUTHOR]

For real algebraic varieties whose real algebraic cohomology group is maximal, a canonical homomorphism is constructed from the cohomology group of the set of complex points into the cohomology group of the set of real points, and then it is proved that this homomorphism is an isomorphism. [ABSTRACT FROM AUTHOR]

The article offers an alternative elementary proof to the one published by J. P. May in the paper "Munshi's proof of the Nullstellensatz," in which some of the algebraic technicalities are avoided. The alternative proof requires a simple version of the Noether normalization lemma. The proof needs the resultant of two polynomials but in such a simple manner that only one property of it is required. The analysis is confined to the weak form of Nullstellensatz because the strong form is easily derived from the weak form using the Rabinowitsch trick.

In this paper we investigate the mathematical properties of the integrability of the symmetric six-vertex model towards the view of algebraic geometry. We show that the algebraic variety originated from Baxter’s commuting transfer method is birationally isomorphic to a ubiquitous threefold known as Segre cubic primal. This relation makes it possible to present the most generic solution for the Yang–Baxter triple associated to this lattice model. The respective -matrix and Lax operators are parameterized by three independent affine spectral variables. [ABSTRACT FROM AUTHOR]