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We characterize all possible relative positions between a circular hyperboloid of one sheet and a sphere through the roots of a characteristic polynomial associated to these quadrics. As an application, this provides a method to detect contact between the 2 surfaces by a simple calculation in many real world applications. [ABSTRACT FROM AUTHOR]

In this article we extend Voevodsky's nilpotence conjecture from smooth projective schemes to the broader setting of smooth proper dg categories. Making use of this noncommutative generalization, we then address Voevodsky's original conjecture in the following cases: quadric fibrations, intersection of quadrics, linear sections of Grassmannians, linear sections of determinantal varieties, homological projective duals, and Moishezon manifolds. [ABSTRACT FROM AUTHOR]

We give a characterization of the parabolic quadricQ(4,q). [ABSTRACT FROM PUBLISHER]

In a paper of 2002 Stefania Ferri gave a characterization of a quadric cone inPG(3,q). The statement of the theorem is true, but the first part of the proof does not work. The same author in a paper of 2004 gave a new correct, but quite tedious, proof of that part. In this short note we prove the first part in a simpler way. [ABSTRACT FROM PUBLISHER]

In this paper it is shown that Fano double quadrics of index 1 and dimension at least 6 are birationally superrigid if the branch divisor has at most quadratic singularities of rank at least 6. Fano double cubics of index 1 and dimension at least 8 are birationally superrigid if the branch divisor has at most quadratic singularities of rank at least 8 and another minor condition of general position is satisfied. Hence, in the parameter spaces of these varieties the complement to the set of factorial and birationally superrigid varieties is of codimension at least () + 1 and () + 1 respectively. [ABSTRACT FROM AUTHOR]

First we introduce the notion of commuting and parallel Ricci tensor for real hypersurfaces in the complex quadric Q m = S O m + 2 / S O 2 S O m . Then, according to the A -isotropic unit normal N , we give a complete classification of real hypersurfaces in Q m = S O m + 2 / S O 2 S O m with commuting and parallel Ricci tensor. [ABSTRACT FROM AUTHOR]

A generalized line star with respect to an elliptic quadric contained in the Klein quadric gives rise to a regular parallelism in real projective 3-space. This was shown by Betten and Riesinger (Results Math 47:226-241, ) using the Thas-Walker construction. They remark that the resulting description of the parallelism is equivalent to a much simpler one, but again the proof is hard. In practice, they always work with that simpler construction. We show that the Thas-Walker approach is not needed here. In fact, one can derive all relevant properties of the parallelism from the simple construction. We also give coordinate-free, short proofs of the topological properties of regular parallelisms in general and of those obtained from generalized line stars in particular. [ABSTRACT FROM AUTHOR]

For any three-dimensional projective space $${\mathbb{P}(V)}$$ , where V is a vector space over a field F of arbitrary characteristic, we establish a one-one correspondence between the Clifford parallelisms of $${\mathbb{P}(V)}$$ and those planes of $${\mathbb{P} (V \wedge V)}$$ that are external to the Klein quadric representing the lines of $${\mathbb{P}(V)}$$ . We also give two characterisations of a Clifford parallelism of $${\mathbb{P}(V)}$$ , both of which avoid the ambient space of the Klein quadric. [ABSTRACT FROM AUTHOR]

In a previous work, we introduced the notion of uniform generators of the Brauer group and proved that general diagonal cubic surfaces do not have such generators. In this paper, we prove that a similar non-existence result holds for affine diagonal quadrics. [ABSTRACT FROM AUTHOR]

We construct a family of ( q − 1 ) 2 -ovoids of Q ( 4 , q ) , the parabolic quadric of PG ( 4 , q ) , for q ≡ 3 ( mod 4 ) . The existence of ( q − 1 ) 2 -ovoids of Q ( 4 , q ) was only known for q = 3 , 7 , or 11. Our construction provides the first infinite family of ( q − 1 ) 2 -ovoids of Q ( 4 , q ) . Along the way, we also give a construction of ( q + 1 ) 2 -ovoids in Q ( 4 , q ) for q ≡ 1 ( mod 4 ) . [ABSTRACT FROM AUTHOR]

In this paper, we introduce a one-form Φ associated to a Lagrangian immersion f from Riemann surface M into the complex hyperquadric Q 2 . It is proved that f is minimal if and only if Φ is vanishing and f is H-minimal if and only if Φ is holomorphic. As an application, a family of S 1 -equivariant minimal Lagrangian tori in Q 2 are obtained. [ABSTRACT FROM AUTHOR]

We give necessary and sufficient criteria for a smooth Enriques surface S ⊂ ℙr to be scheme-theoretically an intersection of quadrics. Moreover we prove in many cases that, when S contains plane cubic curves, the intersection of the quadrics containing S is the union of S and the 2-planes spanned by the plane cubic curves. We also give a new (very quick) proof of the projective normality of S if deg S ≥ 12. [ABSTRACT FROM AUTHOR]

We study the geometry of the birational map between an intersection of a web of quadrics in P7 that contain a plane and the double octic branched along the discriminant of the web. [ABSTRACT FROM AUTHOR]

We continue to investigate pseudotoric geometry. Namely, we present a type of nontoric manifolds with pseudotoric structures. As an application, we construct a new pseudotoric structure on two-dimensional complex quadrics. [ABSTRACT FROM AUTHOR]

First we introduce the notion of Reeb invariant shape operator for real hypersurfaces in the complex quadric Q m = SO m + 2 / SO m SO 2 . Next we give a complete classification of real hypersurfaces in Q m = SO m + 2 / SO m SO 2 with invariant shape operator. [ABSTRACT FROM AUTHOR]

Abstract: We study affine hypersurfaces M which have isotropic difference tensor. Note that any surface always has isotropic difference tensor. Therefore, we may assume that . Such hypersurfaces have previously been studied by the first author and M. Djoric in [1] under the additional assumption that M is an affine hypersphere. Here we study the general case. As for affine spheres, we first show that isotropic affine hypersurfaces which are not congruent to quadrics are necessarily 5, 8, 14 or 26 dimensional. From this, we also obtain a complete classification in dimension 5. [Copyright &y& Elsevier]

We examine the solubility of a diagonal, translation invariant, quadratic equation system in arbitrary (dense) subsets $\mathcal{A}$ ⊂ ℤ and show quantitative bounds on the size of $\mathcal{A}$ if there are no non-trivial solutions. We use the circle method and Roth's density increment argument. Due to a restriction theory approach we can deal with equations in s ≥ 7 variables. [ABSTRACT FROM PUBLISHER]

We investigate basic features of Bianchi’s Bäcklund transformation of quadrics and obtain it under weaker minimal assumptions: the only Bäcklund transformation with defining surface is Bianchi’s Bäcklund transformation of quadrics. [ABSTRACT FROM AUTHOR]

Given an intersection of two quadrics $X\subset { \mathbb{P} }^{m- 1} $, with $m\geq 9$, the quantitative arithmetic of the set $X( \mathbb{Q} )$ is investigated under the assumption that the singular locus of $X$ consists of a pair of conjugate singular points defined over $ \mathbb{Q} (i)$. [ABSTRACT FROM AUTHOR]

We propose an elaborate geometry approach to explain the group law on twisted Edwards curves which are seen as the intersection of quadric surfaces in place. Using the geometric interpretation of the group law, we obtain the Miller function for Tate pairing computation on twisted Edwards curves. Then we present the explicit formulae for pairing computation on twisted Edwards curves. Our formulae for the doubling step are a little faster than that proposed by Arène et al. Finally, to improve the efficiency of pairing computation, we present twists of degrees 4 and 6 on twisted Edwards curves. [ABSTRACT FROM AUTHOR]

Abstract: By applying results on canal surfaces, we study exact rational parametrizations of fixed radius rolling ball blends of pairs of natural quadrics. We classify all configurations where this kind of rational parametrization is possible, and describe a general algorithm for parametrizing fixed radius rolling ball blends. The algorithm is then applied to parametrize the fixed radius rolling ball blends of pairs of natural quadrics. [Copyright &y& Elsevier]

Abstract: We consider two varieties associated to a web of quadrics W in . One is the base locus and the second one is the double cover of branched along the determinant surface of W. We show that small resolutions of these varieties are Calabi–Yau manifolds. We compute their Betti numbers and show that they are not birational in the generic case. The main result states that if the base locus of W contains a plane then in the generic case the two varieties are birational. [Copyright &y& Elsevier]

Abstract: We determine the groups of automorphisms and their orbits for nilpotent Lie algebras of class 2 and small dimension, over arbitrary fields (including the characteristic 2 case). [Copyright &y& Elsevier]

Abstract: A configuration of linear spaces in a projective space is a finite collection of linear subspaces. In this paper we study the degree 2 part of the ideal of such objects. More precisely, for a generic configuration of linear spaces Λ we determine , i.e. the Hilbert function of Λ in degree 2. [Copyright &y& Elsevier]

Abstract: The goal of this work is to prove that all Segre varieties , , (unless the quadric surface in ) support families of arbitrarily large dimension and rank of simple Ulrich (and hence ACM ) vector bundles. Therefore, they are of wild representation type. [Copyright &y& Elsevier]

In [8], a characterization of the finite quadric Veronesean νn2n by means of properties of the set of its tangent spaces is proved. These tangent spaces form a regular generalised dual arc. We prove an extension result for regular generalised dual arcs. To motivate our research, we show how they are used to construct a large class of secret sharing schemes. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we give a complete answer to the isotropy of an Albert bilinear form over the function field of a quadric in characteristic 2. As a consequence, we complete the classification of nongood bilinear forms of height and degree 2 given in Laghribi and Rehmann (2009) . [Copyright &y& Elsevier]

The problem of the ellipsoidal approximation of the sum of two ellipsoids optimal with respect to the minimum of multidimensional volume is considered. It is solved without the use of affinities and representation as a conditional optimization problem. The case of simultaneous degeneracy of the ellipsoids is considered. A geometrical interpretation of the approximation is given. Results of the numerical modeling are presented. [ABSTRACT FROM AUTHOR]

Abstract: White has conjectured that the toric ideal of a matroid is generated by quadric binomials corresponding to symmetric basis exchanges. We prove a stronger version of this conjecture for lattice path polymatroids by constructing a monomial order under which these sets of quadrics form Gröbner bases. We then introduce a larger class of polymatroids for which an analogous theorem holds. Finally, we obtain the same result for lattice path matroids as a corollary. [Copyright &y& Elsevier]

In this paper, we describe noncommutative versions of . These contain the class of noncommutative deformations of . [ABSTRACT FROM PUBLISHER]

In this paper, we study quadric surfaces in a Euclidean 3-space. Furthermore, we classify quadric surfaces in a Euclidean 3-space in terms of the Gaussian curvature and the mean curvature. [ABSTRACT FROM AUTHOR]

Abstract: We present a complete, exact and efficient implementation to compute the edge-adjacency graph of an arrangement of quadrics, i.e. surfaces of algebraic degree 2. This is a major step towards the computation of the full 3D arrangement. We enhanced an implementation for an exact parameterization of the intersection curves of two quadrics, such that we can compute the exact parameter value for intersection points and from that the edge-adjacency graph of the arrangement. Our implementation is complete in the sense that it can handle all kinds of inputs including all degenerate ones, i.e. singularities or tangential intersection points. It is exact in that it always computes the mathematically correct result. It is efficient measured in running times, i.e. it compares favorably to the only previous implementation. [ABSTRACT FROM AUTHOR]

This paper studies the CR-version of Obata theorem on a pseudo-Hermitian CR-manifold $ (M,\theta)$ with contact form $ \theta=\frac{1}{2i} (\partial \rho_A - \bar{\partial} \rho_A)$ $ \rho_A(z)=\vert z\vert^2 + \mathrm{Re} \sum^n_{j=1}A_jz^2_j-1$ $ A_j \in (-1,1).$ [ABSTRACT FROM AUTHOR]

In this paper we study the normal bundle of the embedding of subvarieties of dimension n -- 1 in the Grassmann variety of lines in . Making use of some results on the geometry of the focal loci of congruences ([[Arrondo, Bertolini and Turrini, Asian J. of Math. 5: 535--560, 2001]] and [[Arrondo, Bertolini and Turrini, Asian J. of Math. 9: 449--472, 2005]]), we give some criteria to decide whether the normal bundle of a congruence is ample or not. Finally we apply these criteria to the line congruences of small degree in . [ABSTRACT FROM AUTHOR]

Here we study curves C over $${\mathbb{F}_q}$$ contained in a hyperbolic quadric surface and such that $${\sharp (C(\mathbb{F}_q))}$$ is large. [ABSTRACT FROM AUTHOR]

In this article, we begin with arcs in PG(2, q) and show that they correspond to caps in PG(2 n, q) via the André/Bruck-Bose representation of PG(2, q) in PG(2 n, q). In particular, we show that a conic of PG(2, q) that meets ℓ in x points corresponds to a ( q + 1 − x)-cap in PG(2 n, q). If x = 0, this cap is the intersection of n quadrics. If x = 1 or 2, this cap is contained in the intersection of n quadrics and we discuss ways of extending these caps. We also investigate the structure of the n quadrics. [ABSTRACT FROM AUTHOR]

Quasi-quadricswere introduced by Penttila, De Clerck, O'Keefe andHamilton in [2]. They are defined as point sets which have the same intersection numbers with respect to hyperplanes as non-singular quadrics. We extend this definition in two ways. The first extension is to quasi-Hermitian varieties, which are point sets which have the same intersection numbers with respect to hyperplanes as non-singular Hermitian varieties. The second one is to singular quasi-quadrics, i.e. point sets κ which have the same intersection numbers with respect to hyperplanes as singular quadrics. Our starting point was to investigate whether every singular quasiquadric is a cone over a non-singular quasi-quadric. This question is tackled in the case of a point set K with the same intersection numbers with respect to hyperplanes as a point over an ovoid. [ABSTRACT FROM AUTHOR]

Abstract: We characterize the convexity properties of the tangent injectivity domain on an the ellipsoid of revolution in the oblate case. [ABSTRACT FROM AUTHOR]

Stillman posed a question as to whether the projective dimension of a homogeneous ideal $ I$. More recently, motivated by work on local cohomology modules in characteristic $ p$ is bounded by the sum of the degrees of the generators. We define a family of homogeneous ideals in a polynomial ring over a field of arbitrary characteristic whose projective dimension grows exponentially if the number and degrees of the generators are allowed to grow linearly. We therefore answer Zhang's question in the negative and provide a lower bound to any answer to Stillman's question. We also describe some explicit counterexamples to Zhang's question including an ideal generated by 7 quadrics with projective dimension 15. [ABSTRACT FROM AUTHOR]

Abstract: In this note, we give an improvement on the Bergman kernel for the domain . As an application, we describe how the zeroes of the kernel depend on the defining parameters . We also consider the domain . [Copyright &y& Elsevier]

Abstract: Let be an Artinian standard graded -algebra defined by quadrics. Assume that and that is algebraically closed, of characteristic . We show that is defined by a Gröbner basis of quadrics with, essentially, one exception. The exception is given by where is a complete intersection of three quadrics not containing a square of a linear form. [Copyright &y& Elsevier]

Abstract: We discuss rational parameterizations of surfaces whose support functions are rational functions of the coordinates specifying the normal vector and of a given non-degenerate quadratic form. The class of these surfaces is closed under offsetting. It comprises surfaces with rational support functions and non-developable quadric surfaces, and it is a subset of the class of rational surfaces with rational offset surfaces. We show that a particular parameterization algorithm for del Pezzo surfaces can be used to construct rational parameterizations of these surfaces. If the quadratic form is diagonalized and has rational coefficients, then the resulting parameterizations are almost always described by rational functions with rational coefficients. [Copyright &y& Elsevier]

Abstract: This paper deals with the problem of finding, for a given parametrization of an algebraic variety of arbitrary dimension, another parametrization with coefficients over a smaller field. We proceed adapting, to the parametric case, a construction introduced by A. Weil for implicitly given varieties. We find that this process leads to the consideration of new varieties of a particular kind (ultraquadrics, in the terminology of this paper) in order to check, algorithmically, several interesting properties of the given variety , such as the property of being reparametrizable over the smaller field. [Copyright &y& Elsevier]

Abstract.  This paper using a geometric approach produces vanishing and nonvanishing results concerning the spaces of twisted symmetric differentials $$H^{0}(X, S^{m}\Omega^{1}_{X} \bigotimes \mathcal {O}_{X}(k))$$ on subvarieties $$X \subset {\mathbb{P}}_{ N}$$, with k ≤ m. Emphasis is given to the case of k = m which is special and whose nonvanishing results on the dimensional range dim X > 2/3(N − 1) are related to the space of quadrics containing X and the variety of all tangent trisecant lines of X. The paper ends with an application showing that the twisted symmetric plurigenera, $$Q _{\alpha, m}(X_{ t}) = {\rm dim} H^{0}(X, S^{m}(\omega^{1}_{X_t} \bigotimes \alpha K_{X_t} ))$$ along smooth families of projective varieties Xt are not invariant even for α arbitrarily large. [ABSTRACT FROM AUTHOR]

Abstract: We construct a semiorthogonal decomposition of the derived category of coherent sheaves on a quadric fibration consisting of several copies of the derived category of the base of the fibration and the derived category of coherent sheaves of modules over the sheaf of even parts of the Clifford algebras on the base corresponding to this quadric fibration generalizing the Kapranov''s description of the derived category of a single quadric. As an application we verify that the noncommutative algebraic variety , where is the universal sheaf of even parts of Clifford algebras, is Homologically Projectively Dual to the projective space in the double Veronese embedding . Using the properties of the Homological Projective Duality we obtain a description of the derived category of coherent sheaves on a complete intersection of any number of quadrics. [Copyright &y& Elsevier]

We prove birational rigidity and calculate the group of birational automorphisms of a nodal ℚ-factorial double cover X of a smooth three-dimensional quadric branched over a quartic section. We also prove that X is ℚ-factorial provided that it has at most 11 singularities; moreover, we give an example of a non-ℚ-factorial variety of this type with 12 simple double singularities. [ABSTRACT FROM AUTHOR]

The general solution of the electroelastic problem for a transversely isotropic hyperboloid of revolution is used to find the stress concentration near a hyperboloidal neck in a piezoceramic body subjected to bending. The solution is a sum of four partial solutions for the case where the forces and the normal component of the induction vector on the neck surface are equal to zero. Numerical examples are given for specific external loads and properties of the body. The stress components and normal component of the induction vector near the neck vertex are plotted as a function of the external load and neck curvature [ABSTRACT FROM AUTHOR]