Projective Quadrics Research Articles

Commutative actions on smooth projective quadrics

By a commutative action on a smooth quadric Qn in we mean an effective action of a commutative connected algebraic group on Qn with an open orbit. We show that for all commutative actions on Qn are additive actions described by Sharoiko in 2009. So there is a unique commutative action on Qn up to equivalence. For n = 2 there are three commutative actions on Q 2 up to equivalence, for n = 1 there are two commutative actions on Q 1 up to equivalence.

Affine quadrics and the Picard group of the motivic category

In this paper we study the subgroup of the Picard group of Voevodsky’s category of geometric motives$\operatorname{DM}_{\text{gm}}(k;\mathbb{Z}/2)$generated by the reduced motives of affine quadrics. Our main tools here are the functors of Bachmann [On the invertibility of motives of affine quadrics, Doc. Math.22(2017), 363–395], but we also provide an alternative method. We show that the group in question can be described in terms of indecomposable direct summands in the motives of projective quadrics over$k$. In particular, we describe all the relations among the reduced motives of affine quadrics. We also extend the criterion of motivic equivalence of projective quadrics.

Witt groups of smooth projective quadrics

Let k be a commutative ring containing 12. In this paper, we construct exact sequences connecting Witt groups of smooth projective quadrics over k and Clifford algebras. The exact sequences have an application to a classical problem in quadratic form theory: the Witt kernel of function fields of quadrics. We also apply the exact sequences to compute Witt groups of certain kinds of smooth projective quadrics.

On intersections of two real quadrics

This paper is devoted to describing properties of complete intersections of two real projective quadrics. For brevity we call such varieties biquadrics. One of the main sections is devoted to real projective spaces on real biquadrics. In another main section we study the topology of the real parts of biquadrics. The other sections are auxiliary.

On the splitting of quasilinear p-forms

Abstract Using an analogue of Knebusch's generic splitting tower invariant in the theory of non-singular quadratic forms, we study the splitting behaviour of quasilinear (or Fermat-type) forms of degree p over fields of characteristic p > 0. Several new applications of our main results to the theory of quasilinear quadratic forms are provided, including an analogue of a theorem of Vishik relating to the existence of outer excellent connections in the motives of smooth projective quadrics over fields of characteristic different from 2, partial results towards a quasilinear version of Karpenko's theorem on the possible values of the first higher Witt indices of non-singular quadratic forms in characteristic not 2, and a proof of a conjecture of Hoffmann concerning quadratic forms with maximal splitting in the quasilinear case.

Quadratic forms and quadrics of space over local rings

Relatively recent and new results on the enumeration of classes of projective congruent quadrics and classes of projective equivalent quadrics of projective spaces over a local ring R = 2R with nilpotent principal maximal ideal are reflected in the paper. For the case of the basic ring R with nonprincipal maximal ideal, the enumerations of quadrics of projective plane are given up to projective equivalence.

Hassett–Tschinkel correspondence: Modality and projective hypersurfaces

In 1999, B. Hassett and Yu. Tschinkel introduced a remarkable correspondence between generically transitive actions of the commutative unipotent algebraic group G a n and finite-dimensional local algebras. In this paper we develop Hassett–Tschinkel correspondence and calculate modality of generically transitive G a n -actions on projective spaces, classify actions of modality one, and characterize generically transitive G a n -actions on projective hypersurfaces of given degree. In particular, actions on degenerate projective quadrics are studied.

Some Comments on Projective Quadrics Subordinate to Pseudo-Hermitian Spaces

We study in some detail the structure of the projective quadric Q′ obtained by taking the quotient of the isotropic cone in a standard pseudo-hermitian space H p,q with respect to the positive real numbers \({\mathbb R^{+}}\) and, further, by taking the quotient \({\tilde Q = Q^\prime /U(1)}\). The case of signature (1, 1) serves as an illustration. Q̃ is studied as a compactification of \({\mathbb R \times H_{p-1,q-1}}\)

On Conformal Infinity and Compactifications of the Minkowski Space

Using the standard Cayley transform and elementary tools it is reiterated that the conformal compactification of the Minkowski space involves not only the "cone at infinity" but also the 2-sphere that is at the base of this cone. We represent this 2-sphere by two additionally marked points on the Penrose diagram for the compactified Minkowski space. Lacks and omissions in the existing literature are described, Penrose diagrams are derived for both, simple compactification and its double covering space, which is discussed in some detail using both the U(2) approach and the exterior and Clifford algebra methods. Using the Hodge * operator twistors (i.e. vectors of the pseudo-Hermitian space H_{2,2}) are realized as spinors (i.e., vectors of a faithful irreducible representation of the even Clifford algebra) for the conformal group SO(4,2)/Z_2. Killing vector fields corresponding to the left action of U(2) on itself are explicitly calculated. Isotropic cones and corresponding projective quadrics in H_{p,q} are also discussed. Applications to flat conformal structures, including the normal Cartan connection and conformal development has been discussed in some detail.

Notes on the representational possibilities of projective quadrics in four dimensions

Notes on the representational possibilities of projective quadrics in four dimensions

Holes in [formula omitted

Let F be an arbitrary field of characteristic ≠2. We write W ( F ) for the Witt ring of F, consisting of the isomorphism classes of all anisotropic quadratic forms over F. For any element x ∈ W ( F ) , its dimension dim x is defined as the dimension of a quadratic form representing x. The elements of all even dimensions form an ideal denoted by I ( F ) . The filtration of the ring W ( F ) by the powers I ( F ) n of this ideal plays a fundamental role in the algebraic theory of quadratic forms. The Milnor conjectures, recently proved by Voevodsky and Orlov–Vishik–Voevodsky, describe the successive quotients I ( F ) n / I ( F ) n + 1 of this filtration, identifying them with Galois cohomology groups and with the Milnor K-groups modulo 2 of the field F. In the present article we give a complete answer to a different longstanding question concerning I ( F ) n , asking about the possible values of dim x for x ∈ I ( F ) n . More precisely, for any n ⩾ 1 , we prove that (*) dim I n = { 2 n + 1 − 2 i | i ∈ [ 1 , n + 1 ] } ∪ ( 2 Z ∩ [ 2 n + 1 , + ∞ ) ) , where dim I n is the set of all dim x for all x ∈ I ( F ) n and all F. Previously available partial informations on dim I n include the classical Arason–Pfister theorem (saying that ( 0 , 2 n ) ∩ dim I n = ∅ ) as well as a recent Vishik's theorem on ( 2 n , 2 n + 2 n − 1 ) ∩ dim I n = ∅ (the case n = 3 is due to Pfister, n = 4 to Hoffmann). The proof of ( * ) is based on computations in Chow groups of powers of projective quadrics (involving the Steenrod operations); the method developed can be also applied to other types of algebraic varieties.

Pseudo unitary conformal groups and Clifford algebras for standard pseudo hermitian spaces

This paper, self-contained, deals with pseudo-unitary spin geometry. First, we present pseudo-unitary conformal structures over a 2n-dimensional complex manifold V and the corresponding projective quadrics \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{H}_{{p,q}} \) for standard pseudo-hermitian spaces Hp,q. Then we develop a geometrical presentation of a compactification for pseudo-hermitian standard spaces in order to construct the pseudo-unitary conformal group of Hp,q. We study the topology of the projective quadrics \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{H}_{{p,q}} \) and the “generators” of such projective quadrics. Then we define the space S of spinors canonically associated with the pseudo-hermitian scalar product of signature (2n−1, 2n−1). The spinorial group Spin U(p,q) is imbedded into SU(2n−1, 2n−1). At last, we study the natural imbeddings of the projective quadrics \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{H}_{{p,q}}. \)

Common Transversals and Tangents to Two Lines and Two Quadrics in P

We solve the following geometric problem, which arises in several {\mbox three-dimensional} applications in computational geometry: For which arrangements of two lines and two spheres in ${\Bbb R}^3$ are there infinitely many lines simultaneously transversal to the two lines and tangent to the two spheres? We also treat a generalization of this problem to projective quadrics. Replacing the spheres in ${\bf R}^3$ by quadrics in projective space ${\Bbb P}^3$, and fixing the lines and one general quadric, we give the following complete geometric description of the set of (second) quadrics for which the two lines and two quadrics have infinitely many transversals and tangents: in the nine-dimensional projective space ${\Bbb P}^9$ of quadrics, this is a curve of degree 24 consisting of 12 plane conics, a remarkably reducible variety.

ON THE LEVEL OF A QUATERNION ALGEBRA

In [6] D. W. Lewis proved the existence of a quaternion division algebra of level 2 n or 2 n + 1 for any integer n ≥ 1. In this note, we present news examples of quaternion division algebras of such levels. We also recover a result of P. Koprowski [4]. Our method is different and based on function field of projective quadrics.

Criteria of motivic equivalence for quadratic forms and central simple algebras

We give a short, elementary, and characteristic independent proof of the criterion for motivic isomorphism of two projective quadrics discovered by A. Vishik [24]. We also give a criterion for motivic isomorphism of two Severi-Brauer varieties.

Motivic equivalence of quadratic forms

Let X_\phi and X_\psi be projective quadrics corresponding to quadratic forms \phi and \psi over a field F . If X_\phi is isomorphic to X_\psi in the category of Chow motives, we say that \phi and \psi are motivic isomorphic and write \phi\stackrel{m}\sim\psi . We show that in the case of odd-dimensional forms the condition \phi\stackrel{m}\sim\psi is equivalent to the similarity of \phi and \psi . After this, we discuss the case of even-dimensional forms. In particular, we construct examples of generalized Albert forms q_1 and q_2 such that q_1\stackrel{m}\sim q_2 and q_1\not\sim q_2 .

Pin structures and the modified Dirac operator

General theorems on pin structures on products of manifolds and on homogeneous (pseudo-) Riemannian spaces are given and used to find explicitly all such structures on odd-dimensional real projective quadrics, which are known to be non-orientable (Cahen et al. 1993). It is shown that the product of two manifolds has a pin structure if and only if both are pin and at least one of them is orientable. This general result is illustrated by the example of the product of two real projective planes. It is shown how the Dirac operator should be modified to make it equivariant with respect to the twisted adjoint action of the Pin group. A simple formula is derived for the spectrum of the Dirac operator on the product of two pin manifolds, one of which is orientable, in terms of the eigenvalues of the Dirac operators on the factor spaces.

On 0-distance-preserving permutations of affine and projective quadrics

Two pointsX, Y of a quadricF in an affine or projective space are called 0-distant, writtenX ≈Y, if they coincide or if their connecting line is a subset ofF. In this paper, answers are given to the question under which conditions an arbitrary permutation ϕ ofF satisfyingX ≈Y ⇔Xϕ ≈Yϕ ∀X,Y e F can be extended to a collineation of the given space.

Spin structures on real projective quadrics

Projective quadrics are known to be conformal compactifications of Euclidean spaces. In particular, the (projective) real quadric Q p,q = (S p × S q )/ Z 2 is associated, in this manner, with the flat space R p+q endowed with a metric tensor of signature ( p, q). For p and q positive, the quadric Q p,q is orientable iff p + q is even. The quadric has two natural metrics, invariant with respect to the action of O( p + 1) × O( q + 1): a proper Riemannian one and a pseudo-Riemannian metric of signature ( p,q). This paper contains an explicit description of spin structures on real, even-dimensional quadrics for both metrics, whenever these structures exist. In particular, it is shown that, for p and q even positive, the proper (pseudo-Riemannian) metric gives rise to two inequivalent spin structures iff p + q ≡ 2 (mod 4) ( p + q ≡ 0 (mod 4)). If p and q are odd and > 1, then there is no spin structure for their metric whenever p + q ≡ 0 (mod 4); otherwise, there are two spin structures for each of the metrics. There always exist spin structures on real quadrics with a Lorentzian metric, i.e., when p and q are odd and p or q = 1.