Affine Quadrics Research Articles

Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics

We establish the Hardy-Littlewood property (à la Borovoi-Rudnick) for Zariski open subsets in affine quadrics of the form q(x1,⋯,xn)=m, where q is a non-degenerate integral quadratic form in n⩾3 variables and m is a non-zero integer. This gives asymptotic formulas for the density of integral points taking coprime polynomial values, which is a quantitative version of the arithmetic purity of strong approximation property off infinity for affine quadrics.

On Infinite Effectivity of Motivic Spectra and the Vanishing of their Motives

This paper is dedicated to the study of the kernel of the motivization functor $M_{k}^c:SH^c(k)\to DM^c(k)$ (i.e., we try to describe those compact objects of $SH(k)$ whose associated motives vanish. Moreover, we study the question when the $m$-connectivity of $M^c_{k}(E)$ ensures the $m$-connectivity of $E$ itself (with respect to the corresponding homotopy t-structures). We prove that the kernel of $M_{k}^c$ vanishes and the corresponding connectivity detection statement is also valid if and only if $k$ is a non-orderable field; this is an easy consequence of the corresponding results of T. Bachmann (who considered the case where the $2$-adic cohomological dimension of $k$ is finite). We also sketch a deduction of these statements from the slice-convergence results of M. Levine. Moreover, for a general $k$ we prove that this kernel does not contain any $2$-torsion; the author also suspects that all its elements are odd torsion. Besides we prove that the kernel in question consists exactly of infinitely effective (in the sense of Voevodsky's slice filtration) objects of $SH^c(k)$ (assuming that the exponential characteristic of $k$ is inverted in the coefficient ring). These result allow (following another idea of Bachmann) to carry over his results on the tensor invertibility of certain motives of affine quadrics to the corresponding motivic spectra whenever $k$ is non-orderable. We also generalize a theorem of A. Asok.

EXTENSIONS OF ISOMORPHISMS OF SUBVARIETIES IN FLEXIBLE VARIETIES

Let X be an algebraic variety isomorphic to the complement of a closed subvariety of dimension at most n − 3 in \( {\mathbbm{A}}_{\mathrm{k}}^n \). We find some conditions under which an isomorphism of two closed subvarieties of X can be extended to an automorphism of X. We also study the similar problem for subvarieties of affine quadrics and SL(n, k).

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.

Motivic equivalence of affine quadrics

In this article we show that the motive of an affine quadric {q=1} determines the respective quadratic form.

Power-free values of polynomials on symmetric varieties

Given a symmetric variety Y defined over the rationals and a non-zero polynomial with integer coefficients, we use techniques from homogeneous dynamics to establish conditions under which the polynomial can be made r-free for a Zariski dense set of integral points on Y. We also establish an asymptotic counting formula for this set. In the special case that Y is a quadric hypersurface, we give explicit bounds on the size of r by combining the argument with a uniform upper bound for the density of integral points on general affine quadrics.

On the Invertibility of Motives of Affine Quadrics

We show that the reduced motive of a smooth affine quadric is invertible as an object of the triangulated category of motives DM(k,Z[1/e]) (where k is a perfect field of exponential characteristic e). We also establish a motivic version of the conjectures of Po Hu on products of certain affine Pfister quadrics. Both of these results are obtained by studying a novel conservative functor on (a subcategory of) DM (k Z[1/e]), the construction of which constitutes the main part of this work.

An explicit KO‐degree map and applications

The goal of this note is to study the analog in unstable A 1 -homotopy theory of the unit map from the motivic sphere spectrum to the Hermitian K-theory spectrum, that is, the degree map in Hermitian K-theory. We show that ‘Suslin matrices’, which are explicit maps from odd-dimensional split smooth affine quadrics to geometric models of the spaces appearing in Bott periodicity in Hermitian K-theory, stabilize in a suitable sense to the unit map. As applications, we deduce that K i M W ( F ) = G W i i ( F ) for i ⩽ 3 , which can be thought of as an extension of Matsumoto's celebrated theorem describing K 2 of a field. These results provide the first step in a program aimed at computing the sheaf π n A 1 ( A n ∖ 0 ) for n ⩾ 4 .

On the Brauer group of the product of a torus and a semisimple algebraic group

Let T be a torus (not assumed to be split) over a field F, and denote by nH et 2 (X,{ie375-1}) the subgroup of elements of the exponent dividing n in the cohomological Brauer group of a scheme X over the field F. We provide conditions on X and n for which the pull-back homomorphism nH et 2 (T,{ie375-2}) → n H et 2 (X × F T, {ie375-3}) is an isomorphism. We apply this to compute the Brauer group of some reductive groups and of non-singular affine quadrics. Apart from this, we investigate the p-torsion of the Azumaya algebra defined Brauer group of a regular affine scheme over a field F of characteristic p > 0.

Algebraic vector bundles on spheres

We determine the first non-stable A 1 -homotopy sheaf of SL n . Using techniques of obstruction theory involving the A 1 -Postnikov tower, supported by some ideas from the theory of unimodular rows, we classify vector bundles of rank at least d - 1 on split smooth affine quadrics of dimension 2 d - 1 . These computations allow us to answer a question posed by Nori, which gives a criterion for completability of certain unimodular rows. Furthermore, we study compatibility of our computations of A 1 -homotopy sheaves with real and complex realization.

Isolated Hypersurface Singularities and Special Polynomial Realizations of Affine Quadrics

The research is supported by the Australian Research Council. The fourth author is supported by the Russian Foundation for Basic Research and grant no. NSh-3476.2010.1 of the Leading Scientific Schools program.

The automorphism group of an affine quadric

We determine the automorphism group for a large class of affine quadrics over a field, viewed as affine algebraic varieties. The proof uses a fundamental theorem of Karpenko's in the theory of quadratic forms [13], along with some useful arguments of birational geometry. In particular, we find that the automorphism group of the n-sphere {x02+···+xn2=1} over the real numbers is just the orthogonal group O(n+1) whenever n is a power of 2. It is not known whether the same is true for arbitrary n. This result is reminiscent of Wood's theorem that when n is a power of 2, every real polynomial mapping from the n-sphere to a lower-dimensional sphere is constant [22].

On 1-isometries of affine quadrics over finite fields

Letq be a regular quadratic form on a vector space (V,\(\mathbb{F}\)) and assume\(4 \leqslant dim V \leqslant \infty \wedge |\mathbb{F}| \in \mathbb{N}\). A 1-isometry of the central quadric\(\mathcal{F}: = \{ x \in V|q(x) = 1\}\) is a permutation ϕ of\(\mathcal{F}\) such that $$q(x - y) = \nu \Leftrightarrow q(x^\varphi - y^\varphi ) = \nu \forall x,y \in \mathcal{F}$$ (*) holds true for a fixed element ν of\(\mathbb{F}\). For arbitraryν ∈\(\mathbb{F}\) we prove thatϕ is induced (in a certain sense) by a semi-linear bijection\((\sigma ,\varrho ):(V,\mathbb{F}) \to (V,\mathbb{F})\) such thatq oσ =ϱ oq, provided\(\mathcal{F}\) contains lines and the exceptional case\((\nu = 2 \Lambda |\mathbb{F}| = 3 \Lambda \dim V = 4 \Lambda |\mathcal{F}| = 24)\) is excluded. In the exceptional case and as well in case of dim V = 3 there are counterexamples. The casesν ≠ 2 and v=2 require different techniques.

Reductive group actions on affine quadrics with 1-dimensional quotient: Linearization when a linear model exists

We study reductive group actions on complex affine quadrics. Such an action is called linearizable if it is equivalent to the restriction of a linear orthogonal action in the ambient affine space of the quadric. A linear model for a given action is a linear orthogonal action with the same orbit types and equivalent slice representations. We prove that if a reductive group action on an affine quadric with a 1-dimensional quotient has a linear model, then the action is linearizable. As a consequence, the action is linearizable if certain topological conditions are satisfied.

Linear models for reductive group actions on affine quadrics

Linear models for reductive group actions on affine quadrics

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.

Quadrics as hyperplanes in finite affine geometries

In PG( n, q), n even, the number of points on a nondegenerate quadric is (q n − 1) (q − 1) , the same as the cardinality of the hyperplanes. In a previous article we showed that PG( n, q), n even, q odd, possesses a family of nondegenerate quadrics that act as hyperplanes, in the sense that the intersections of the former have the same cardinalities and structure as those of the latter. This leads to a family of ( q + 1)-caps which are the “lines” of a new incidence structure on the points of the original geometry. In the present article we describe the situation in AG( n, q), q odd. A family of “affine quadrics” is presented, whose members have q n−1 points, the same as the cardinality of the hyperplanes. A natural one-to-one correspondence arises between the two families. The intersections of these affine quadrics provide a family of q-sets which behave like lines and which are either q-caps or lines in the original geometry. To arrive at this construction, one starts with a symmetric matrix Q over GF( q), of size n + 1 and with minimal polynomial ( x − λ) n+1 . This leads to a ring of matrices of form c 0 I + c 1 Q + … + c n Q n , c i ϵ GF( q), and they define the quadrics in question.

Quadrics as hyperplanes in finite affine geometries

In PG( n , q ), n even, the number of points on a nondegenerate quadric is (q n − 1) (q − 1) , the same as the cardinality of the hyperplanes. In a previous article we showed that PG( n , q ), n even, q odd, possesses a family of nondegenerate quadrics that act as hyperplanes, in the sense that the intersections of the former have the same cardinalities and structure as those of the latter. This leads to a family of ( q + 1)-caps which are the “lines” of a new incidence structure on the points of the original geometry. In the present article we describe the situation in AG( n , q ), q odd. A family of “affine quadrics” is presented, whose members have q n −1 points, the same as the cardinality of the hyperplanes. A natural one-to-one correspondence arises between the two families. The intersections of these affine quadrics provide a family of q -sets which behave like lines and which are either q-caps or lines in the original geometry. To arrive at this construction, one starts with a symmetric matrix Q over GF( q ), of size n + 1 and with minimal polynomial ( x − λ ) n +1 . This leads to a ring of matrices of form c 0 I + c 1 Q + … + c n Q n , c i ϵ GF( q ), and they define the quadrics in question.