Pure motives with representable Chow groups

Let k be an algebraically closed field. We show using Kahn's and Sujatha's theory of birational motives that a Chow motive over k whose Chow groups are all representable (in the sense of Definition 2.1) belongs to the full and thick subcategory of motives generated by the twisted motives of curves.     

Transcendence degree of division algebras

We define a transcendence degree for division algebras by modifying the lower transcendence degree construction of Zhang. We show that this invariant has many of the desirable properties one would expect a noncommutative analogue of the ordinary transcendence degree for fields to have. Using this invariant, we prove the following conjecture of Small. Let k be a field, let A be a finitely generated k-algebra that is an Ore domain, and let D denote the quotient division algebra of A. If A does not satisfy a polynomial identity, then GKdim(K) ≤ GKdim(A) − 1 for every commutative subalgebra K of D.     

On the support of complete intersection zero-cycles

On an algebraic varietyY ? ? _N we will call complete intersection a 0-cycle when it is the intersection of Y with a codimension n complete intersection of ? _N . We consider the following problem: Let E?Y be given. Does E contain the support of a complete intersection 0-cycle? The two main theorems shown in this article give the answers in some cases: first, a negative answer for E some “big” subset of a singular irreducible algebraic variety; secondly, a positive answer for some “small” subset, on any algebraic variety.     

Transcendence degree one function fields over a finite field with many automorphisms

Let $\mathbb{K}$ be the algebraic closure of a finite field ${\mathbb{F}}_q$ of odd characteristic p. For a positive integer m prime to p, let $F=\mathbb{K}(x,y)$ be the transcendence degree 1 function field defined by $y^q+y=x^m+x^{?m}$. Let $t=x^{m(q?1)}$ and $H=\mathbb{K}(t)$. The extension $F|H$ is a non-Galois extension. Let K be the Galois closure of F with respect to H. By Stichtenoth [20], K has genus $\mathfrak{g}(K)=(qm?1)(q?1)$, p-rank (Hasse–Witt invariant) $\gamma (K)={(q?1)}^2$ and a $\mathbb{K}$-automorphism group of order at least $2q^{2}m(q?1)$. In this paper we prove that this subgroup is the full $\mathbb{K}$-automorphism group of K; more precisely ${\text{Aut}}_{\mathbb{K}}(K)=\mathrm{\Delta }?D$ where Δ is an elementary abelian p-group of order $q^2$ and D has an index 2 cyclic subgroup of order $m(q?1)$. In particular, $\sqrt{m}|{\text{Aut}}_{\mathbb{K}}(K)|>\mathfrak{g}{(K)}^{3/2}$, and if K is ordinary (i.e. $\mathfrak{g}(K)=\gamma (K)$) then $|{\text{Aut}}_{\mathbb{K}}(K)|>{\mathfrak{g}}^{3/2}$. On the other hand, if G is a solvable subgroup of the $\mathbb{K}$-automorphism group of an ordinary, transcendence degree 1 function field L of genus $\mathfrak{g}(L)\ge 2$ defined over $\mathbb{K}$, then $|{\text{Aut}}_{\mathbb{K}}(K)|\le 34{(\mathfrak{g}(L)+1)}^{3/2}<68\sqrt{2}\mathfrak{g}{(L)}^{3/2}$; see [15]. This shows that K hits this bound up to the constant $68\sqrt{2}$.Since ${\text{Aut}}_{\mathbb{K}}(K)$ has several subgroups, the fixed subfield $F^N$ of such a subgroup N may happen to have many automorphisms provided that the normalizer of N in ${\text{Aut}}_{\mathbb{K}}(K)$ is large enough. This possibility is worked out for subgroups of Δ.     

Chow groups and Borel-Moore schemes

Summary In this paper we study the Chow groups of schemes for which the class map to Borel-Moore homology is an isomorphism. Then we determine the Chow groups of the scheme Cop^k P ⁿ parametrizing finite coplanary subschemes of lenght k ofP ⁿ and of the variety of «complete S-tuples» of Le Barz.The authors were partially supported by the DGICYT.     

Cylinder Homomorphisms and Chow Groups

Let X be a projective algebraic manifold of dimension n (over C), CH¹(X) the Chow group of algebraic cycles of codimension l on X, modulo rational equivalence, and A¹(X) ? CH¹(X) the subgroup of cycles algebraically equivalent to zero. We say that A¹(X) is finite dimensional if there exists a (possibly reducible) smooth curve T and a cycle z∈CH¹(Γ × X) such that z_*:A¹(Γ)-A¹(X) is surjective. There is the well known Abel-Jacobi map λ₁:A¹(X)-J(X), where J(X) is the lth Lieberman Jacobian. It is easy to show that A¹(X)→J(X) A¹(X) finite dimensional. Now set with corresponding map A^*(X)→J(X). Also define Level . In a recent book by the author, there was stated the following conjecture: where it was also shown that (?) in (**) is a consequence of the General Hodge Conjecture (GHC). In this present paper, we prove A^*(X) finite dimensional ?? Level (H^*(X)) ≤ 1 for a special (albeit significant) class of smooth hypersurfaces. We make use of the family of k-planes on X, where ([…] = greatest integer function) and d = deg X; moreover the essential technical ingredients are the Lefschetz theorems for cohomology and an analogue for Chow groups of hypersurfaces. These ingredients in turn imply very special cases of the GHC for our choice of hypersurfaces X. Some applications to the Griffiths group, vanishing results, and (universal) algebraic representatives for certain Chow groups are given.     

Geometric motives and the h-topology

The aim of this paper is to give a detailed proof of a comparison of Voevodsky’s categories of geometric motives with and without transfers, respectively. The latter category is defined by means of h-topology introduced by Voevodsky, a topology essentially given by Zariski coverings, finite coverings and blowups.     

Bounded enumeration reducibility and its degree structure

We study a strong enumeration reducibility, called bounded enumeration reducibility and denoted by ≤_be, which is a natural extension of s-reducibility ≤_s. We show that ≤_s, ≤_be, and enumeration reducibility do not coincide on the ${\Pi^0_1}$ –sets, and the structure ${\boldsymbol{\mathcal{D}_{\rm be}}}$ of the be-degrees is not elementarily equivalent to the structure of the s-degrees. We show also that the first order theory of ${\boldsymbol{\mathcal{D}_{\rm be}}}$ is computably isomorphic to true second order arithmetic: this answers an open question raised by Cooper (Z Math Logik Grundlag Math 33:537–560, 1987).     

A Presentation for the Chow Ring of

We give a presentation for the Chow ring of the moduli space of degree 2 stable maps from 2-pointed rational curves to the projective line. Also, integrals of all degree 4 monomials in the hyperplane pullbacks and boundary divisors of this ring are computed using equivariant intersection theory. Finally, the presentation is used to give a new computation of the (previously known) values of the genus zero, degree 2, 2-pointed gravitational correlators of the projective line.     

On the Chow Ring of a Flag

Let G be a reductive linear algebraic group over an algebraically closed field K, let P? be a parabolic subgroup scheme of G containing a Borel subgroup B, and let P = P?_red ? P? be its reduced part. Then P is reduced, a variety, one of the well known classical parabolic subgroups. For char(K) = p > 3, a classification of the P?'s has been given in [W1]. The Chow ring of G/P only depends on the root system of G. Corresponding to the natural projection from G/P to G/P? there is a map of Chow rings from A(G/P?) to A(G/P). This map will be explicitly described here. Let P = B, and let p > 3. A formula for the multiplication of elements in A(G/P?) will be derived. We will prove that A(G/P?) ? A(G/P) (abstractly as rings) if and only if G/P ? G/P? as varieties, i. e., the Chow ring is sensitive to the thickening. Furthermore, in certain cases A(G/P?) is not any more generated by the elements corresponding to codimension one Schubert cells.     

The Computational Complexity of the Chow Form

We present a bounded probability algorithm for the computation of the Chowforms of the equidimensional components of an algebraic variety. In particular, this gives an alternative procedure for the effective equidimensional decomposition of the variety, since each equidimensional component is characterized by its Chow form. The expected complexity of the algorithm is polynomial in the size and the geometric degree of the input equation system defining the variety. Hence it improves (or meets in some special cases) the complexity of all previous algorithms for computing Chow forms. In addition to this, we clarify the probability and uniformity aspects, which constitutes a further contribution of the paper. The algorithm is based on elimination theory techniques, in line with the geometric resolution algorithm due to M. Giusti, J. Heintz, L. M. Pardo, and their collaborators. In fact, ours can be considered as an extension of their algorithm for zero-dimensional systems to the case of positive-dimensional varieties. The key element for dealing with positive-dimensional varieties is a new Poisson-type product formula. This formula allows us to compute the Chow form of an equidimensional variety from a suitable zero-dimensional fiber. As an application, we obtain an algorithm to compute a subclass of sparse resultants, whose complexity is polynomial in the dimension and the volume of the input set of exponents. As another application, we derive an algorithm for the computation of the (unique) solution of a generic overdetermined polynomial equation system.     

Compatibility of weak approximation for zero-cycles on products of varieties

The Brauer-Manin obstruction is conjectured to be the only obstruction to weak approximation for zero-cycles on proper smooth varieties defined over number fields. We prove that the conjecture is compatible for products of rationally connected varieties, K3 surfaces, Kummer varieties and one curve.