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11
votes
0answers
169 views

Weil conjectures for higher dimensional cycles?

Let $X$ be a smooth projective variety over $\mathbb{F}_{q}$. For each pair of positive integers $n$ and $d$, let $\text{Chow}_{n,d}(X)$ denote the (coarse) moduli space of $n$-cycles of degree $d$ on ...
8
votes
0answers
162 views

Motivic cohomology of a point

I was wondering how much is known about the integral motivic cohomology groups of $\mathrm{Spec}\, k$, $H^{n,p}_{\mathrm{mot}}(\mathrm{Spec}\, k,\mathbb{Z})$. One knows that ...
4
votes
0answers
135 views

Motivic Interpretation of Rationally Trivial Cycles

The Chow groups are defined by taking groups of cycles modulo rationally trivial cycles. One then has a cycle class map to etale cohomology (over the base field), and for a number field, one expects ...
3
votes
1answer
169 views

Proper pushforward of algebraic cycles

Let $f:X\to Y$ be a finite surjective morphism of smooth integral projective varieties over an algebraically closed field $k$ of characteristic 0. Denote by $CH_i(W):=Z_i(W)/\sim$ the Chow group of ...
4
votes
0answers
95 views

Non-universally trivial Chow group of zero-cycles on Fano hypersurfaces

Let $X$ be a smooth projective variety over a field $k$. By (one) definition, the Chow group of zero-cycles $CH_0(X)$ is universally trivial if, for every field extension $k \subset K$, the degree map ...
4
votes
1answer
173 views

$l$-dependence of the group of homologically zero cycles

Consider the class map $$cl:CH^i(X)\to H^{2i}_{cont}(X,\mathbb{Z}_l(i))$$ where the RHS is the continuous etale cohomology(defined by Jannsen in his paper "Continuous etale cohomology"). In this paper ...
3
votes
1answer
184 views

About the decomposition of a Chow group of a variety

I would like that someone helps me to find an article on the net treating the following decomposition : $ \mathrm{CH}^k (X)_{ \mathbb{Q} } = \displaystyle\bigoplus_{ i + j = k } \mathrm{CH}^{i,j} (X) ...
4
votes
0answers
193 views

Chow group over function field and algebraic equivalence

It is known that for smooth projective varieties $X,Y$ over $k=\bar k,$ $$CH^d(X_{k(Y)})=\varinjlim_{U\subset Y\ open}CH^d(X\times_k U)$$ I was wondering whether there was such an equality with ...
3
votes
2answers
159 views

Obstruction to get a galois invariant cycle

Let $X$ be a smooth projective variety over a finite field $k$, $G=Gal(\bar{k}/k)$ and $\Gamma\in CH^i(\bar{X})$ such that: $cl(\Gamma) \in H_{et}^{2i}(\bar{X},\mathbb{Z}_l(i))^G$ and $\exists$ ...
1
vote
1answer
159 views

Chow groups of locally trivial affine fibrations

Let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic $0$. A locally trivial $\mathbf{A}^n$-fibration is a morphism $\pi \colon Y \to X$ such that $\pi^{-1}(U)\cong ...
4
votes
1answer
307 views

When is the pullback in Chow groups defined?

This is the first time I ask a question on Mathoverflow, so I apologize in advance if it is not suitable/a duplicate/otherwise inappropriated. I am thinking about Voevodsky's category of motives and ...
2
votes
1answer
229 views

Self-intersection and generic point

The Wikipedia entry on intersection theory contains the following statement: [for C a curve, on a surface] "the self-intersection points of C is the generic point of C, taken with multiplicity C · ...
0
votes
0answers
120 views

Chow group of a product

Let $X$ and $Y$ be smooth varieties over $k$. I was wondering if there is a decomposition of the Chow group $CH(X\times Y)$ in terms of $CH(X)$ and $CH(Y)$ similar to the Kunneth decomposition of ...
0
votes
0answers
158 views

What can one say about zero-cycle groups for products of Chow motives

What can one say about the Chow group of zero-cycles (up to rational equivalence) for a product of smooth projective varieties and Chow motives (so, I am interested in the kernel $Chow_0(P)\otimes ...
1
vote
1answer
187 views

Deformation space form the point of view of intersection theory

I'm interested in deformations of subvarieties of a toric variety $X$. Suppose we know a subvariety $V$ in the Chow group of $X$, for example, $V$ is a linear combination of powers of hypersurfaces. ...
4
votes
2answers
308 views

on the Zariski sheafification of Quillen's K-theory

Let $X$ be a smooth k-variety and denote by $K_n$ Quillen's K-theory sheaf, that is: the Zariski sheaf on $X$ associated to the presheaf $U \mapsto K_n(U)$. The Bloch-Quillen formula says that ...
18
votes
1answer
1k views

For which varieties is the natural map from the Chow ring to integral cohomology an isomorphism?

My apologies if this question is too naive. Let $X$ be a smooth projective complex variety. There is a natural map $A^{\bullet}(X) \to H^{2\bullet}(X)$ of graded rings from the Chow ring of $X$ to ...
0
votes
1answer
141 views

Can generalization of Mumford’s theorem imply Mumford’s theorem for surface?

Mumford’s theorem for surface says that for a surface $S$ with $p_g(S)\neq0$ ,$\text{CH}_0(S)$ is not representable(or infinite-dimensional). But in Voisin's LECTURES ON THE HODGE ...
0
votes
1answer
263 views

About generalized Bloch conjecture

Conjecture((generalized Bloch conjecture)): Suppose that $\text{H}^{k,0}(X)=0$ for all $k>0$. Then $\text{CH}_0(X)=\mathbb{Z}.$ Is generalized Bloch conjecture known for complete intersections of ...
8
votes
1answer
346 views

Commutativity of the Chow ring in positive characteristic

I was looking in Ravi Vakil's notes on Intersection Theory, Class 20, where he introduces the bivariant intersection theory, in particular the Chow ring $A^\ast (X)$. On p. 2, he writes the following ...
7
votes
0answers
773 views

problematic proof of the moving lemma, second part ?

I have heard several times that the proof of the second part of the Chow's moving lemma (of algebraic geometry), is problematic; and that this is the reason Fulton, Intersection theory, does not use ...
0
votes
0answers
245 views

Vanishing of motivic cohomology with finite coefficients in negative degrees

I wonder whether the "finite coefficient version" of Beilinson-Soule conjecture i.e. the following statement holds or not. STATEMENT: Let $X$ be a smooth and projective scheme over a finite field ...
2
votes
1answer
341 views

Chow group of a (particular) motive [+ reference request]

I have two (not unrelated) questions. Let me first give a short introduction. Introduction For a general overview of the setup I refer to the introduction (§1) of [Zhang]. Let $k$ be a number field ...
2
votes
1answer
297 views

Is There a Mayer-Vietoris Spectral Sequence of Motivic Cohomology for Closed Coverings?

For etale cohomology, there is a spectral sequence of the following form ("Mayer-Vietories spectral sequence for closed covers"): $E_{1}^{p,q}=\oplus_{i_{0}< \cdots < i_{p}} H_{ Y_{i_{0} \cdots ...
3
votes
1answer
311 views

examples of Chow rings of surfaces

Can somone provide me (articles/literature) with examples of Chow rings of surfaces? (e.g. here: http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf Chapter 9) What I want is a list of (smooth ...
1
vote
1answer
171 views

Putting two complete varieties in a family over the projective line

Let $X$ and $Y$ be two proper varieties of dimension $n$ over a field $k$. I'm looking for "reasonable" conditions, under which, there exists a proper and dominant morphism $f:V\to \mathbb{P}^1_k$, ...
1
vote
1answer
221 views

Schubert problems to cycle class in Grassmanian

Say I have a family of linear spaces, and that I can solve all Schuber problems of that family (that is, how many members of the family pass through a set $S$ of linear spaces, where we consider all ...
3
votes
0answers
177 views

Generalized linear systems

Let $X$ be an algebraic variety and let $Z\subset X$ be a subvariety. Let $[Z]$ be the class defined by $Z$ in the Chow group. Let $L(Z)$ be set of effective algebraic cycles on $X$ linearly ...
2
votes
1answer
356 views

Zero-cycles on an arithmetic surface

Could anyone give a reference for the following statement, which I believe is true. "Let X be a regular scheme, flat over $Spec( \mathbb{Z}) $, with fiber dimension $1$. Then the Chow group $CH^2(X)$ ...
12
votes
1answer
613 views

Are Chow groups generated by local complete intersections?

Let $X$ be a smooth projective variety over an algebraically closed field. The Chow group $\mathbb Q\mathrm{CH}^d(X)$ is $\mathbb Q$--linearly generated by irreducible subvarieties $Z \subseteq X$ of ...
8
votes
1answer
451 views

Do we know the Chow groups of spheres?

Let $k$ be a field (of char. not $2$) and $X_k=\text{Spec} (k[x_1,\cdots,x_n]/(x_1^2+\cdots +x_n^2-1))$. Do we know the Chow groups $A_i (X_k)$? I could not find any references, even for $X_{\mathbb ...