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### 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 ...

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### $l$-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 ...

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174 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) ...

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### 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 ...

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### 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
$\exists$ ...

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156 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 ...

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253 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 ...

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224 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 · ...

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107 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 ...

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153 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 ...

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### 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. ...

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303 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 ...

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**1**answer

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### 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 ...

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139 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 ...

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### 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 ...

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334 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 ...

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### 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 ...

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233 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 ...

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334 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 ...

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292 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 ...

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306 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 ...

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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 $f:V\to \mathbb{P}^1_k$, ...

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### 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 ...

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### 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 ...

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### 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 $CH^2(X)$ ...

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586 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 ...

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### 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 ...