Questions tagged [chow-groups]

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Constancy of the degree of a top-dimensional product of Chern classes in smooth proper families

Let $S$ be a Dedekind scheme. Let $f:X\to S$ be a smooth proper morphism with connected $n$-dimensional fibers. Assume that Hodge symmetry holds for all the closed fibers of $f$. Are the degrees of $\...
4
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2answers
253 views

Are Chow groups invariant under universal homeomorphisms?

Let $f:X\to Y$ be a universal homeomorphism of schemes, $R$ a coefficient ring. Which assumptions on $f$ and $R$ are suffient to ensure that the pullback map $f^*$ of $R$-linear Chow groups is ...
3
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169 views

Relations between rational algebraic K-theory and Chow groups

A consequence of Grothendieck's Riemann-Roch Theorem is the fact that the Chern character induces an isomorphism between algebraic $ch: K_{0}(X) \otimes \mathbb{Q} \stackrel{\cong}{\rightarrow} C H^{*...
2
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1answer
295 views

Looking for examples of not injective maps and not surjective maps of the form $ A_{k} (X) \to H_{2k} ( X , \mathbb{Z} ) $

Here: https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/alggeom-2002-c9.pdf, on pages: $ 1 $ and $ 2 $, we find the following paragraph: For any scheme of finite type over a ground field ...
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134 views

Degree of varieties swept out by linear spaces (Eisenbud & Harris' 3264 and All That)

I have a couple of questions on statements from Harris' and Eisenbud's lecture "3264 and All That" at page 145, Section 4.2.3: Varieties swept out by linear spaces. The contents of 4.2.3 answers ...
6
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0answers
169 views

Chow ring of $E_7$ varieties

Consider a split algebraic group $G$ of type $E_7$ over a field of characteristic zero. It is known that some subgroups $P_i$ of $G$, which are called parabolic, have the property that the object $G/...
4
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1answer
164 views

Question regarding intersection product in Chow group of $\mathbb{P}^n\times\mathbb{P}^m $

Assume we are working over complex numbers. Let $\pi:\mathbb{P}^n\times\mathbb{P}^m\rightarrow \mathbb{P}^n (m,n>0)$ be the first projection. Suppose we are given a vector bundle $E$ over $\mathbb{...
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0answers
69 views

Numerical and rational equivalences on intersection of divisors

Let $X$ be a smooth projective variety over a finite field. Since $Pic^0(X)$ is finite and $Pic^0(X)$ can be identified with numerically equivalent to zero divisors this implies that for divisors on $...
2
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1answer
200 views

Question regarding Chow group of a blow-up

Let $X$ be a smooth complex projective variety, and $Y\hookrightarrow X$ be a smooth projective subvariety. Let $\pi:\tilde{X}\rightarrow X$ be the blow-up along $Y$, and let $j:E\hookrightarrow \...
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193 views

Few questions about the algebraic cycles and the conjectures of Beilinson and Tate

I have three slightly related questions about algebraic cycles which I am just going to list them. I'd really appreciate any answers: 1) Is there any example of a smooth projective variety $X$ over a ...
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80 views

Defining pull-back of Chow groups under a morphism of special type

Let $X$ be a normal complex projective variety (not necessarily smooth), and let $Y$ be a smooth complex projective variety. Let $Z\subset X$ be a smooth closed subvariety. Let $\pi : Y\rightarrow X$...
2
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0answers
113 views

Computing Chow group of a variety which is almost a blow-up of another variety

Let $X$ be a normal complex projective variety (not necessarily smooth), and let $Y$ be a smooth complex projective variety. Let $Z\subset X$ be a smooth closed subvariety. I have a morphism which is ...
2
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0answers
160 views

Vanishing of Chow groups in high codimension

Let $X$ be a smooth affine variety of dimension $n>2$ over $\mathbb{C}$. From the examples I have seen (admittedly very little) it seems to me that these varieties don't have torsion classes a ...
2
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69 views

Chow group of a pair

In a paper by S. Landsburg the (higher) Chow groups of a pair $(X,Y)$ are defined when $Y$ is a smooth closed subvariety of a smooth variety $X$ as follows. We consider the sub-complex $z^{*}(X;.)_{Y}...
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34 views

Calculations of residue homomorphisms in cycle modules

In the proof of Proposition 2.2 and Theorem 2.3 in Chow groups with coefficients https://eudml.org/doc/233731 written by M. Rost, he wrote $\mathbb{A}^{1}={\rm Spec}F[u], \mathbb{A}^{2}={\rm Spec}F[...
4
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1answer
157 views

Proper locally trivial bundle is injective on Chow groups

If $X\to Y$ is a map of varieties that is Zariski-locally isomorphic to a projection $U\times P\to U$ with $P$ (smooth) proper, I think the pullback $A_{\bullet}(Y)\to A_{\bullet}(X)$ is supposed to ...
3
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0answers
218 views

The Chow ring of a blow-up along a badly embedded subscheme

Let $X$ be a variety and $i:Y\hookrightarrow X$ be a regularly embedded closed subvariety, and write $\tilde X$ for $Bl_Y X$ the blow-up of $X$ along $Y$. Let $A^*(X)$ refer to the Fulton-Macpherson ...
2
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1answer
163 views

Linear sections of $Gr(V,2)$

Let $V$ be a vector space, and consider $G=Gr(V,2)\subset \mathbb{P}^N$ embedded via the Plucker embedding. Let $W\subset \mathbb{P}^N$ be a linear subspace. I want to find the class $[W\cap G]\in A(G)...
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122 views

homologically trivial $1$-cycles and surfaces

Let $X$ be a smooth (complex) threefold and $\gamma\in {\rm CH}_1(X)$ a homologically trivial $1$-cycle. Is there a way to construct a (singular) surface $S\subset X$ supporting $\gamma$ such that, ...
3
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1answer
533 views

A question on Grothendieck Riemann Roch

As an exercise for myself I wanted to check GRR in the following situation. Consider $P:X \rightarrow B$ to be an Weierstrass elliptic fibration with a section, and $X\times_B X$ be the fiber product ...
3
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0answers
321 views

Chern class map and the exponential sequence

Let $X$ be a smooth projective variety over the complex numbers, and $$c^1_X : \text{NS}(X)\to H^2_{\rm Betti}(X,\mathbf{Z}(1))$$ the first cycle map to Betti cohomology. The cokernel $\text{coker}(c^...
2
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0answers
289 views

Specialization maps for Chow groups

Let $S$ be a finite type regular integral affine scheme of finite type over $\text{Spec}(\mathbf{Z})$, and $\mathcal{X}\to S$ a smooth projective morphism. Let $\eta$ be the generic point of $S$, $s\...
5
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0answers
201 views

Divisibility properties of Chow groups (beyond Roitman's theorem)

For affine varieties over separably closed fields, there are classical vanishing theorems for cohomology. For an affine variety $X$ of dimension $d$ over $\mathbb{C}$, we have ${\rm H}^i_{\rm sing}(X(\...
1
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1answer
430 views

Pull-back of algebraic cycles

Since today is the Chow-variety day, I'm going to ask my question here. Suppose I have a smooth projective variety $X$ over a field of characteristic zero, and a smooth hyperplane $p: H\...
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0answers
177 views

Codimension restrictions on intersections

This is a question I stumbled across earlier this week. I see a similar one has been asked here. Suppose I have a smooth quasi projective variety $X$ over a field $K$, and I call $\text{Chow}^r(X\...
2
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1answer
136 views

Effective cycles of codimension 1 and field extensions

Let $X$ be a smooth quasi-projective variety over a field $k$, with pure dimension $d$, $K/k$ an arbitrary field extension. For any algebraic cycle $\eta$ of codimension $1$ on $X_K$ ($\eta\in Z^1(...
4
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311 views

Homotopical enhancements of cycle class maps

Fix a smooth projective variety $X$ over the complex numbers. We write $H^n(X,\mathbf{Z}(d)) = \text{CH}^d(X, 2d-n)$ for Bloch's higher Chow groups. Notation For a field $k$, recall $\Delta^n_{k} :=...
4
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1answer
442 views

Chow group and base change

Let $k$ be a field with algebraic closure $\overline{k}$. Let $f\colon X\to k$ be a smooth projective variety(geometrically connected) over $k$. Is the base change map $$\phi_i\colon \mathrm{CH}^i(X)...
2
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1answer
220 views

Chow ring of product of Brauer-Severi Varieties

Let $K$ be a field, $\alpha, \beta \in \mathrm{Br}(K)$, let $X,Y$ be their Brauer-Severi Varieties, is there a way to calculate $A^*(X\times Y)$? For example, if $\alpha,\beta$ both has degree $5$, $...
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0answers
318 views

Motivic $\mathbf{Z}(1)$

I know that the Bloch higher Chow complex, $\mathbf{Z}(i)_{\mathcal{M}}$, on a smooth scheme over a field $k$, reads, in degree $1$: $$\mathbf{Z}(1)_{\mathcal{M}}\simeq\mathbf{G}_m[-1].$$ How to see ...
7
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1answer
314 views

Higher Chow groups for complete smooth intersections?

Let $F$ be a smooth complete intersection of $r$ hypersurfaces of degree $d_{1},\dots,d_{r}$ in $\mathbb{P}^{n+r}$ over an algebraic closed field. A classical result of A. Roitman says that the group ...
6
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1answer
668 views

Higher Chow groups revisited

Let $X$ be an algebraic variety over a field $k$. Bloch defines the "algebraic singular complex" using the algebraic simplices $$\Delta^n = \text{Spec}(k[x_0,\dots,x_n]/(x_0+x_1+\dots+x_n=1) \subset ...
7
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1answer
655 views

Algebraic cycles, Chow spaces and Hilbert-Chow morphisms

In the sequel, let $S$ be a scheme, and $X$ a locally of finite type algebraic space over $S$. In his thesis ([R1-R4]), David Rydh introduces, among several others, the notion of relative cycles on $...
6
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137 views

Why is $[l_1]+[l_2]+[l_3]$ a constant in $CH_0(F(X))$, where $F(X)$ is the Fano variety of lines of a cubic fourfold?

Let $X\subset \mathbb P^5$ be a cubic fourfold and $F(X)$ be its Fano varieties of lines. Let $\mathbb P^2$ be a plane such that the intersection $\mathbb P^2\cap X$ consists of three lines $L_1,L_2,$ ...
11
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1answer
485 views

Chow ring of Hilbert scheme of 4 points in $\mathbb{P}^2$

What is known about the Chow ring of the Hilbert scheme of length 4 subschemes of $\mathbb{P}^2$? I know there is work on cycles on Hilbert schemes in the literature, but I don't know what can be ...
12
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1answer
623 views

Chow rings of smooth toric varieties

In his 1987 article The geometry of toric varieties Danilov gives a combinatorial presentation of Chow rings of complete smooth toric varietes. Given a complete unimodular fan $\Sigma$ we have $$ A^*(...
9
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453 views

Relative Chow groups

Most cohomologies have the notion of cohomology with support on a closed subspace, and also cohomology with compact support. In general, for any morphism $f\colon Y\to Z$ the inverse image fits into a ...
5
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152 views

Fulton's pullback vs. Pullback via Gersten complexes vs. Pullback coming from motivic homotopy SH(k)

Let $f: X \rightarrow Y$ be a morphism of smooth projective $k$-schemes (let's assume $f$ flat or even smooth). There is pullback in Fulton's style $f^*_{Ful}: CH^p(Y) \rightarrow CH^p(X)$ given by $[...
18
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1answer
503 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 ...
9
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0answers
581 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 $H^{n,n}_{\mathrm{mot}}(\...
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0answers
207 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
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1answer
496 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 $i$...
5
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0answers
240 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
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1answer
217 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 ...
4
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1answer
273 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
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1answer
292 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
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2answers
225 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 $\exists$ $...
2
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1answer
239 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 ...
6
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1answer
1k 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
308 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 · C."...