Questions tagged [chow-groups]
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113 questions
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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/...
- 1,479
4
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1
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255
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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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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 $...
- 5,901
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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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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 ...
- 5,901
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239
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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 ...
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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 ...
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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}...
- 1,566
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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[...
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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 ...
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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 ...
- 528
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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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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, ...
- 1,463
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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 ...
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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^...
user95222
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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\...
user119470
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352
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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(\...
- 17.4k
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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\...
user113393
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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\...
user113452
2
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1
answer
172
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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(...
user92332
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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} :=...
user97068
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1
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1k
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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 $$\phi_i\colon \mathrm{CH}^i(X)...
user39380
2
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1
answer
339
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Chow ring of product of Brauer-Severi Varieties
Let $K$ be a field, $\alpha, \beta \in \mathrm{Br}(K)$, let $X,Y$ be their Brauer-Severi Varieties, is there a way to calculate $A^*(X\times Y)$?
For example, if $\alpha,\beta$ both has degree $5$, $...
user39380
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Motivic $\mathbf{Z}(1)$
I know that the Bloch higher Chow complex, $\mathbf{Z}(i)_{\mathcal{M}}$, on a smooth scheme over a field $k$, reads, in degree $1$:
$$\mathbf{Z}(1)_{\mathcal{M}}\simeq\mathbf{G}_m[-1].$$
How to see ...
user113393
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1
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449
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Higher Chow groups for complete smooth intersections?
Let $F$ be a smooth complete intersection of $r$ hypersurfaces of degree $d_{1},\dots,d_{r}$ in $\mathbb{P}^{n+r}$ over an algebraic closed field. A classical result of A. Roitman says that the group ...
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Higher Chow groups revisited
Let $X$ be an algebraic variety over a field $k$.
Bloch defines the "algebraic singular complex" using the algebraic simplices
$$\Delta^n = \text{Spec}(k[x_0,\dots,x_n]/(x_0+x_1+\dots+x_n=1) \subset ...
user97068
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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 $...
user87684
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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 $L_1,L_2,$ ...
- 600
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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 ...
- 1,537
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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^*(...
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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 ...
- 2,871
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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 $[...
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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 ...
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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}}(\...
- 596
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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 ...
- 15.4k
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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$...
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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))$$ where the RHS is the continuous etale cohomology(defined by Jannsen in his paper "Continuous etale cohomology"). In this paper ...
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About the decomposition of a Chow group of a variety
I would like that someone helps me to find an article on the net treating the following decomposition : $ \mathrm{CH}^k (X)_{ \mathbb{Q} } = \displaystyle\bigoplus_{ i + j = k } \mathrm{CH}^{i,j} (X) $...
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Chow group over function field and algebraic equivalence
It is known that for smooth projective varieties $X,Y$ over $k=\bar k,$ $$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 ...
- 61
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254
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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
$\exists$ $...
- 101
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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 ...
- 1,121
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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 ...
- 16.5k
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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."...
- 3,212
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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 $H(X\...
- 299
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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 ...
- 16.9k
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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. ...
- 5,063
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2
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491
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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 $CH^n(...
- 43
31
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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 ...
- 118k
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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 ANDGROTHENDIECK–...
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