All Questions
17 questions with no upvoted or accepted answers
4
votes
0
answers
167
views
Is the group of homologically trivial cycles in a variety over a finite field torsion?
Let $X$ be a smooth projective variety over $\mathbb{F}_q$. Is any cycle in the Chow group $CH^i(X)$ which is trivial in $\ell$-adic cohomology automatically torsion? For abelian varieties I believe ...
- 531
4
votes
0
answers
135
views
Specialization map Chow groups preserves algebraic equivalence
Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$.
Let $\pi\colon X\rightarrow \text{Spec}(R)$ be a smooth projective morphism with geometrically integral fibers.
In ...
- 984
4
votes
0
answers
350
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} :=...
user97068
3
votes
0
answers
152
views
Locus where a family of cycles is rationally trivial is countable union of closed subvarieties?
Following up on this question which received a negative answer, I wonder if something weaker is true.
We work in the same set-up as the previous question. Let $B$ be a smooth quasi-projective variety ...
- 984
3
votes
0
answers
340
views
Pushforward and pullback on the level of Chow varieties
Let $X$ and $Y$ be complex projective varieties. Let's assume we have a finite flat morphism $f:X\rightarrow Y$ of degree $k$. We know that it is possible to pullback and also pushforward algebraic
...
- 5,901
3
votes
0
answers
556
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^...
user95222
2
votes
0
answers
239
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 ...
- 737
2
votes
0
answers
656
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\...
user119470
2
votes
0
answers
261
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\...
user113452
1
vote
0
answers
137
views
Locus where a family of cycles is rationally trivial is closed?
Let $B$ be a smooth quasi-projective variety over a field of characteristic zero.
Let $\pi\colon \mathcal{X} \rightarrow B$ be a smooth and projective morphism with geometrically integral fibres. Let $...
- 984
1
vote
0
answers
162
views
Bloch's higher Chow group as relative ordinary Chow group
If X is a variety and $Y\subset X$ is a closed subscheme then one can define relative Chow group. The definition is follows: there is subcomplex $\psi_Y\colon z^r_Y(X,*)\hookrightarrow z^r(X,*)$ of ...
- 219
1
vote
0
answers
254
views
A question on the Chow group on stacks
Let $X$ be a separated Deligne-Mumford stack finite type over the ground field. Then there is a Chow group $A_*(X)$ of $X$ which is well-behaved under flat pull-back, defined as follows.
Let $\...
- 565
1
vote
0
answers
102
views
About Definition 2 in Roĭtman's Paper
Let $A(X)$ be the Chow group of $0$-cycles on a nonsingular irreducible projective variety $X$ over an uncountable algebraically closed field of characteristic zero.
In Definition 2 of Roĭtman's paper ...
- 519
1
vote
0
answers
77
views
Non-torsion infinitely divisible elements in the Chow group
It was shown in "Clemens, Herbert, Homological equivalence, modulo algebraic equivalence, is not finitely generated.", that the Chow group mod algebraic equivalence of smooth complex ...
- 5,901
0
votes
0
answers
80
views
Chow moving lemma with additional property
All varieties are over algebraically closed field of characteristic zero. Let $S$ be a smooth projective surface. Let $D$ be an irreducible divisor on $S$, $H$ be another divisor and $Z\subset S$ be a ...
- 219
0
votes
0
answers
145
views
Chow countability argument
I would like to know what the "Chow countability argument or HIlbert schemes countability argument" says in order to finish an exercise. Any reference will also be very useful :)!
- 519
0
votes
0
answers
288
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 ...
- 16.9k