Skip to main content

All Questions

Filter by
Sorted by
Tagged with
0 votes
0 answers
79 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 ...
Galois group's user avatar
1 vote
0 answers
97 views

Weil restriction of cycles and norm algebra

This question is on a concrete descrption of weil restricton of an affine algebra. Let L/K be a Galois extension. Since I only care about the quadratic case, we may assume that $\Gamma:=\operatorname{...
Guangzhao Zhu's user avatar
1 vote
0 answers
80 views

Computing Chow groups of affine, simplicial toric varieties

Let $k$ be an algebraically closed field. Let $X$ be an $n$-dimensional affine, simplicial toric variety over $k$. There exists an $n$-dimensional simplicial cone $\sigma$ in $\mathbb{R}^n$ such that $...
Boris's user avatar
  • 639
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 ...
Bma's user avatar
  • 531
1 vote
0 answers
134 views

Paper request: Alberto Collino, Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic Jacobians

I am trying to locate a copy of the paper by Alberto Collino titled "Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic Jacobians". I can neither download nor purchase ...
user6419's user avatar
  • 441
5 votes
0 answers
157 views

Motives with compact support, Chow groups and proper pushforward maps

In Motivic cohomology of smooth geometrically cellular varieties (1999), Corollary 3.5, Bruno Kahn proves the following statement. Consider a cellular variety $X$ (i.e. it admits a filtration by ...
Bruno Stonek's user avatar
  • 3,004
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 ...
Jef's user avatar
  • 984
1 vote
0 answers
89 views

Vanishing of chow group of 0-cycles for affine, simplicial toric varieties

Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be an affine, simplicial toric variety over $k$. If $X$ has dimension one, then it is the affine line over the field $k$, so ...
Boris's user avatar
  • 639
1 vote
0 answers
96 views

Chow ring of simplicial toric varieties

Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be a simplicial toric variety over $k$. In the 2011 book Toric Varieties by Cox, Little and Schenck, there is a theorem that ...
Boris's user avatar
  • 639
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 ...
Jef's user avatar
  • 984
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 $...
Jef's user avatar
  • 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 ...
Galois group's user avatar
1 vote
0 answers
136 views

Universal properties for Bloch's higher Chow groups

I work in the category of varieties over some field of characteristic zero. Assume that for any variety I can define the group $\widetilde{CH}^r(X,n)$ which behave like classical Bloch's higher Chow ...
Galois group's user avatar
2 votes
0 answers
292 views

Tate's conjecture for arithmetic schemes

Tate's conjecture is about a map from Chow groups of a smooth projective variety $X$ to the $l$-adic cohomology i.e. $CH^n(X)\rightarrow (H^{2n}(\bar{X}, \mathbb{Q}_l(n)))^G$ where $G$ is the Galois ...
user127776's user avatar
  • 5,901
1 vote
2 answers
279 views

Correspondences acting on cohomology groups $H^*(X)$ & splittings

Let $X$ be a smooth connected proper scheme over field $k$. It is known that correspondences $\alpha \subset X \times X$ regarded as objects in Chow groups $\text{CH}^*(X \times X)$ act on cohomology $...
JackYo's user avatar
  • 619
2 votes
0 answers
60 views

Composition of correspondences pulled back to $\mathrm{CH}_0$

Let $X,Y,Z$ be varieties. Given two correspondences $\Gamma_1 \subset X \times Y$ and $\Gamma_2 \subset Y \times Z$ there is a composition, $$ [\Gamma_1] \circ [\Gamma_2] = \pi_{13 *} (\pi_{12}^* [\...
Ben C's user avatar
  • 3,730
3 votes
0 answers
248 views

Do Weil cohomology theories for schemes over arbitrary rings exist, and do the standard theorems (Lefschetz fixed point, Tr. Formula etc.) still hold?

A Weil cohomology theory is a functor that assigns to a smooth projective variety $X$ of dimension $d$ over a field $k$ a graded ring of cohomology groups with values in a field $K$ of characteristic $...
The Thin Whistler's user avatar
3 votes
0 answers
338 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 ...
user127776's user avatar
  • 5,901
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 $\...
Kim's user avatar
  • 565
0 votes
0 answers
75 views

General fiber and the symmetric product of an ample hypersurface

Let $Sym^m(X)$ be the $m$th symmetric product of a smooth projective variety $X$, $n=\dim(X)$, $Y_1$ an ample hypersurface of $X$, and $CH_0(X)_{hom}$ the Chow groups of $0$-cycles of degree $0$....
Roxana's user avatar
  • 519
1 vote
1 answer
217 views

Meaning of torsion points in a Roitman's theorem

I am having some problems to understand the meaning of the following theorem due to Roitmann. I found this theorem in Voisin's book: Hodge Theory and Complex Algebraic Geometry, Volume II, page ...
Roxana's user avatar
  • 519
2 votes
0 answers
125 views

About finite dimensionality of Chow groups of zero cycles

Let $S$ be a connected smooth complex projective surface. Let $Sym^{d}(S)$, $d\in \mathbb{Z}^+_0$, be the $d$-th symmetric product of $S$ parametrizing $0$-cycles of degree $d$. Let $Sym^{d,d}(S)=...
Roxana's user avatar
  • 519
4 votes
0 answers
114 views

Chow group of different reductions of a smooth projective variety

Let $(R,\mathfrak{m},k)$ be a discrete valuation ring with fraction field $K$. Let $X/K$ be a smooth projective variety. Let $\mathcal{X},\mathcal{X}'$ be smooth projective models of $X$ over $R$. Let ...
user avatar
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 ...
Roxana's user avatar
  • 519
1 vote
0 answers
213 views

Algebraic correspondence as morphisms in Betti cohomology

$\newcommand{\sing}{\mathrm{sing}}$Take a commutative ring $R$ and smooth projective complex varieties $X$ and $Y$. An element $\alpha\in CH^*(X\times Y)_R$ induces the algebraic correspondence for ...
OOOOOO's user avatar
  • 349
1 vote
1 answer
373 views

Singularities of Chow varieties

Let $X$ be a smooth complex projective variety. The Chow variety of degree $d$, $r$ dimensional subvarieties is denoted by $C_{d,r}(X)$. The Chow variety can have many topologically connected ...
user127776's user avatar
  • 5,901
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 ...
user127776's user avatar
  • 5,901
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 :)!
Roxana's user avatar
  • 519
0 votes
1 answer
224 views

Notation on a Mumford's paper

I am reading the paper " D. Mumford. Rational equivalence of $0$-cycles on surfaces. J. Math. Kyoto Univ. 9 (1969) 195 - 204" and I do not understand a notation in bottom of page 197. It ...
Roxana's user avatar
  • 519
2 votes
0 answers
64 views

Motivic complexes associated to adequate equivalence relations

Motivic cohomology sheaves $\mathbb{Z}(n)$ are homotopy invariant sheaves with transfers (under finite maps) and they satisfy excision long exact sequence when everything is smooth. The motivic ...
user127776's user avatar
  • 5,901
0 votes
0 answers
172 views

Pullback of algebraic $K$-theory along the surjection of abelian varieties

Given a surjective homomorphism of abelian varieties $f:A\rightarrow B$ where $\text{dim}(A)>\text{dim}(B)$, does $f^*$ induce a rational injection of algebraic $K$-theory? According to the ...
user127776's user avatar
  • 5,901
2 votes
0 answers
274 views

Generic rank of proper pushforward of the trivial line bundle

Given a proper surjective morphism $f:X\rightarrow Y$ where $X$ and $Y$ are smooth projective varieties. The proper pushforward $f_!$ is the homomorphism that sends the class of a coherent sheaf $M$ ...
user127776's user avatar
  • 5,901
1 vote
1 answer
452 views

The Ogus conjecture for crystalline cohomology

How is the Ogus conjecture explicitly stated, which is a variant of the Hodge and the Tate conjectures for crystalline cohomology ? How do we build its class cycle map, and how do we formulate its ...
Angel65's user avatar
  • 595
2 votes
0 answers
90 views

Homological and rational adequate equivalences for product of curves

There is a variant of Standard Conjecture D for projective varieties over finite fields. It claims that rational and homological equivalences are equivalent on cycles after tensoring with $\mathbb{Q}$....
user127776's user avatar
  • 5,901
5 votes
1 answer
264 views

Moduli of rational equivalent classes of 0-cycles

Let $X$ be a smooth variety over a field $k$ and I'd like to think about $CH_0(X)$ the 0-Chow group i.e. the group of rational equivalent classes of 0-cycles. I'm wondering if there is any reasonable ...
wkf's user avatar
  • 647
0 votes
0 answers
129 views

Injectivity of pushforward of rational Chow groups

I'd like to know whether there is a known counter-example to the following statement. Let $X$ be a smooth projective variety over a finite field. Let $Z$ be a codimension $2$ smooth subvariety which ...
user127776's user avatar
  • 5,901
2 votes
1 answer
409 views

Invariance of Chow groups of projective bundles under automorphisms of bundles

Suppose $X$ is a smooth scheme, $E=O_X^{\oplus n}$ and $\varphi\in SL_n(E)$, i.e. $\varphi$ has trivial determinant and is an isomorphism. Is the morphism $$\mathbb{P}(\varphi)^*:CH^{\bullet}(\mathbb{...
Nanjun Yang's user avatar
5 votes
0 answers
191 views

Applications of Chow rings of classifying spaces in algebraic geometry

For an algebraic group $G$, the Chow ring of its classifying space $BG$, in the sense of Totaro, The Chow ring of a classifying space has been computed in many cases. Is there any interesting ...
Xing Gu's user avatar
  • 935
3 votes
0 answers
280 views

Descent and Chow groups

One of the features of the $\mathbb{A}^1$-homotopy theory is the existence of the motivic Eilenberg-MacLane space $K(\mathbb{Z}(n),2n)$ such that for $k$-schemes $X$, we have $$[X,K(\mathbb{Z}(n),2n)]\...
curious math guy's user avatar
2 votes
0 answers
239 views

Localization of Chow groups and flat base change

For any flat morphism $f:X\rightarrow Y$, we have a flat pullback of Chow groups $$Ch^i(Y)\rightarrow Ch^i(X).$$ A particular example of this is of course an open immersion $U\rightarrow X$. In that ...
curious math guy's user avatar
9 votes
0 answers
276 views

Which field extensions do not affect Chow groups?

Let $X$ be a (say, smooth projective) variety over a field $k$. For which $K$ it is known that the ("ordinary", that is, not higher) Chow groups of $X$ map onto that of $X_K$ bijectively? ...
Mikhail Bondarko's user avatar
4 votes
2 answers
458 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 ...
Mikhail Bondarko's user avatar
3 votes
0 answers
914 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^{*...
Zhiyu's user avatar
  • 6,622
2 votes
1 answer
585 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 ...
YoYo's user avatar
  • 325
6 votes
0 answers
203 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/...
nxir's user avatar
  • 1,479
4 votes
1 answer
255 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{...
Hajime_Saito's user avatar
1 vote
0 answers
116 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 $...
user127776's user avatar
  • 5,901
2 votes
1 answer
542 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 \...
Hajime_Saito's user avatar
2 votes
0 answers
331 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 ...
user127776's user avatar
  • 5,901
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 ...
Hajime_Saito's user avatar