# Questions tagged [algebraic-cycles]

The algebraic-cycles tag has no usage guidance.

170
questions

11
votes

2
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### Does Poincaré duality preserve algebraic cycles?

Let $X$ be a smooth, projective (complex) variety of dimension $n$ and $Z \subset X$ be a subvariety of codimension $k$ (if necessary assume $Z$ is non-singular). We know the cohomology class $[Z]$ of ...

2
votes

2
answers

149
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### Decompose complete directed graph with n vertices into n edge-disjoint cycles with length n−1

I want to know how to decompose a complete directed graph with $n$ nodes into $n$ edge-disjoint cycles with length $n-1$. I found this result was proved in Bermond and Faber - Decomposition of the ...

1
vote

0
answers

75
views

### Behaviour of cycles modulo algebraic equivalence on an etale covering

I found a neat result in Beauville's paper "VARIÉTÉS DE PRYM ET JACOBIENNES INTERMÉDIAIRES" : if $U \subset \mathbb{P}^n$ is an open and $V \to U$ is a conic bundle whose fibres are all ...

3
votes

0
answers

117
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### A Galois equivariant Weil cohomology theory with coefficients in the rational numbers and a variation of the Tate/Hodge conjecture

A well-known example of Serre shows that there can be no Weil cohomology theory with $\mathbb Q$ coefficients for schemes over $\mathbb F_{p^2}$. However, this example is no obstruction to a Weil ...

2
votes

1
answer

181
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### Finite flat pullback of the diagonal

Let $X, Y$ be smooth projective connected complex varieties of the same pure dimension $d$ and $f : X\to Y$ a finite flat surjective morphism.
Let $\Delta_X$ be the closed subscheme of $X\times X$ ...

1
vote

1
answer

136
views

### Cohomology classes fixed by algebraic automorphism subgroups

Let $X$ be a smooth projective complex variety and $t\in H^{2p}(X,\mathbf{Q})(p)$ a rational cohomology class.
Assume that there exist
$$t_1,\ldots,t_N\in H^{2*}(X,\mathbf{Q})(*)$$
algebraic classes (...

2
votes

1
answer

180
views

### Cup products and correspondences

Suppose $X$ is a smooth projective complex variety, connected of dimension $n$.
Let $a$ be an algebraic correspondence in $A^n(X\times X)$, the group of cycles modulo homological equivalence in $H^{2n}...

2
votes

1
answer

144
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### Reference for torsion-freeness of the group of correspondences on a smooth projective variety

In Beauville's "Variétés de Prym et jacobiennes intermédiaires", Proposition 3.5, it is claimed that $\textrm{Corr}(T)$ is torsion-free for a smooth projective variety $T$. Here $$\textrm{...

2
votes

0
answers

112
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### Pullback of algebraic cycles and homological triviality

Let $X$ be a smooth, projective variety of dimension $2m$ for some $m \ge 2$ and $D$ be a non-singular divisor in $X$ such that the dual of the normal bundle of $D$ in $X$ is very ample. We have the ...

2
votes

0
answers

89
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### Filtration of Chow variety based on the dimension of the intersection of cycles

Let $X$ be a complex projective variety. Let $Y$ be subvariety of $X$. Consider the Chow variety of $r$ cycles on $X$ denoted by $C_r(X)$ (We can assume it is the Chow of variety of irreducible cycles)...

2
votes

0
answers

91
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### Generic set theoretic intersection of two high codimensional varieties

Let $X$ be a complex projective variety, let $Y$ be a fixed subvariety. Consider all effective irreducible algebraic $r$-cycles where $r$ is fixed and chosen in a way such that $r \ll \text{Codim}_X(Y)...

12
votes

2
answers

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### What exactly do the standard conjectures in characteristic zero refer to?

As the title suggest it seems standard conjectures mean different things depending on the context. I had the impression that in characteristic 0 they are a list of conjectures about varieties over an ...

2
votes

1
answer

96
views

### Cycles that are graphs of morphisms

Assume we have an irreducible algebraic cycle $Z$ on $X\times Y$ where $X$ and $Y$ are projective varieties ($X$ is smooth) such that restriction of $Z$ to $U\times Y$ where $U\subset X$ is a Zariski ...

3
votes

0
answers

86
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### Pushing cycles out of singularity

Let $X$ be a smooth complex quasi-projective variety and $Y$ a projective complex variety. Let's assume $Z$ is the singular locus of $Y$. We assume that $\text{codim}_Y(Z)\gg \text{dim}(X)$. Let's ...

18
votes

0
answers

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### Cycles in algebraic de Rham cohomology

Let $F$ be a number field, $S$ a finite set of places, and $X$ a smooth projective $\mathscr{O}_{F,S}$-scheme with geometrically connected fibers. For each point $t\in \text{Spec}(\mathscr{O}_{F,S})$, ...

3
votes

0
answers

195
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### 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
...

3
votes

0
answers

160
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### Boundedness indices in Voevodsky's smash nilpotence conjecture in family

Let $X$ be a smooth projective variety over an algebraically closed field $k$. Voevodsky introduced the following notion : an algebraic cycle $Z$ in $X$ is smash nilpotent if there exist $N>0$ such ...

1
vote

0
answers

155
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### Is an algebraic cycle always contained some smooth ample hypersurface?

Let $X$ be a smooth projective threefold over $\mathbb C$. Let $Z$ be an algebraic 1-cycle of $X$, so $Z$ consists of curves in $X$.
Is it always possible to find a smooth ample hypersurface $Y$ of $X$...

4
votes

1
answer

179
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### Higher Chow cycles

Recall that the higher Chow groups $CH^k(X,m)$ are defined as the homology of the complex $Z^k(X,\bullet)$, where $Z^k(X,m)$ is the subgroup of codimension $k$ cycles of $X\times \Delta^m$ which meet ...

2
votes

1
answer

217
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### Algebraic and homological equivalence relations for $0$-cycles

Let $X$ be a connected smooth projective variety. Let $Z_0(X)_{alg}$ be the group of $0$-cycles algebraically equivalent to $0$ and $Z_0(X)_{\hom}$ be the group of $0$-cycles homologically equivalent ...

1
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0
answers

232
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### 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 $\...

4
votes

0
answers

579
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### Is there any relation between the pushforward of a divisor and the pushforward of its line bundle?

Given a morphism between normal varieties $f: X \to Y$, we can push forward a Cartier divisor $D$ to get a cycle $f_* D$. On the other hand, we can form the line bundle $\mathscr{L}(D)$ and push that ...

1
vote

0
answers

113
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### Algebraic deformation retract of subvarieties

Let $X$ be a complex projective variety. Let $V_1$ and $V_2$ be two closed subvarieties of $X$. We say $V_2$ admits an algebraic deformation retract to $V_1$ in $X$, if the closed immersion $V_1\...

1
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0
answers

100
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### 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 ...

5
votes

0
answers

92
views

### Torsion in homology of Chow variety

For the sake of this question for simplicity you can assume $X$ is a hypersurface in the complex projective space (not necessarily smooth). Let $C_{r,d}(X)$ be the quasi-projective Chow variety ...

4
votes

0
answers

198
views

### Motives based on Hodge cycles vs algebraic cycles

I am not a specialist of motives. I am afraid my questions are rather naive.
We have the category of (pure) motives based on Hodge cycles by Deligne. In his articles with Milne, morphisms between ...

1
vote

0
answers

63
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 ...

0
votes

0
answers

142
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### 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 :)!

3
votes

1
answer

532
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### Subrings of Chow rings

Let $X$ be a smooth projective variety over $\mathbf{F}_p$, call $\overline{X}$ the base change to $\overline{\mathbf{F}}_p$, and denote by $F$ the base change to $\overline{X}$ of the absolute ...

2
votes

1
answer

106
views

### Cycle of non-equidimensional scheme

In Fulton's intersection theory, example 1.7.1, he mentioned an example that contradicts to the splitting of cycles with respect to irreducible components. Consider the subscheme $X$ in $\mathbb{A}^3$ ...

0
votes

0
answers

180
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### Definition of rational equivalence

In the book "C. Voisin. Hodge Theory and Complex Algebraic Geometry. Volume II. Cambridge studies in advanced mathematics 76 (2002)" page 247, definition 9.4, says:
Definition 9.4.
The ...

0
votes

0
answers

90
views

### Closed algebraic subset dominating a curve

In the book "C. Voisin. Hodge Theory and Complex Algebraic Geometry. Volume II. Cambridge
studies in advanced mathematics 76 (2002)" page 228 says:
Let $X$ be a smooth projective variety. If ...

0
votes

1
answer

218
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 ...

3
votes

0
answers

191
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### Are cubical higher Chow groups of field $CH^{n-1}(F,n)$ generated by linear cycles?

In the paper "The linearization of higher Chow cycles of dimension one" W. Gerdes proved that Higher Chow homology group $CH^{n-1}(F,n)=H^{n}(z^{n-1}(F,*))$ are generated by linear cycles.
...

5
votes

0
answers

198
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### algebras of algebraic cycles satisfying Hard Lefschetz

June Huh and Botong Wang considired (arxiv:1609:08808) algebras of algebraic classes generated by (all) divisor classes and constructed examples of smooth projective varieties for which this algebra ...

3
votes

1
answer

364
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### Cycle class map for singular varieties

I am reading the cycle class map for singular projective varieties as mentioned by Laterveer in this article (see Definition $1$). The article does not define the map but refers to an article of ...

4
votes

0
answers

116
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### Looking for a crystalline analogue of , $\mathcal {Z}_{\sim}^* (X)_F \simeq \mathcal{Z}_{\sim}^* (X_{k ' })_F^{\mathrm{Gal} ( k ' / k )} $

Is there a crystalline analogue for the following formula, using the crystalline Frobenius $ F_v $ instead of the absolute Galois group $ \mathrm{Gal} (\overline{k} / k) $ ?
Here is the formula, which ...

1
vote

1
answer

407
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### 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 ...

2
votes

0
answers

194
views

### (1,1)-form that does not come from a divisor

Let $M$ be projective complex manifold. The Lefschetz (1,1)-theorem says that the cycle map
$$
\text{cl}:\operatorname{Pic}(M) \to \text{Hod}^1(M)
$$
is surjective.
Question. Is there an interesting ...

1
vote

1
answer

169
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### Find the representative cycle location in the persistence diagram

I know that some toolbox such as Dionysus can also return the representative points which are on the boundaries of the cycles (topological features of a point cloud). I can clearly extract these ...

9
votes

1
answer

304
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### Matrix obtained by recursive multiplication and a cyclic permutation

Have you ever seen this matrix? Each row is obtained from the previous one by multiplying each element by the corresponding element of the next cyclic permutation of $(a_1,\dots, a_n)$:
$$\left(
\...

2
votes

1
answer

269
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### Meaning of "combinatorial data"

I saw several times that often some data describing certain
algebraic objects,
eg the set of cells of a simplical complex
or a Cech cycle of a chosen coving of a variety
are called "combinatorial ...

4
votes

2
answers

383
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 ...

1
vote

1
answer

182
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### Confusion regarding a definition of cycles

For a projective variety $X$ over $\mathbb{C}$, let us denote by $CH_k(X)$ the Chow group of $k$-cycles of $X$, modulo rational equivalence. Also, let $CH_k(X)_{hom}$ denote $k$-cycles modulo ...

1
vote

0
answers

95
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### Why is $\Delta - p_0 - p_{2}$ a projector?

I apologize in advance, since I am probably doing a very naive mistake in my computation. I am learning about pure (Chow / Grothendieck) motives. One of the first steps is to consider the category ...

8
votes

1
answer

563
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### Why can Euler systems constructed from algebraic cycles only be anticyclotomic?

In a footnote to the 2018 Zerbes-Loeffler lecture notes from the Arizona Winter School, it's stated that Euler systems constructed from algebraic cycles "cannot give a full Euler system, only an ...

3
votes

0
answers

337
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### Rational equivalence and ''$\mathbb{A}^1$-equivalence" for zero cycles

Let $k$ be a field, let $X\subset\mathbb{P}^r$ be a smooth projective algebraic variety defined over $k$.
We define an equivalence relation on the group of zero cycles $Z_0(X)$ in the following way:
...

4
votes

0
answers

452
views

### Chern classes of torsion-free sheaves

Let $X$ be a smooth projective variety and $Z$ a closed subvariety of co-dimension $k$. The first $k-1$ chern classes of the ideal sheaf of $Z$ vanishes and the $k$-th chern class is given by ...

3
votes

0
answers

121
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### Extra Algebraic $(1,1)$ cycles on a complex surface

Suppose $x,y,w,z$ are homogeneous coordinates of $\mathbb{CP}^3$, and
\begin{eqnarray}
X_t := \left(F_t = x f_2 +y g_2 +t F_3 = 0 \right)
\end{eqnarray}
be a family of degree 3 hypersurfaces in $\...

2
votes

0
answers

183
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 ...