Questions tagged [algebraic-cycles]

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A relative Abel-Jacobi map on cycle classes

I have a question about relativizing a classical cohomological construction that I think should be easy for someone well versed in such manipulations. Background: Suppose $X$ is a smooth projective ...
Asvin's user avatar
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A relative cycle class map

Suppose I have a smooth projective morphism $p: X \to S$ between varieties, and a relative cycle $Z \subset X \to S$ which is assumed to be as nice as can be (rquidimensional with fibers of dimension $...
Asvin's user avatar
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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
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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
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Multiplicity and the perfect projective line

Let $\mathbf{F}_p$ be the field with $p$ elements, and $X = (\mathbf{P}^1_{\overline{\mathbf{F}}_p})^\text{perf}$ the inverse perfection of the projective line over $\mathbf{F}_p$. Let $\Gamma$ be the ...
Tim's user avatar
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0 answers
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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
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Pull and push formula for degree for non-flat morphism

Let $\varphi\colon X_1\to X_2$ be dominant proper morphism of finite degree (in particular $\dim X_1=\dim X_2$) between varieties. Let $D \subset X_2$ be a Cartier divisor. Is it true that $$\varphi_*...
Galois group's user avatar
12 votes
2 answers
1k views

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 ...
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2 votes
2 answers
193 views

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 ...
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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 ...
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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 ...
Asvin's user avatar
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2 votes
1 answer
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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 $X\times X$ ...
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1 vote
1 answer
138 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 (...
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2 votes
1 answer
213 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}...
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2 votes
1 answer
150 views

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{...
TCiur's user avatar
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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 ...
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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)...
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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)...
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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 ...
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2 votes
1 answer
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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 ...
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3 votes
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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 ...
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18 votes
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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 $t\in \text{Spec}(\mathscr{O}_{F,S})$, ...
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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 ...
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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 ...
Libli's user avatar
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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 $X$...
AG learner's user avatar
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4 votes
1 answer
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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 ...
Monsieur Periné's user avatar
2 votes
1 answer
238 views

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 ...
Roxana's user avatar
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0 answers
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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 $\...
Kim's user avatar
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4 votes
0 answers
798 views

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 ...
Kim's user avatar
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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\...
user127776's user avatar
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1 vote
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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 ...
Roxana's user avatar
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0 answers
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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 ...
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4 votes
0 answers
225 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 ...
Takahiro Matsuda's user avatar
1 vote
0 answers
70 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
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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 :)!
Roxana's user avatar
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1 answer
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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 ...
user avatar
2 votes
1 answer
114 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 $\mathbb{A}^3$ ...
Peter Liu's user avatar
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0 answers
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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 ...
Roxana's user avatar
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0 answers
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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 ...
Roxana's user avatar
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1 answer
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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
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3 votes
0 answers
195 views

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. ...
user avatar
5 votes
0 answers
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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 ...
Dima Sustretov's user avatar
3 votes
1 answer
421 views

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 ...
user45397's user avatar
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4 votes
0 answers
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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 ...
Bradley04's user avatar
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1 vote
1 answer
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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 ...
Bradley04's user avatar
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2 votes
0 answers
197 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 ...
user145752's user avatar
1 vote
1 answer
199 views

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 ...
abolfazl taghribi's user avatar
9 votes
1 answer
339 views

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( \...
SandraS's user avatar
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2 votes
1 answer
284 views

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 ...
user267839's user avatar
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4 votes
2 answers
416 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