Unanswered 'intersection-theory+arithmetic-geometry+ag.algebraic-geometry' Questions

8 votes

0 answers

569 views

Bloch Ogus spectral sequence

Let $X$ be a smooth projective variety over $\mathbf{C}$, and $p : X_{\rm an}\to X_{\rm Zar}$ the obvious map of sites. The Leray spectral sequence $$H^r(X_{\rm Zar}, R^sp_*\mathbf{C})\Rightarrow H^{...

user92332

asked Apr 2, 2018 at 0:39

5 votes

0 answers

397 views

Vector bundles vs algebraic cycles

For integral schemes, the Picard group is isomorphic to the group of Cartier divisors modulo linear equivalence. What is the correct analog of this isomorphism, for higher codimension Chow groups vs ...

user119470

asked Jan 12, 2018 at 20:07

5 votes

0 answers

486 views

Computing intersection number of two arithmetic line bundles

I have some questions in Arithmetic Arakelov geometry Let $\mathcal X\to Spec(\mathcal O_K)=C$ be an arithmetric projective variety over $C$ , where $\mathcal O_K$, ring of number filed $K$ and $\...

user21574

asked Feb 4, 2016 at 15:42

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

531

asked Jul 11 at 22:55

4 votes

0 answers

261 views

Motives up to homological equivalence

Let $X$ be a smooth projective variety over a field $k$ finitely generated over its prime field, and $M_{hom}(X)$ the category of motives modulo $\ell$-adic homological equivalence. (1) Is $M_{hom}(...

user120812

asked Mar 15, 2018 at 15:33

4 votes

0 answers

245 views

Hard Lefschetz for cycles

Let $X$ be a smooth projective variety over a field $k$. It is known by work of Deligne, that the Lefschetz operator: $$ L^k:H^{2n-2k}\left(X_{\overline{k}},\mathbf{Q}_{\ell}\right)\to H^{2n+2k}\left(...

user92332

asked Mar 15, 2018 at 2:26

4 votes

0 answers

347 views

Complete intersections in projective spaces

Let $X$ be an arbitrary smooth projective variety over a field $k$. Do there exist: a smooth complete intersection $X'$ in a projective space. a surjective morphism of $k$-varieties $X'\to X$ ?

user119470

asked Feb 24, 2018 at 13:34

3 votes

0 answers

409 views

Non algebraizable formal abelian schemes

I'd like to collect a good number of examples of formal schemes over $\text{Spf}(\mathbf{Z}_p)$, whose special fiber is a projective variety over $\mathbf{F}_p$, but that are not algebraizable. If ...

user97068

asked May 18, 2018 at 18:37

3 votes

0 answers

114 views

Multiplicative structure on Deligne cohomology

Let $X$ be a smooth projective variety over the complex numbers, and $\mathbf{Z}(p)_{\mathcal{D}}$ the Deligne complex on $X$: $$\mathbf{Z}(p)_{\mathcal{D}} : \ \ \mathbf{Z}(p)\to\mathcal{O}_X\to\...

user92332

asked Mar 12, 2018 at 20:53

2 votes

0 answers

219 views

Liftability of varieties, after fpqc base change

Let $X$ be a smooth projective variety over a finite field, with a closed immersion to some other smooth projective variety $S$, with $S$ liftable. Suppose there exists an fpqc cover $S'\to S$, such ...

user124171

asked May 15, 2018 at 2:25

2 votes

0 answers

256 views

Neron Severi under specialization

Let $X$ be a smooth projective variety over $\mathbf{Q}$, and $\mathcal{X}$ a smooth projective model over $\mathbf{Z}[1/N]$ for $N$ large enough. Call $\eta$ the generic point $\text{Spec}(\mathbf{Q}...

user95222

asked Mar 16, 2018 at 23:46

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

asked Feb 21, 2018 at 16:00

2 votes

0 answers

239 views

Group completion of Chow varieties

Let $X$ be a quasi-projective variety over a perfect field $k$. Given a projective embedding $j : X\to \mathbf{P}(\mathscr{E})$, the Chow variety $\text{Chow}_r(X, j)$ is a quasi-projective variety ...

user113393

asked Feb 8, 2018 at 22:42

2 votes

0 answers

228 views

On a class of loci in Chow varieties

Let $k$ be a field, $i:X\hookrightarrow \mathbf{P}(\mathscr{E})$ be a fixed projective embedding of a smooth projective $k$-variety $X$, whose dimension is pure and equals $d\ge 0$. For $0\le p\le d$,...

user95222

asked Jan 14, 2018 at 11:01

1 vote

0 answers

145 views

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

85

asked Oct 29, 2023 at 23:12

1 vote

0 answers

34 views

Hodge classes generated in degree $1$

Let $X$ be a smooth projective variety over the complex numbers, and $\text{Hdg}^p(X)_{\mathbf{Q}}$ the abelian group of Hodge classes in $H^p(X,\mathbf{Q}(p))$. Denote by $\text{Hdg}^*(X)$ the ...

user95222

asked Mar 14, 2018 at 18:58

1 vote

0 answers

290 views

Coniveau in étale motivic cohomology

Let $X$ be a smooth variety over a field. Is there a spectral sequence: $$E_1^{p,q} := \bigoplus_{x\in X^{(p)}}H^{q-p}(\kappa(x)_{\rm ét},\mathbf{Z}(n))\Rightarrow H^{q-p}(X_{\rm ét},\mathbf{Z}(n))$$...

user92332

asked Mar 13, 2018 at 1:31

1 vote

0 answers

118 views

Torsion homologically trivial cycles

Is there an example of a smooth projective variety $X$ over the complex numbers, such that $$\ker(\text{CH}^2(X)\to H^4(X,\mathbf{Z}(2))$$ is not torsion?

user119470

asked Mar 11, 2018 at 19:12

1 vote

0 answers

118 views

Filtrations and the Betti cycle map

Let $X$ be a smooth projective complex variety. Calling $f : X_{\rm an}\to X_{\rm Zar}$ the "change-of-topology" morphism of sites induced by sending a Zariski open $U$ of $X$ to its analytification, ...

user119470

asked Mar 11, 2018 at 0:05

1 vote

0 answers

231 views

Lower bound for intersection number

The base scheme is an algebraically closed field. Let $X\to \mathbf{P}^1$ be an arithmetic surface over $\mathbf{P}^1$ and let $P$ be a section of $X\to \mathbf{P}^1$. Let $D$ be an effective (edited)...

Taicho

225

asked Oct 9, 2011 at 8:50

0 votes

0 answers

405 views

Twisted sheaves on tower of $\mathbb{P}^n$

Take the projective space $\mathbb{P}^n$ over a ring $W$. We call $\mathcal{O}(q)$ the usual twisted line bundle. Now take the map $f: \mathbb{P}^n\to\mathbb{P}^n$ defined by $$[x_0,\ldots, x_n]\...

user124171

asked May 19, 2018 at 20:22

0 votes

0 answers

148 views

Is sum $(E_i, E_j)$ non-positive, with $E_i$'s the exceptional components of a desingularization

Let $Y$ be an integral normal 2-dimensional scheme and let $X\longrightarrow S$ be a flat projective morphism, where $S$ is a Dedekind scheme. Let $f:X\longrightarrow Y$ be a minimal resolution of ...

Frederick

1

asked Aug 27, 2011 at 11:51

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All Questions

Bloch Ogus spectral sequence

Vector bundles vs algebraic cycles

Computing intersection number of two arithmetic line bundles

Is the group of homologically trivial cycles in a variety over a finite field torsion?

Motives up to homological equivalence

Hard Lefschetz for cycles

Complete intersections in projective spaces

Non algebraizable formal abelian schemes

Multiplicative structure on Deligne cohomology

Liftability of varieties, after fpqc base change

Neron Severi under specialization

Specialization maps for Chow groups

Group completion of Chow varieties

On a class of loci in Chow varieties

Multiplicity and the perfect projective line

Hodge classes generated in degree $1$

Coniveau in étale motivic cohomology

Torsion homologically trivial cycles

Filtrations and the Betti cycle map

Lower bound for intersection number

Twisted sheaves on tower of $\mathbb{P}^n$

Is sum $(E_i, E_j)$ non-positive, with $E_i$'s the exceptional components of a desingularization

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