All Questions
Tagged with ag.algebraic-geometry intersection-theory
329 questions
2
votes
0
answers
323
views
Surjectivity of pushforward on Chow rings for stacks
Let $f:X\rightarrow Y$ be a proper morphism of smooth Deligne-Mumford stacks of finite type over $\mathbb{C}$ that is birational, but not flat. The coarse spaces of $X$ and $Y$ are both not smooth. Is ...
- 776
2
votes
0
answers
335
views
Functoriality of Chern-Fulton's class
Let $X$ be a, possibly singular, algebraic variety embedded as a closed subvariety of a manifold $M$ with map $i : X \rightarrow M$, and $\pi : \tilde{M} \rightarrow M$ be a proper birational map with ...
- 151
1
vote
0
answers
74
views
Is it possible to represent a closed substack as a fundamental cycle?
Let $X$ be an Artin stack and $Z \subset X$ be a closed substack. Can we represent $Z$ as a fundamental cycle? i.e. $[Z] = \sum_i a_i [Z_i]$ where $Z_i$ are integral substacks of $X$. In other word, ...
- 421
2
votes
1
answer
249
views
Vector bundles on henselian schemes
Let $X$ be a smooth and projective scheme over $\mathbf{Z}_p$.
We call $\mathfrak{X}$ the ringed space whose topological space is the topological space of the special fiber of $X$, and whose ...
- 181
2
votes
0
answers
186
views
Splittings in the difference bundle construction of Atiyah-Hirzebruch
I'm reading the construction of difference bundle(generalized) in the paper of Atiyah-Hirzebruch(Analytic cycles on complex manifolds)
There is one thing I cannot understand. The followings are in ...
- 421
3
votes
1
answer
185
views
Equivalence relations among algebraic cycles
In the book 3264 and All That by Eisenbud & Harris, the authors claim that for smooth projective varieties admitting an affine stratification, the algebraic equivalence relation and the rational ...
- 1,252
3
votes
1
answer
495
views
On connectedness of intersection of subgroups
I am quite interested in any partical answer to the following general (maybe a little bit vague) question: Is there some criterion about the connectedness of the intersection of two connected ...
- 31
1
vote
0
answers
177
views
Continuity of Intersection Pairing on Chow monoids
Let $X$ be a smooth irreducible complex projective variety. As we know, if $\alpha,\beta$ are two cycles intersecting properly in $X$, we can define, via Serre's Intersection Formula, their ...
- 1,252
1
vote
0
answers
39
views
Counting Zeros Under Unitary Action
Assume we have two polynomials $f_1$ and $f_2$ with Newton polytopes $A_1 , A_2 \in \mathbb{Z}^2$. Also suppose that coefficients of $f_1$ and $f_2$ are generic. Then we pick a unitary matrix $Q$ and ...
- 351
1
vote
0
answers
121
views
What happens to a variety after a change of variables?
Suppose I have an irreducible affine variety $X \subseteq \mathbb{A}^n_k$.
Let us denote $X = \{ x \in k^n : f_j(x) = 0 \ (1 \le j \le M) \}$. $k$ is an algebraically closed field. Let $a_i \in k$, $...
- 3,625
2
votes
1
answer
663
views
A "boundary map" for the algebraic equivalence relation of cycles
In what follows by projective variety I will mean the zero locus of homegeneous polynomials in some projective space. Anyway, feel free to deal with a sufficiently good scheme if you want.
Let $X$ be ...
- 1,252
4
votes
1
answer
322
views
Pairing on arithmetic surfaces
Let $f: X \to S$ be an arithmetic surface, where $S=\operatorname{Spec } O_K$ for a number field $K$. It is well known that if we want to introduce a reasonable intersection theory on $X$ we have to ...
- 321
1
vote
0
answers
45
views
Hypersurface whose "square" level sets intersect all linear subspace of "high" dimension
Let $k$ be an infinite perfect field (e.g. I'm happy to assume that $k$ has characteristic $0$. On the other hand, the algebraically closed case is not interesting for this question). The question is ...
- 1,017
4
votes
1
answer
221
views
Proper locally trivial bundle is injective on Chow groups
If $X\to Y$ is a map of varieties that is Zariski-locally isomorphic to a projection $U\times P\to U$ with $P$ (smooth) proper, I think the pullback $A_{\bullet}(Y)\to A_{\bullet}(X)$ is supposed to ...
- 323
6
votes
1
answer
717
views
Fulton's deformation to the normal cone vs Verdier's
Let $X$ be a smooth variety over a field $k$, and let $Y$ be a smooth subvariety. In the literature, I've seen two versions of the deformation to the normal cone:
Verdier's version: $\tilde{X}_Y^\...
- 3,079
6
votes
0
answers
355
views
Schubert cycles on Grassmannian bundles
Let $X$ be smooth variety and let $\mathcal{E}$ be a vector bundle on $X$ of rank $n$. On the total space of the Grassmannian bundle $\pi:G(k,\mathcal{E})\to X$ we have the tautological exact sequence ...
- 735
3
votes
0
answers
440
views
The Chow ring of a blow-up along a badly embedded subscheme
Let $X$ be a variety and $i:Y\hookrightarrow X$ be a regularly embedded closed subvariety, and write $\tilde X$ for $Bl_Y X$ the blow-up of $X$ along $Y$. Let $A^*(X)$ refer to the Fulton-Macpherson ...
- 528
3
votes
1
answer
434
views
Intersections with a Power of an Ample Divisor on an Abelian Variety
Let $A$ be a $g$-dimensional, complex abelian variety, let $H$ be an ample divisor, let $D\in Pic^0(A)$, and let $0\leq k\leq g$.
Question 1: Does $D^k\neq 0\in CH^k(A,\mathbb{Q})$ imply that $H^{g-k}...
- 776
0
votes
0
answers
95
views
$ch(L f^*\epsilon)$
I know the famous principle (or theorem depending on how you define) of Chern classes for locally free sheaves and a morphism $f: X'\rightarrow X$,
$ch(f^* \epsilon)=f^* ch(\epsilon)$.
But if $f$ ...
4
votes
2
answers
347
views
Cycle class of zeroes of a global section
Let $\mathcal{F}$ be a locally free sheaf of rank $n$ on an $n$ dimensional complex manifold $X$. If the zero locus of a generic global section of $\mathcal{F}$ is $0$ dimensional, then its cycle ...
- 2,788
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
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
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
3
votes
1
answer
190
views
Projective embeddings and quasi-compactness
Let $X$ be a projective scheme over a ring $R$, and $p : X\to\mathbf{P}^n_R$ a projective embedding.
Does there exist $n$ large enough so that the complement $U\subset \mathbf{P}^n_R$ of $p(X)$ in ...
user124171
4
votes
2
answers
764
views
Linear sections of Segre varieties and rational normal scrolls
In a projective space $\mathbb{P}^{k+2}$ consider two complementary subspaces $\mathbb{P}^1,\mathbb{P}^k$, and let $C\subset\mathbb{P}^k$ be a degree $k$ rational normal curve. Fixed an isomorphism $\...
user114666
2
votes
0
answers
136
views
Explanation of proposition 6.7 (a) of Fulton's Intersetion Theory
Suppose $X$ is a smooth variety over a field $k$ of characteristic zero, and $Z$ is a smooth subvariety of codimension d. Now let $\tilde{X}$ be the blow-up of $X$ at $Z$, and let the exceptional ...
- 2,971
1
vote
0
answers
142
views
intersect a subvariety with a Schubert variety
Let $Y$ be an irreducible subvariety inside $Gr(r,n)$ (Grassmannian of $r$-plane inside $\mathbb{C}^n$) and $X_\lambda$ be a Schubert variety corresponding to $\lambda$. Assume that $codim(Y)+codim(X_\...
- 849
2
votes
1
answer
236
views
Linear sections of $Gr(V,2)$
Let $V$ be a vector space, and consider $G=Gr(V,2)\subset \mathbb{P}^N$ embedded via the Plucker embedding. Let $W\subset \mathbb{P}^N$ be a linear subspace. I want to find the class $[W\cap G]\in A(G)...
- 2,788
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
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
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
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:
$$
L^k:H^{2n-2k}\left(X_{\overline{k}},\mathbf{Q}_{\ell}\right)\to H^{2n+2k}\left(...
user92332
2
votes
1
answer
328
views
Lefschetz standard conjecture under specialization/generization
Let $S$ be a smooth connected noetherian scheme (not necessarily over a field) with residue fields that are all of finite type over their prime field.
Let $f: \mathcal{X}\to S$ be a smooth projective ...
user92332
1
vote
0
answers
34
views
Hodge classes generated in degree $1$
Let $X$ be a smooth projective variety over the complex numbers, and $\text{Hdg}^p(X)_{\mathbf{Q}}$ the abelian group of Hodge classes in $H^p(X,\mathbf{Q}(p))$.
Denote by $\text{Hdg}^*(X)$ the ...
user95222
1
vote
0
answers
290
views
Coniveau in étale motivic cohomology
Let $X$ be a smooth variety over a field.
Is there a spectral sequence:
$$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))$$...
user92332
3
votes
0
answers
114
views
Multiplicative structure on Deligne cohomology
Let $X$ be a smooth projective variety over the complex numbers, and $\mathbf{Z}(p)_{\mathcal{D}}$ the Deligne complex on $X$:
$$\mathbf{Z}(p)_{\mathcal{D}} : \ \ \mathbf{Z}(p)\to\mathcal{O}_X\to\...
user92332
2
votes
1
answer
295
views
Chow groups modulo homological equivalence for abelian varieties
Let $X$ be an abelian variety over a field $k$.
Let $A^p_{\rm hom}(X)$ be the $p$-th Chow group of cycles modulo homological equivalence ($\ell$-adic, if $k$ is of char $p$).
Do we have $$A^p_{\rm ...
user97068
1
vote
0
answers
118
views
Torsion homologically trivial cycles
Is there an example of a smooth projective variety $X$ over the complex numbers, such that
$$\ker(\text{CH}^2(X)\to H^4(X,\mathbf{Z}(2))$$
is not torsion?
user119470
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
5
votes
1
answer
552
views
A quite puzzling question on Deligne cohomology sheaves and cycle maps
Intro. I would be deeply grateful if someone could please clarify the following to me.
The question. (the main point is (4))
Let $X$ be a smooth projective variety over $\mathbf{C}$, and $\mathbf{Z}(...
user119470
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
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
1
vote
1
answer
435
views
Properties of codimension under pull back
If I pull back a cycle of codimension $c$ along a morphism of schemes I can easily see that the codimension can stay the same or the codimension can drop to all the way to $0$. But intuitively it ...
- 3,968
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
2
votes
1
answer
735
views
Pull-back of algebraic cycles
Since today is the Chow-variety day, I'm going to ask my question here.
Suppose I have a smooth projective variety $X$ over a field of characteristic zero, and a smooth hyperplane $p: H\...
user113393
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
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
5
votes
0
answers
398
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
7
votes
1
answer
470
views
Example of a smooth family of projective surfaces with non-vanishing integrals of Todd classes
Motivation:
Let $\pi\colon S \rightarrow B$ be smooth projective morphism of relative dimension 2 over a smooth projective scheme $B$. If the stucture sheaves of the fibres do not have higher ...
- 76
4
votes
1
answer
509
views
Intersection number of divisors on abelian surfaces and its invariance under translation by 2-Torsion points
I am reading Fulton's but I cannot find a useful result for my problem that seems to be something well known.
Let $\mathcal{J}$ be the Jacobian of a hyperelliptic curve of genus $2$. Consider $D_1,...
