Questions tagged [intersection-theory]
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376 questions
3
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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
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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
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136
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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
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0
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142
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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
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1
answer
236
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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
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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
$$H^r(X_{\rm Zar}, R^sp_*\mathbf{C})\Rightarrow H^{...
user92332
2
votes
0
answers
256
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Neron Severi under specialization
Let $X$ be a smooth projective variety over $\mathbf{Q}$, and $\mathcal{X}$ a smooth projective model over $\mathbf{Z}[1/N]$ for $N$ large enough.
Call $\eta$ the generic point $\text{Spec}(\mathbf{Q}...
user95222
4
votes
0
answers
261
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Motives up to homological equivalence
Let $X$ be a smooth projective variety over a field $k$ finitely generated over its prime field, and $M_{hom}(X)$ the category of motives modulo $\ell$-adic homological equivalence.
(1) Is $M_{hom}(...
user120812
4
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0
answers
245
views
Hard Lefschetz for cycles
Let $X$ be a smooth projective variety over a field $k$. It is known by work of Deligne, that the Lefschetz operator:
$$
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
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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
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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
655
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
434
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
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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
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
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
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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,...
7
votes
1
answer
403
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Family of zero dimensional subschemes
While reading Fulton's Intersection theory, I came across the following comment.
Let $X$ be a projective scheme over an algebraically closed field. Assume we have been given a map $g : \mathbb{P}^1 \...
- 443
4
votes
0
answers
265
views
Is there an analogy of Sumihiro's equivariant Chow's lemma for DM stack?
There is an analogy of Chow's lemma for a DM stack $X$ written in the Laumon's book 'Champ algebrique'. There exists a generically finite, proper surjective morphism $Y \to X$ from a quasi-projective ...
- 421
1
vote
1
answer
200
views
Gysin map for projective sub-bundles of exceptional divisors
Let $X$ be a smooth, projective variety, $Y \subset X$ a smooth, projective subvariety of codimension $3$. Denote by $\pi:\tilde{X} \to X$ the blow-up of $X$ along $Y$ and by $E$ the exceptional ...
- 2,126
1
vote
1
answer
2k
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Intersection number of divisors with its pull back and its push forward
I am in an ideal situation but I would appreciate a hint. First here is the scenario.
Let $\mathcal{J}$ be an the abelian variety obtained from the Jacobian of a genus $2$ curve $\mathcal{H}/k$ ...
7
votes
1
answer
449
views
Higher Chow groups for complete smooth intersections?
Let $F$ be a smooth complete intersection of $r$ hypersurfaces of degree $d_{1},\dots,d_{r}$ in $\mathbb{P}^{n+r}$ over an algebraic closed field. A classical result of A. Roitman says that the group ...
- 578
1
vote
1
answer
827
views
Self-intersection of divisors and Chern class
Let $X$ be a smooth, projective variety and $Y \subset X$ a smooth, effective divisor. Consider now the natural map $i^\ast:H^2(X) \to H^2(Y)$. Then,
When is the image of $c_1(\mathcal{O}_X(Y)) \in H^...
- 2,126
7
votes
1
answer
945
views
Push-forward of nef divisors via finite morphisms
Let $f:X\rightarrow Y$ be a finite morphism between smooth projective varieties, and let $D$ be an effective nef but not ample divisor on $X$.
Consider the divisor $f_{*}D$ on $Y$. Is $f_{*}D$ nef ...
user58018
6
votes
0
answers
313
views
Intersection of curves in abelian varieties
Fix an abelian variety $A$ of dimension at least $3$ and a (smooth, projective, irreducible) curve $C$ inside $A$ that generates $A$ as an algebraic group, over an algebraically closed field. Now, fix ...
- 30.5k
1
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0
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289
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Intersection with very ample divisor and linear equivalence
Let $X$ be a smooth, projective variety and $D, E$ two effective divisors of $X$ which correspond to distinct elements on the cohomology group $H^2(X,\mathbb{Q})$. Denote by $H$ a very ample divisor ...
- 2,126
4
votes
1
answer
572
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Pushforward of curves
Let $Z$ be a subvariety of an irreducible projective variety $X$, and let $i:Z\rightarrow X$ be the inclusion.
Let $N_1(X),N_1(Z)$ be the $\mathbb{Q}$-vector spaces of curves in $X$ and $Z$ ...
user114666
7
votes
1
answer
482
views
Why Green functions and not Neron functions?
Arakelov constructed a nice intersection theory on arithmetic surfaces. A key point is the notion of Green function for a Riemann surface, which will be involved in the ''part at infinity'' of the ...
- 321
6
votes
1
answer
1k
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Higher Chow groups revisited
Let $X$ be an algebraic variety over a field $k$.
Bloch defines the "algebraic singular complex" using the algebraic simplices
$$\Delta^n = \text{Spec}(k[x_0,\dots,x_n]/(x_0+x_1+\dots+x_n=1) \subset ...
user97068
4
votes
1
answer
389
views
Gysin map and blow up
Let $X$ be a smooth projective variety and $W \subset X$ a smooth, projective subvariety. Let $\pi:\tilde{X} \to X$ be the blow-up of $X$ along $W$. Let $E$ be the exceptional divisor of $\pi$ and $i:...
- 2,126
5
votes
1
answer
2k
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Intersections of quadratic planes as elliptic curves
An elliptic curve defined over a field $k$ is a smooth projective curve of genus $1$, plus a $k$-rational point. Every elliptic curve can be written in a Weierstrass form, i.e. as a plane cubic curve ...
- 3,432
1
vote
0
answers
188
views
First computations of intersection products – a formula in Fulton
The book is Fulton, Intersection Theory. My question pertains to Examples 6.1.4 and 6.1.5. In 6.1.4, we are looking at effective Cartier divisors $A,B$ and $D$ on a nonsingular surface $X$ with $A$ ...
- 1,217
4
votes
2
answers
2k
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How to define the intersection multiplicity of a projective variety and a complete intersection?
In the appendix of Algebaric Geometry by Hartshorne, he shows us that how Serre defines the intersection number in a more general case:$$i(X,Y;Z)=\sum(-1)^i\cdot\bigl(\operatorname{length} \...
- 41
9
votes
1
answer
475
views
About Riemann-Roch without denominators
The Riemann-Roch without denominators can be expressed as follows:
Let $f: X\rightarrow Y$ be a closed embedding of quasi-projective smooth $k$-varieties of codimension $d$ for some field $k$. Let $E$ ...
- 1,906
3
votes
0
answers
215
views
Is algebraic geometry related to conical intersection in potential energy surface of molecules?
I post this question here because it seems that the equations describing conical intersections in molecular potential energy surface are similar to what algebraic geometry research concerns.
...
- 157
4
votes
0
answers
181
views
Why is flatness needed for the Segre classes of a family of cones to be equal in the Chow ring of the base
Let $X$ be an algebraic scheme and $\mathscr C$ a cone on $X\times\mathbf A^1$ and $C_t$ denote the restriction of $\mathscr C$ to $X\times\{t\}$ ($t=0,1$, or whatever). The claim in Fulton's ...
- 1,217
11
votes
1
answer
737
views
Chow ring of Hilbert scheme of 4 points in $\mathbb{P}^2$
What is known about the Chow ring of the Hilbert scheme of length 4 subschemes of $\mathbb{P}^2$?
I know there is work on cycles on Hilbert schemes in the literature, but I don't know what can be ...
- 1,537
1
vote
0
answers
248
views
Irreducible components of normal cone $C_{X/Y}$ dominates X?
Assume $X$ is a subscheme of $Y$ and $X,Y$ are irreducible.
Then every irreducible component of the normal cone $C_{X/Y}$ dominates $X$?
- 421