Questions tagged [intersection-theory]
The intersection-theory tag has no usage guidance.
361
questions
2
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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 $$A^p_{\rm ...
1
vote
0
answers
118
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Torsion homologically trivial cycles
Is there an example of a smooth projective variety $X$ over the complex numbers, such that
$$\ker(\text{CH}^2(X)\to H^4(X,\mathbf{Z}(2))$$
is not torsion?
1
vote
0
answers
116
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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, ...
5
votes
1
answer
532
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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}(...
4
votes
0
answers
330
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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$ ?
2
votes
0
answers
595
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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\...
1
vote
1
answer
388
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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 ...
2
votes
0
answers
233
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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 ...
2
votes
1
answer
666
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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\...
2
votes
0
answers
257
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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\...
2
votes
0
answers
220
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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$,...
5
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0
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371
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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 ...
7
votes
1
answer
460
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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 ...
4
votes
1
answer
484
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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
372
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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 \...
4
votes
0
answers
255
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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 ...
1
vote
1
answer
194
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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 ...
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
422
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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 ...
1
vote
1
answer
757
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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^...
7
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1
answer
823
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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 ...
6
votes
0
answers
306
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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 ...
1
vote
0
answers
270
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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 ...
4
votes
1
answer
547
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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$ ...
7
votes
1
answer
462
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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 ...
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 ...
4
votes
1
answer
369
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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:...
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 ...
1
vote
0
answers
167
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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 $A$ ...
4
votes
2
answers
1k
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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} \...
9
votes
1
answer
452
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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 $E$ ...
3
votes
0
answers
207
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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.
...
4
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0
answers
166
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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 ...
11
votes
1
answer
695
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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
vote
0
answers
240
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 $X$?
6
votes
0
answers
563
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Blow-up and the Chow group of zero cycles
Let $\tilde{X}\to X$ be a blow-up of a variety $X$ (over an algebraically closed field).
Is it true that the Chow group of zero cycles of $\tilde{X}$ is isomorphic to that of $X$? What if $X$ is a ...
1
vote
0
answers
230
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Do A-infinity algebra(in Floer theory)have some kind of intersection theory and Poincare duality?
In Lagrangian Floer theory, we can define an A-infinity algebra. It is by first choosing a subset $X_L$ of chains in the Lagrangian submanifold $L$, and then defining boundary maps on(Actually, sum of ...
2
votes
0
answers
127
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intersections between closed curves on surfaces
I would like to find a result telling me that two simple closed curves $\alpha$ and $\beta$ (on a non-orientable surface $S$) are in minimal position if and only if there is not a disk in $S$ whose ...
8
votes
1
answer
761
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What is the main failure in using Naive Chow group in Artin Stack
I'm reading Andrew Kresch's paper, Cycle groups in Artin Stacks.
The author defined Chow groups of Artin stacks by very technical way, instead of ordinary ways which he called 'naive chow group', ...
0
votes
0
answers
195
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Linear section of an algebraic variety
Let $\pi$ be a linear subspace of $\mathbb{P}^n$ and $X$ a reduced, irreducible variety of $\mathbb{P}^n$. Suppose that $\pi \cap X$ is reducible, hence $\pi \cap X=Y_1\cup Y_2 \cup \cdots Y_k$. When ...
0
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0
answers
111
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Reducible sections of algebraic varieties
Let $X$ be an irreducible variety. Is there some necessary condition on a hyperplane $H$ such $X\cap H$ is reducible? Also, suppose that $H\cap X$ is reducible, i.e., $H\cap X=Y_1\cup Y_2 \cup \cdots \...
1
vote
1
answer
210
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Zero dimensional components of an intersection
Let $X$ be a smooth projective algebraic variety over an algebraically closed field and let $A,B$ be closed irreducible subvarieties of complementary codimension in $X$. Let $n$ denote their ...
2
votes
0
answers
185
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Top intersections on the Hilbert scheme of points on a surface
The Picard group of $S^{[n]}$ is generated by the Picard group of $S$ (via a map $L \mapsto L_n$) and $E$, where $E = -\frac{B}{2}$, where $B$ is the exceptional divisor of the Hilbert Chow morphism.
...
1
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0
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103
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Degree of an isogeny in the endomorphism ring of the jacobian of a curve and self intersection index in its ring of correspondences
I hope this question is not too basic.
Let $C/\bar{k}$ be a nonsingular irreducible curve of genus $g$ and $\mathfrak{C}(C\times C)\cong \text{CH}^1(C\times C)$ be its ring of correspondences.
I am ...
0
votes
1
answer
299
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Intersections of divisors in blow-ups of $\mathbb{P}^n$
Let $p_1,p_2,p_3\in\mathbb{P}^n$ be three general points, $X$ the blow-up of $\mathbb{P}^n$ at $p_1,p_2,p_3$, then along the lines $\left\langle p_i,p_j\right\rangle$, and finally along the plane $\...
4
votes
1
answer
199
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Segre Classes of reducible variety
Suppose I have a singular projective variety $X\subset \mathbb{P}^n$ that is reducible with $X=\bigcup_i X_i$ smooth irreducible components. That is, the irreducible components are smooth but $X$ is ...
1
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0
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273
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How to think about the quotient field of an integral stack?
This is the definition given in Vistoli's paper.
Let $F$ be an integral stack. A rational function of $F$ is a morphism $G \rightarrow A^1_S$ defined on a nonempty open substack $G$ of $F$.
...
2
votes
0
answers
226
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Computing higher dimensional intersection numbers for complete intersections of $\mathbb P^n$
Let $X_1,X_2$ be two smooth hypersurfaces of degree $d$ in $\mathbb P^{n}$. Let $B=X_1\cap X_2$. Assume $B$ is smooth. Let $\mathcal N_{B/\mathbb P^n}$ be the normal bundle to $B$. Let $H$ be the ...
6
votes
2
answers
341
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Nonempty intersection in Grassmannian
Where can I find a proof of the following fact:
If $X_1$ and $X_2$ are subvarieties of $\mathbb{G}(k,n)$ of codimension $c_1$ and $c_2$ satisfying $c_1+c_2<n+1-2k$, then the intersection $X_1\cap ...
2
votes
0
answers
96
views
Class of the locus where two sections are proportional
Let $X$ be a smooth (complex) projective $n$-dimensional variety ($n\geq 3$) and $\mathcal E$ a vector bundle of rank $r<n$ generated by its global sections on $X$. Let $\sigma\in H^0(\mathcal E)$ ...