Questions tagged [intersection-theory]

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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 ...
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1 vote
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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?
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1 vote
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116 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, ...
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5 votes
1 answer
532 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}(...
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4 votes
0 answers
330 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 $X'\to X$ ?
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2 votes
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595 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\...
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1 vote
1 answer
388 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 ...
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233 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 ...
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2 votes
1 answer
666 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\...
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2 votes
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257 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\...
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2 votes
0 answers
220 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 $0\le p\le d$,...
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371 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 ...
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7 votes
1 answer
460 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 ...
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4 votes
1 answer
484 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,...
Eduardo R. Duarte's user avatar
7 votes
1 answer
372 views

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 \...
random123's user avatar
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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 ...
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1 answer
194 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 ...
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1 answer
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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 $\mathcal{H}/k$ ...
Eduardo R. Duarte's user avatar
7 votes
1 answer
422 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 ...
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1 answer
757 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^...
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7 votes
1 answer
823 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 ...
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6 votes
0 answers
306 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 ...
Felipe Voloch's user avatar
1 vote
0 answers
270 views

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 ...
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4 votes
1 answer
547 views

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 $Z$ ...
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7 votes
1 answer
462 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 ...
manifold's user avatar
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1 answer
1k views

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 ...
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4 votes
1 answer
369 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:...
Ron's user avatar
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5 votes
1 answer
2k views

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 ...
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0 answers
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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 $A$ ...
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4 votes
2 answers
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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} \...
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9 votes
1 answer
452 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 $E$ ...
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3 votes
0 answers
207 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. ...
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4 votes
0 answers
166 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 ...
Tomo's user avatar
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11 votes
1 answer
695 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 ...
DCT's user avatar
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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 $X$?
keaton's user avatar
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6 votes
0 answers
563 views

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 ...
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1 vote
0 answers
230 views

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 ...
Wenfeng Jiang's user avatar
2 votes
0 answers
127 views

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 ...
user104820's user avatar
8 votes
1 answer
761 views

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', ...
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0 votes
0 answers
195 views

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 ...
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0 answers
111 views

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 \...
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1 vote
1 answer
210 views

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 ...
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2 votes
0 answers
185 views

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. ...
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1 vote
0 answers
103 views

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 ...
Eduardo R. Duarte's user avatar
0 votes
1 answer
299 views

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 $\...
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4 votes
1 answer
199 views

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 ...
IMeasy's user avatar
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1 vote
0 answers
273 views

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 $F$. ...
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  • 231
2 votes
0 answers
226 views

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 ...
Jesus Martinez Garcia's user avatar
6 votes
2 answers
341 views

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 ...
DCT's user avatar
  • 1,537
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 $\sigma\in H^0(\mathcal E)$ ...
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