Questions tagged [intersection-theory]

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36
votes
3answers
5k views

What do higher Chow groups mean?

Let $z^i(X, m)$ be the free abelian group generated by all codimension $i$ subvarieties on $X \times \Delta^m$ which intersect all faces $X \times \Delta^j$ properly for all j < m. Then, for each i,...
21
votes
0answers
935 views

Is there a functor of points approach to algebraic cycles and intersection theory?

Motivation Most of the algebraic geometry I have done so far was concerned with group schemes (e.g., abelian schemes, tori, unipotent groups). In that part of the field the "functor of points POV" is ...
20
votes
1answer
834 views

$8$-ary operation $(\mathbb{P}^2)^8 \text{ }-\to \mathbb{P}^2$, can we say anything about what this formula would look like?

My friend, who is currently taking an algebraic geometry course from an unnamed prolific poster on MO, told me about the following bonus question on one of his problem sets a few weeks ago. ...
16
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3answers
5k views

Survey article on Intersection Theory

Does anybody knows about good overview on intersection theory. The book of Fulton has very hard language. Does there exist simple overview on this topic with many examples?
16
votes
1answer
2k views

Geometric examples of the Serre intersection formula

The Serre intersection formula, as an alternating sum of contributions from Tor-groups, is something that combines a lot of ingredients that I'm interested in, but I've never really felt that I have a ...
16
votes
1answer
2k views

Deformation to the normal cone

Deformation to the normal cone appears in several places including Intersection theory and Verdier specialisation of construtible sheaves or D-modules. I'd like to understand what happens when we ...
15
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6answers
3k views

Curves with negative self intersection in the product of two curves

I wonder if the following is known: Are there two compact curves C1 and C2 of genus>1 defined over complex numbers, such that their product contains infinite number of irreducible curves of negative ...
15
votes
3answers
2k views

Can a curve intersect a given curve only at given points?

Clearly the question in the title has a positive answer for analytic (or smooth, or continuous ...) curves, but what about the algebraic category? More specifically, given an irreducible polynomial ...
15
votes
1answer
832 views

Link: Serre's intersection formula <-> Bloch-Quillen Thm / When only intersecting divisors, is there 'shorter' approach of proof known?

In very short: When proving the equivalence of intersection theory constructed through (Milnor) K-sheaves and their product vs. defining the product via Serre's local multiplicity formula + moving, I ...
14
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2answers
2k views

Is there a Serre Tor formula for nonproper intersections?

Background: Let $X$ be a smooth complex projective algebraic variety, and let $V$ and $W$ be closed subvarieties. For simplicity, let's assume that $\dim V+\dim W=\dim X$. Now Serre's famous Tor ...
14
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1answer
1k views

Bezout's Theorem for weighted homogeneous polynomials

Bezout's Theorem states that for two homogeneous polynomials $f(x,y,z), g(x,y,z)$ over an algebraically closed field of degrees $m,n$ respectively, such that the two polynomials do not share a common ...
14
votes
2answers
844 views

What does taking the graded algebra do to the Grothendieck group, and its relation to the Chow ring?

Let $X$ be a nonsingular variety. (Perhaps some/all of this works over more general smooth schemes, but let's stick to the simple case.) In, e.g., Fulton's Intersection Theory chapter 15, and Soule's ...
14
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1answer
926 views

Are Chow groups generated by local complete intersections?

Let $X$ be a smooth projective variety over an algebraically closed field. The Chow group $\mathbb Q\mathrm{CH}^d(X)$ is $\mathbb Q$--linearly generated by irreducible subvarieties $Z \subseteq X$ of ...
13
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1answer
480 views

Intersection of subvarieties versus ranks of Chow groups modulo numerical equivalences

A nice property of $\mathbb P^n$ is: Property 1: Two subvarieties $U,V$ such that $\operatorname{dim} U +\operatorname{dim} V \geq n$ always intersect. (for example, any 2 curves in $\mathbb P^2$ ...
12
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4answers
2k views

Context for intersection theory

This is a pretty basic question. Hartshorne defines "intersection multiplicity" for any two divisors on a surface. Fulton has an impressive framework of generalizing this in his book (my understanding ...
12
votes
3answers
1k views

A nontrivial surface on which any two curves intersect

One interesting property of the projective plane is that any two plane curves intersect. (More generally, if $V$ and $W$ are subvarieties of any projective space, and codim $V$ + codim $W \geq 0$, ...
11
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4answers
2k views

Question on Kähler/ample cone, cone of curves…

Assume $X$ is smooth "simply connected" complex projective variety and $Y\subset X$ a smooth hyperplane section. ( $Y= X\cap H$, $H\subset \mathbb{P}^n$). Let's $NE(X)$ be the cone of effective 1-...
11
votes
1answer
513 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 ...
11
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2answers
2k views

Examples of excess intersection theory?

Let $M$ be a smooth manifold of dimension $m$ and $\pi:E\rightarrow M$ a vector bundle of rank $e$. Given a section $s$ of the bundle $\pi:E\rightarrow M$, we expect that the zero locus $Z(s)$ of $s$ ...
11
votes
1answer
681 views

Schemes with no nonconstant maps to lower dimensional schemes

Fix an algebraically closed field $k$ (arbitrary characteristic), all schemes will be of finite type over $k$. (Property *): I'm interested in (classes of) examples of schemes $X$ (irreducible, of ...
10
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3answers
2k views

Chern classes of a blow-up at a point

Let $X$ be a nonsingular projective variety over $\mathbb{C}$, and let $\widetilde{X}$ be the blow-up of X at a point $p\in X$. What relationships exist between the degrees of the Chern classes of $X$...
10
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1answer
4k views

Self-intersection and the normal bundle

Let $X/k$ be a surface nonsingular and proper over an algebraically closed field $k$. Let $C \subset X$ be a nonsingular curve. Then it is clear that the self-intersection $(C \cdot C)_X$ is $\textrm{...
10
votes
0answers
367 views

The virtual fundamental class as derived intersection

Say $X$ is a smooth projective variety and $\beta\in H_2(X)$ is a class. Then there is a finite-type proper scheme (or in general, stack) $SM : = \overline{\mathcal{M}}_{g,n}(X,\beta)$ of stable maps ...
10
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0answers
300 views

Smooth, complete varieties on which “zero is effective”

I will say zero is effective on a complete, smooth variety $X$ if some positive linear combination of irreducible varieties is rationally equivalent to zero. In other words, zero is effective if there ...
9
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1answer
462 views

Commutativity of the Chow ring in positive characteristic

I was looking in Ravi Vakil's notes on Intersection Theory, Class 20, where he introduces the bivariant intersection theory, in particular the Chow ring $A^\ast (X)$. On p. 2, he writes the following ...
9
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1answer
2k views

Reference for the Hodge Bundle

For the purposes of this question, let the Hodge bundle $\lambda$ be the bundle on a fibration of abelian varieties $X\to B$ with fiber over $b\in B$ the space of 1-forms on $X_b$, or the pullback to $...
9
votes
1answer
361 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 $E$ ...
8
votes
1answer
634 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', ...
8
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1answer
1k views

What is the relation between Lefschetz fixed point theorem and Poincare-Hopf theorem on vector fields?

In Dubrovin/Fomenko/Novikov Modern geometry--Methods and applications, Part II, the (Poincare-)Hopf theorem is treated in section 15.2 (see theorem 15.2.7 on page 129), while the Lefschetz theorem on ...
8
votes
2answers
357 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 $\...
8
votes
1answer
521 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 ...
8
votes
1answer
396 views

Interpretation of “27” lines for cubic surface with rational double points

It is well known that a smooth cubic surface has $27$ distinct lines. Explicitly, if we choose a planar representation, i.e., blowup $\mathbb P^2$ at $6$ general points $p_1,...,p_6$, the $27$ lines ...
8
votes
1answer
608 views

Flat morphisms whose fibers are affine spaces

Let $f:X \to Y$ be a flat morphism, such that each fiber is isomorphic to the affine space $\mathbb{A}^n$. Then is is true that $f$ is a Zariski affine bundle? If not, is it at least an ètale affine ...
8
votes
0answers
201 views

Integrality of primary genus $0$ Gromov-Witten invariants of a Fano manifold

Suppose $(X,\omega)$ is a positively monotone compact symplectic manifold, i.e., after a positive scaling of the symplectic form, we have $c_1(T_X) = [\omega]$ in de Rham cohomology ($T_X$ has well-...
8
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0answers
390 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^{...
8
votes
0answers
189 views

Chow ring of extended tropicalizations

In Allermann-Rau '09, the authors define the Chow groups of an arbitrary abstract tropical cycle. In particular, one may take the tropical cycle to be the tropicalization of a subvariety of a torus. ...
7
votes
2answers
3k views

Self-intersection of exceptional divisor

Suppose that $X$ is a smooth threefold, and $C \subset X$ a smooth curve. Let $Y$ be the blowup of $X$ along $C$, with exceptional divisor $E$. What is the intersection number $E^3$ on $Y$? (in ...
7
votes
1answer
371 views

Intersection theory in analytic geometry

This might be a weird/stupid question, but it came to me a couple of times, and I would like to get an answer for that. In some papers I read, constantly the authors define some analytic subspaces, ...
7
votes
1answer
445 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 ...
7
votes
1answer
1k views

Simple description of a Chow ring of blow-ups.

Is there a simple description of a Chow ring of a blow-up of a point on a smooth projective variety? Or at least of successive blow-ups of $\mathbb{P}^n$? Maybe something like $A(\tilde{X})=f^*(A(...
7
votes
1answer
528 views

Local model of virtual fundamental cycle

The following baby version of virtual fundamental cycle is well known: Let $M\subset V$ be the zero locus of a section $s$ of a vector bandle $E \to V$, in general $s$ is not transversal to the zero ...
7
votes
1answer
667 views

Calculating chern numbers yields a contradiction, why?

I am really stuck on this one. Let $Y=\mathbb{P}^n$ be the complex projective space and let $\tilde Y$ be the blow-up of $Y$ along a linear subvariety $X$ of codimension $d$. We get the following blow-...
7
votes
1answer
378 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 ...
7
votes
1answer
309 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 \...
7
votes
1answer
340 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 ...
7
votes
0answers
117 views

Intersection numbers via residue formula

$\newcommand{\sslash}{\mathbin{/\mkern-7mu/}}$With a friend we are trying to understand residue formulas in the article "Cohomology pairings on singular quotients in geometric invariant theory&...
7
votes
0answers
497 views

intersection theory on proper algebraic spaces

I have a question about the second example in Hartshorne's Algebraic Geometry, Appendix B, section 3 (given by Hironaka?). It is an example of a compact complex Moishezon 3-fold $X$ which is not an ...
6
votes
2answers
205 views

Action of an isomorphism in cohomology as the intersection with the class of the graph

Let $X$ and $Y$ be two complex manifolds of dimension 2 and let $\varphi:X\rightarrow Y$ be an isomorphism. I have read that the action of $\varphi^*:H^2(Y,\mathbb{Z})\rightarrow H^2(X,\mathbb{Z})$ ...
6
votes
2answers
294 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 ...
6
votes
2answers
580 views

Graphs of maps between manifolds as cycles and intersection theory

I'm guessing that the answer to this question is well-known, but I'm struggling to find anything to help me. Let $X,Y$ be compact manifolds of dimension $n,m$ respectively. Let $f:X \to Y$ be a ...

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