# Questions tagged [intersection-theory]

The tag has no usage guidance.

302 questions
Filter by
Sorted by
Tagged with
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,...
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 ...
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. ...
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?
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $\mathbb P^2$ ...
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 ...
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 $W \geq 0$, ...
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-...
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 ...
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 $s$ ...
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 ...
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 $X$...
4k views

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 $\textrm{... 0answers 367 views ### The virtual fundamental class as derived intersection Say$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)$of stable maps ... 0answers 300 views ### Smooth, complete varieties on which “zero is effective” I will say zero is effective on a complete, smooth variety$X$if some positive linear combination of irreducible varieties is rationally equivalent to zero. In other words, zero is effective if there ... 1answer 462 views ### Commutativity of the Chow ring in positive characteristic I was looking in Ravi Vakil's notes on Intersection Theory, Class 20, where he introduces the bivariant intersection theory, in particular the Chow ring$A^\ast (X)$. On p. 2, he writes the following ... 1answer 2k views ### Reference for the Hodge Bundle For the purposes of this question, let the Hodge bundle$\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$, or the pullback to$...
361 views

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 $E$ ...
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', ...
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 ...
357 views

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 ...
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-...
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 ...
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 $g : \mathbb{P}^1 \... 1answer 340 views ### Higher Chow groups for complete smooth intersections? Let$F$be a smooth complete intersection of$r$hypersurfaces of degree$d_{1},\dots,d_{r}$in$\mathbb{P}^{n+r}$over an algebraic closed field. A classical result of A. Roitman says that the group ... 0answers 117 views ### Intersection numbers via residue formula$\newcommand{\sslash}{\mathbin{/\mkern-7mu/}}$With a friend we are trying to understand residue formulas in the article "Cohomology pairings on singular quotients in geometric invariant theory&... 0answers 497 views ### intersection theory on proper algebraic spaces I have a question about the second example in Hartshorne's Algebraic Geometry, Appendix B, section 3 (given by Hironaka?). It is an example of a compact complex Moishezon 3-fold$X$which is not an ... 2answers 205 views ### Action of an isomorphism in cohomology as the intersection with the class of the graph Let$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})$... 2answers 294 views ### Nonempty intersection in Grassmannian Where can I find a proof of the following fact: If$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 ...
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 ...