All Questions
Tagged with intersection-theory ag.algebraic-geometry
329 questions
39
votes
3
answers
6k
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,...
- 12.3k
23
votes
0
answers
1k
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 ...
- 5,504
21
votes
1
answer
981
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.
...
user61522
17
votes
2
answers
2k
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 ...
- 693
16
votes
3
answers
6k
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?
- 2,179
16
votes
1
answer
2k
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 ...
- 27k
16
votes
1
answer
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
1
answer
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 ...
- 7,527
15
votes
2
answers
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 ...
- 18.7k
15
votes
6
answers
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 ...
- 28.9k
15
votes
3
answers
2k
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 $W \geq 0$, ...
- 7,338
15
votes
3
answers
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 ...
- 5,359
15
votes
1
answer
1k
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 ...
- 4,015
15
votes
1
answer
995
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 ...
- 904
13
votes
2
answers
3k
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 $s$ ...
- 583
13
votes
1
answer
1k
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 ...
- 3,555
13
votes
1
answer
563
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 $\mathbb P^2$ ...
- 30.6k
12
votes
4
answers
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 ...
- 5,929
12
votes
1
answer
6k
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{...
- 3,555
11
votes
3
answers
3k
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 $X$...
- 1,159
11
votes
2
answers
558
views
Hypersurface of singular plane cubics
In the projective space $\mathbb{P}^9 = \mathbb{P}(\mathbb{C}[x,y,z]_3)$, parametrizing plane cubics, consider the hypersurface $X\subset\mathbb{P}^9$ whose points corresponds to singular cubics. The ...
- 8,998
11
votes
4
answers
4k
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
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
11
votes
0
answers
608
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 ...
- 7,972
10
votes
1
answer
570
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 ...
- 40.3k
10
votes
1
answer
425
views
To whom is Bézout's theorem for varieties due?
The following is a modern, fairly general form of Bézout's theorem. (There are forms that are more general and/or more precise; bear with me.)
Define the degree of a reducible variety to be the sum of ...
- 20.2k
10
votes
1
answer
903
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 ...
- 1,803
10
votes
0
answers
327
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-...
- 1,761
10
votes
0
answers
331
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 ...
- 7,338
9
votes
1
answer
3k
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 $...
- 16k
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
8
votes
1
answer
792
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', ...
- 421
8
votes
2
answers
5k
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 ...
- 83
8
votes
1
answer
635
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 ...
- 503
8
votes
1
answer
837
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 ...
- 185
8
votes
0
answers
569
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^{...
user92332
8
votes
0
answers
204
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. ...
- 166
7
votes
2
answers
219
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 $\varphi^*:H^2(Y,\mathbb{Z})\rightarrow H^2(X,\mathbb{Z})$ ...
- 71
7
votes
1
answer
946
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
7
votes
1
answer
2k
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(...
- 420
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
7
votes
1
answer
2k
views
Chern classes of pushforwards
Let $f:X\to Y$ be a proper morphism of normal varieties (smooth as DM stacks, but I only care about the coarse spaces). The map $f$ is generically finite, but not flat (so no hope of smoothness and ...
- 16k
7
votes
1
answer
714
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-...
- 4,902
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
7
votes
1
answer
403
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 \...
- 443
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
7
votes
0
answers
183
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
0
answers
551
views
Semi-continuity of intersection numbers
I always trusted the following quite vague statement:
If you have a family of effective divisors $D_1(t),\dots , D_k(t)$ on a $k$-dimensional projective variety $X_t$, where $t$ is a paramater say ...
- 2,384
7
votes
0
answers
570
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 ...
- 4,265
6
votes
2
answers
789
views
Reference request: Kleiman's proof of Snapper's Lemma
On page 4 of Nitin Nitsure's paper Construction of Hilbert and Quot Schemes, the author refers to the fact that Hilbert polynomials are indeed polynomials as
a special case of Snapper's Lemma, see &...
- 1,805