Questions tagged [intersection-theory]

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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,...
Peter Arndt's user avatar
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 ...
jmc's user avatar
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20 votes
1 answer
959 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. ...
user avatar
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 ...
peterx's user avatar
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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?
Klim Efremenko's user avatar
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 ...
Stanley Yao Xiao's user avatar
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 ...
AFK's user avatar
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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 ...
John Pardon's user avatar
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15 votes
6 answers
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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 ...
Dmitri Panov's user avatar
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15 votes
3 answers
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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 $W \geq 0$, ...
Charles Staats's user avatar
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 ...
pinaki's user avatar
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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 ...
Xandi Tuni's user avatar
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15 votes
1 answer
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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 ...
olli_jvn's user avatar
  • 894
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 $s$ ...
Koopa's user avatar
  • 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 ...
LMN's user avatar
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13 votes
1 answer
553 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 $\mathbb P^2$ ...
Hailong Dao's user avatar
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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 ...
Makhalan Duff's user avatar
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 $X$...
gio's user avatar
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11 votes
1 answer
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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{...
LMN's user avatar
  • 3,515
11 votes
2 answers
519 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 ...
Puzzled's user avatar
  • 8,832
11 votes
1 answer
694 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
  • 1,537
11 votes
0 answers
538 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 ...
Dmitry Vaintrob's user avatar
10 votes
4 answers
3k 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-...
Mohammad Farajzadeh-Tehrani's user avatar
10 votes
1 answer
540 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 ...
Dan Petersen's user avatar
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10 votes
1 answer
750 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 ...
AG learner's user avatar
  • 1,701
10 votes
0 answers
313 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-...
Mohan Swaminathan's user avatar
10 votes
0 answers
320 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 ...
Charles Staats's user avatar
9 votes
1 answer
384 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 ...
H A Helfgott's user avatar
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9 votes
1 answer
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 $...
Charles Siegel's user avatar
9 votes
1 answer
450 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 $E$ ...
Lao-tzu's user avatar
  • 1,856
8 votes
1 answer
760 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', ...
keaton's user avatar
  • 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 ...
cluelessgrad's user avatar
8 votes
1 answer
2k 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 ...
Mikhail Katz's user avatar
  • 15.1k
8 votes
1 answer
614 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 ...
Xiaobo Zhuang's user avatar
8 votes
1 answer
792 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 ...
Calc's user avatar
  • 185
8 votes
0 answers
520 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^{...
user avatar
8 votes
0 answers
202 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. ...
Max Kutler's user avatar
7 votes
1 answer
820 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 ...
user avatar
7 votes
1 answer
459 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 ...
user'''''''s user avatar
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(...
14555's user avatar
  • 410
7 votes
1 answer
702 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-...
Jesko Hüttenhain's user avatar
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
  • 299
7 votes
1 answer
371 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
  • 411
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 ...
Tsk's user avatar
  • 578
7 votes
0 answers
175 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&...
Nicolas Hemelsoet's user avatar
7 votes
0 answers
1k views

Applications of E8 manifold

The $E_8$ Cartan matrix is given by, $$ K_{E_8}=\begin{pmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\ ...
wonderich's user avatar
  • 10.3k
7 votes
0 answers
187 views

Divisibility of all entries in an intersection form

What are situations where one can conclude that all entries of an intersection form are divisible by a fixed integer? More precisely: $F \subset S$ is a proper connected (usually reducible) half-...
Geordie Williamson's user avatar
7 votes
0 answers
552 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 ...
shenghao's user avatar
  • 4,195
6 votes
1 answer
549 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, ...
Winnie_XP's user avatar
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