Questions tagged [intersection-theory]
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376 questions
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Bounds for intersection multiplicity
Let's for simplicity work in $\mathbb{C}^n$. Suppose that $f_1,\dots, f_n$ are polynomials and $0$ is an isolated solution of the system $f_1(z)=\dots=f_n(z)=0$. I want to bound from below the ...
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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 ...
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Negative self intersection and section of the conormal sheaf for a singular complex curve
Let $M$ be a complex $2$-dimensional manifold and $C$ be a compact complex curve in $M$ (possibly singular).
Let us suppose that there exists a holomorphic function $f\in\mathcal{O}(M)$ such that
$f$ ...
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Relationship between virtual cohomological dimension and tautological rings for moduli spaces of curves
Here's the short version of the question. For $M_{g,n}$, $M_{g,n}^{rt}$, $M_{g,n}^{ct}$ and $\overline M_{g,n}$ it seems that the virtual cohomological dimension is given by the complex dimension plus ...
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Nef classes on abelian varieties in positive characteristic
Thomas Bauer shows in http://arxiv.org/pdf/alg-geom/9712019v1.pdf that for a complex abelian variety a nef line bundle is numerically equivalent to an effective divisor (this is shown in Lemma 1.1). ...
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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 ...
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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. ...
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Chow ring of a $\mu_2$-gerbe
Suppose that $X$ is a stack, and $Y \to X$ is a $\mu_2$-gerbe. Is there any relationship between the integral Chow rings (in the sense of Edidin and Graham) of $X$ and $Y$?
(I assume they become ...
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2
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219
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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 $\varphi^*:H^2(Y,\mathbb{Z})\rightarrow H^2(X,\mathbb{Z})$ ...
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Reference request: Samuel's multiplicity and degree
I am looking for references for the following simple facts.
Let $Y\subset \mathbb{P}^n$ be a variety (or pure-dimensional algebraic set). For $P\in Y$ denote by $e_p(Y)$ the (Samuel's) multiplicity ...
- 2,263
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Axiomatic intersection theory
Is there an axiomatic intersection theory?
What I expect is something like:
An intersection theory is a functor from the category of schemes(or other spaces) to the category of algebras, with well-...
- 35.9k
3
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434
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Bezout's theorem for non-proper intersections?
Is there a version of Bézout's theorem for non-proper intersections?
For my specific problem, the setup is as follows: Let $P_1,P_2,P_3,P_4\in\mathbb{C}[z_1,z_2,z_3,z_4]$, and suppose that (as a ...
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Intersection of curves
Let $f(x,y)=0$ and $g(x,y)=0$ be curves in $\mathbb R^2$. Assume that the origin $(0,0)\in \mathbb R^2$ is a $d$-fold point of $f$ and an $e$-fold point of $g$, respectively. Let $f_d(x,y)$ be the sum ...
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Relation between the index of (the sum of) lattices in euclidean space, and their orthogonal complement.
I am trying to figure out something concerning the index of lattices.
The question came about after reading the paper of W.Fulton and B.Sturmfels, ("Intersection theorey on toric varieties"). To ...
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Non-proper intersection of projective schemes
Let $X, Y$ be projective varieties in $\mathbb{P}^n$ for $n>10$. Assume that dimensions of $X,Y$ are greater than $n/2$. My first question is as follows:
Is there any criterion (...
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283
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Is the zero set of a equivariant polynomial map of minimal degree a union of linear subspaces?
Suppose that a finite group acts on two vector spaces $X$ and $Y$, and that $f:X\longrightarrow Y$ is an equivariant polynomial map which is homogeneous of degree $n$, and that there does not exist ...
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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 ...
- 8,004
5
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2
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755
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Top chern class under finite, unramified, dominant morphism
Situation: Let $\Bbbk$ be an algebraically closed field. Assume that $\pi:Y\to X$ is an finite, dominant, unramified morphism between nonsingular varieties of dimensions $n$. Let $d=\deg(\pi)$.
What ...
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543
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Minimal resolution of Log del Pezzo surfaces
Suppose $X$ is a log del pezzo projective surface of index $l$. As far as I understand it will have a finite number of singular points all of which can be resolved by sucessive blow-ups.
Let $E_i$ be ...
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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 ...
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Lefschetz Fixpoint theorem for non-orientable manifolds
The classical lefschetz fixpoint theorem is stated for oriented compact manifolds $M$ and a smooth map $f:M\to M$ as follows:
the intersection number $I(\Delta, \mathrm{graph}(f))$ is equal to the ...
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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 ...
- 42.9k
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Cohomology of vector bundles via Intersection Theory
Let $X$ be a smooth projective variety over a fixed field $k$ (take $k = \mathbb{C}$ if necessary). For a vector bundle $E$ on $X$, $ch(E)$ will be in the Chow ring.
$\textbf{Question 1: }$ If $\...
- 21.4k
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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 $\mathbb P^2$ ...
- 7,338
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440
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non proper intersection
Let X and Y two smooth closed subschemes of a smooth projective scheme Z over a field.
Let $W:=X\cap Y$.
I suppose that W is non empty and that the intersection of X and Y is non proper, i.e
codim(...
- 3,472
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245
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Segre class of cones and Base change of projective cones
I'm trying to work out a result in Fulton's intersection theory and I think I need the following basic result about base change of projective cones (whose support may not be the entire base scheme).
...
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Topology of K3 as a sum of two abelian fibrations.
Let $E$ be a blow-up of $\mathbb{P}^2$ at 9-points in the bases locus of a pencil of elliptic curves (A $T^2$ fibration over $S^2$).
K3 surfaces is obtained by removing a fiber from two copies of $E$ ...
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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 $s$ ...
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How many zero-constraints can be added to a subspace-restricted matrix before no solution exists?
I'm trying to develop an estimator for the concentration matrix of a Gaussian Graphical Model. I've become stuck in trying to find conditions for the estimator to exist. I have a sufficient ...
CommunityBot
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Intersection form on quotient manifold
Let $E_{1},E_{2}$ be elliptic curves over $\mathbb{C}$. We denote by $\iota_{i}$ the translation by a 2-torsion point on $E_{i}$. Then $G=\mathbb{Z}/2\mathbb{Z}$ acts freely on the the product $E_{1}\...
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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 $X$...
- 34.7k
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where to learn K-group of coherent sheaves modulo numerical equivalence?
I am trying to emerge from my complete ignorance about intersection theory.
I have a bias toward sheaves, so I like the idea of doing intersection theory with the K-group of coherent sheaves. From ...
- 1,889
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2
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is intersection of a curve and a family of curves generically constant as a scheme?
(everything below is defined over an algebraically closed field)
Let $D$ be a (smooth) surface, and let $X \subset T \times D$ be a flat family of curves on $D$, where $T$ is irreducible. Let $E$ be ...
- 149k
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examples of Chow rings of surfaces
Can somone provide me (articles/literature) with examples of Chow rings of surfaces?
(e.g. here: http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf Chapter 9)
What I want is a list of (smooth ...
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Intersections with divisors on moduli of curves
Let $\overline{\mathcal{M}}_{g,n}$ be the moduli stack of stable curves of genus $g$ with $n$ points.
Consider
$0 \neq \gamma \in Pic(\overline{\mathcal{M}}_{g,n})$
the first Chern class of a ...
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1
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Putting two complete varieties in a family over the projective line
Let $X$ and $Y$ be two proper varieties of dimension $n$ over a field $k$. I'm looking for "reasonable" conditions, under which, there exists a proper and dominant morphism $f:V\to \mathbb{P}^1_k$, ...
- 42.9k
3
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2
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Non-vanishing of cup product in cohomology
Let $X$ be a smooth projective variety over $\mathbb{C}$, and let $\alpha \in H^{2k}(X)$ be an algebraic cohomology class. Let us fix an $l$ such that $H^l(X) \neq 0$ and $H^{l+2k}(X) \neq 0$.
The ...
- 35.2k
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What is known about the structure of $\mathbb Z[c_1(\mathcal O_V(1))]$ for a projective $\Bbbk$-variety $V$?
Motivation:
Following Fulton's Intersection Theory, the Chern class of an arbitrary algebraic $\Bbbk$-scheme $X$ can be constructed as follows. First, define the graded by codimension abelian group $...
- 573
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Example of cone of numerically effective curves which is not polyhedral
I think I have seen more than one reference in which the cone of numerically effective curves can be 'not polyhedral', i.e. with an infinite number of extremal rays
I cannot remember where I read ...
- 2,215
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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-...
CommunityBot
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Intersection powers of the exceptional divisor (and the transform of a hyperplane)
In light of my previous question, I am interested in the following scenario: Let $\tilde Y$ be the blow-up of $Y=\mathbb{P}^n$ along a linear subvariety $X\subseteq Y$ of codimension $d$, i.e. $X\cong\...
CommunityBot
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Are there polynomials (almost) all of whose intersection numbers are divisible by some integer?
I've been playing around with some basic intersection theory, and I've wondered the following:
For every two integers $n$ and $m$, and complex numbers $a_1,...,a_n$, are there polynomials $f_1(x),...,...
- 361
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611
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Vanishing associated to a resolution of singularities
Let $\pi: V\to W$ be a resolution of singularities, let $E \subset V$ be the exceptional divisor, and let $F$ be a coherent sheaf such that $R^i\pi_*F=0$ for $i>0$.
Can we conclude that $R^i\...
- 42.9k
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Are there n polynomials for which all intersection multiplicities are at least m?
I don't know whether this is known or not, but I was thinking of the following problem.
Let $n$ and $m$ be natural numbers. Are there $n$ polynomial $f_1,...,f_n\in \mathbb{C}[x]$, such that all of ...
- 1,691
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How many points determine a line?
Consider the affine space $\mathbb C^n$ and then, because of reasons, compactify it to obtain the projective space $\mathbb P^n$. One of the most basic axioms or propositions of geometry is that ...
- 42.9k
2
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2
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Numerically negative exceptional divisor on a surface.
Suppose $S$ is an algebraic surface (possibly projective) over an algebraically closed field $k$. Suppose $D_i$ are irreducible smooth curves (rational, if you want) with negative self-intersection ...
- 118k
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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 ...
CommunityBot
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non degenerate quadratic form on the group of correspondences on an algebraic curve?
Hi,
Given two (smooth, projective) curves $X$ and $Y$ over a field $k$, define a correspondence to be a line
bundle $L$ on $X\times Y$. A trivial correspondence is a correspondence of the form $p_1^*...
- 39.3k
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definition of group operation in elliptic curves
Hi,
Using the isomorphism between an elliptic curve $E$ and its $Pic_1(E)$ group, one can
easily give $E$ the structure of a group variety after choosing a point $O\in E$. The
operation that one gets ...
- 47.4k
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1
answer
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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 ...
- 66.3k