Questions tagged [intersection-theory]
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361
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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 ...
8
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614
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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 ...
5
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713
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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 ...
2
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529
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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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540
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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 ...
3
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499
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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 ...
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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 $\...
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553
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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 $\mathbb P^2$ ...
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0
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436
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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(...
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229
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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 $E$ ...
13
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3k
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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 ...
3
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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 $X$...
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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 ...
3
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274
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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 ...
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473
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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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247
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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 ...
1
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1
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205
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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 $f:V\to \mathbb{P}^1_k$, ...
3
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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 ...
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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 $...
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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 ...
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703
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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-...
2
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1
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426
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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\...
2
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240
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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),...,...
4
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602
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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\...
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309
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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 ...
4
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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 ...
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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 ...
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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 ...
3
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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^*...
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2
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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 ...
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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 ...
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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 ...
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Degree of a real algebraic variety and regular morphisms
I'm reading Fulton's "Intersection theory", which i need for some applied needs.
And i have two questions on general definition of degree used in Fulton.
1)Let us we have a real algebraic variety ...
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1
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264
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Intersection positivity for curves and surfaces
Let $X$ be a smooth complete variety over an algebraically closed field of dimension $\geq3$. Given a divisor $D_1$ on $X$ with $D_1 \cdot C>0$ for every curve $C \subset X$, and a divisor $D_2$ ...
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2
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Dimension of affine variety
Assume that I have $k$ polynomials $f_1(x_1,\ldots x_n),f_2(x_1,\ldots x_n),\ldots f_k(x_1,\ldots x_n)$ in $n>k$ variables. Is it possible to calculate, ,i.e., does there exist a fast algorithm, ...
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if f is birational, is the pushforward map on the numerical groups surjective?
this question was asked on mathunderflow but no one gave a satisfactory answer (perhaps here it will receive more attention?)
Say that one has a morphism of projective algebraic varieties $f: X \to Y$...
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Lower bound for intersection number
The base scheme is an algebraically closed field.
Let $X\to \mathbf{P}^1$ be an arithmetic surface over $\mathbf{P}^1$ and let $P$ be a section of $X\to \mathbf{P}^1$. Let $D$ be an effective (edited)...
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Is there a section disjoint from 0, 1 and infinity on the projective line
Let $K$ be a number field with ring of integers $O_K$. Is there a section of $\mathbf{P}^1_{O_K}$ over $O_K$ whose image is disjoint from $0$, $1$ and $\infty$? If $K=\mathbf{Q}$ this is not possible ...
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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,...
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0
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Is -(E,E) greater or equal to 2 for a minimal resolution
I'm quite confused by the terminology minimal resolution and minimal model.
Let $f:X\longrightarrow Y$ be a minimal resolution of singularities, where $Y$ is a normal surface.
Let $E$ be an ...
4
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820
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intersection number
I vaguely recall the following fact that I'd like to use in my research. It should be easy to see that this holds (if it does) but I can't seem to prove it.
Let $p:X\longrightarrow S$ be a (regular) ...
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6
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3k
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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 ...
15
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1
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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 ...
3
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0
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384
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intersection theory
Assume that $X$ is a smooth 3-fold (over $\mathbb C$). Let $V$ be a smooth divisor on $X$ and let $S_1,S_2$ be prime divisors on $X$. Assume that given a curve $C$ on $X$ not contained in $V$, then
$...
1
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1
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961
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A Theorem in Intersection theory.
Fulton's Book on intersection theory (Pg.223, theorem 12.3) asserts the following result:
For r pure dimensional schemes in P^n, whose co-dimensions add to at most n, the product of their degrees is ...
16
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3
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6k
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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?