Questions tagged [intersection-theory]
In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. (Ref: http://en.m.wikipedia.org/wiki/Intersection_theory). Do not use this tag for elementary problems in linear algebra or geometry. (e.g. determining whether two lines intersect)
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Hartshorne Example V.1.4.1 - Why $C^2=\deg_C(\mathcal{L}(C)\otimes\mathcal{O}_C)$?
In the book Algebraic Geometry by Hartshorne, Example V.1.4.1 says $C^2=\deg_C(\mathcal{L}(C)\otimes\mathcal{O}_C)$ holds due to Lemma V.1.3. Here, $C$ is a nonsingular curve on a nonsingular ...
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Degree of a subvariety of the Grassmannian
This might be a stupid question due to my lack of knowledge in algebraic geometry.
So I have a subvariety $V\subset G_k(\mathbb{C}^n)$ of codimension $k$. The only thing I know about $V$ is its ...
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Intersection multiplicities
Let $X \subset \Bbb P^n$ be a curve defined by $f_1 = \dots = f_r = 0$ and $H$ an hyperplane. How to compute the multiplicity $m$ at a point $p \in X \cap H$ ?
Specific example :
In my case, $X \...
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Calculating intersection multiplicities explicitly with Serre's formula
Let's say I want to explicitly calculate the intersection multiplicity of two subvarieties of $\Bbb A^n_k$ using Serre's Tor-formula (involving as little homological algebra as possible).
A typical ...
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Commutativity of Intersection product
I am reading the proof of the commutativity of intersection product from Ravi Vakil's notes: http://virtualmath1.stanford.edu/~vakil/245/245class8.pdf
This theorem states that if $D$ and $D'$ are ...
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Minimizing cross section area of the intersection between a plane and mesh
I have a 3D triangular mesh and a cutting plane. I want to minimize the area of the cut intersection by rotating the normal of the plane.
The image shows the cross section of the cut and the mesh.
...
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Coordinate ring of irreducible component
In the proof of affine dimension theorem (you can see the proof I am following here), Hartshorne assume that $A(Y)/\mathfrak{p}$ is the coordinate ring of the irreducible component $W$.
Why is this ...
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Intersection of two varieties need not be a variety
I am introducing my self to algebraic geometry, and by now I am reading Algebraic Geometry by Robin Hartshorne. In the exercise $2.16$, I have to prove that the intersection of two varieties need not ...
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Intersection number of complex curves in a complex surface
Suppose $C_1,C_2$ are embedded complex curves in a complex surface $S$, and $C_1,C_2$ have no common component. Assuming $C_1$ and $C_2$ intersect transversally, the intersection number $C_1\cdot C_2$ ...
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Simplifying intersection of 3 sets when 2 are independent
I am working on a problem involving the probability of the intersection of 3 sets:
$P(A \cap B \cap C)$, and am wondering if this can be simplified at all, since we are also given that the sets $A$ ...
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Fulton, example 3.2.16: Application of the splitting principle.
In his "Intersection theory" book Fulton proves in lemma 3.2 the following fact:
if a $E$ is a filtered vector bundle of rank $r$ over $X$ with quotients line bundles $L_i$, $s$ is a section of $E$ ...
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Signature of self-intersection submanifold
Let $S\subset M$ be a (complex) codimension $1$ submanifold. $S\cdot S$ denotes the self-intersection manifold of $S$ in $M$. Then it seems that $\text{Sign}(S\cdot S)$ (the signature of $S\cdot S$) ...
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Intersection of two submanifolds with dimension condition
Consider a manifold of dimension $n$, we have two submanifolds $A$ and $B$, with dimension $n_1$ and $n_2$ that satisfies $n_1+n_2=n$. Can we conclude they only intersect at finite points (or isolated ...
CommunityBot
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Prove the equation for the point of intersection of two vector lines
This is my first question here and I'm not sure if it's the right place but I'm kinda desperate. I was learning about vector lines and parametric equations. My teacher gave me the assignment below and ...
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Fulton Ex. 3.2.22: Counting conics intersecting 8 lines in 3-space
I'm trying to understand Fulton's approach to the question how many plane conics in $\mathbb P^3$ meet 8 general lines. Suppose $V$ is a 4-dimensional vector space, and $\mathbb P^3 = \mathbb P(V)$. ...
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Self intersection of curve on $P^1 \times P^1$
Let $l = \mathbb{P}^1 \times \{0\}$, I want to prove $l \cdot l=0$, where $l \cdot l$ is the self intersection of $l$. I am working out of Ravi Vakil chapter 20, this is exercise 20.2.C. His hint is ...
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Theorem I.7.5 (Hilbert-Serre) Hartshorne uniqueness of Hilbert polynomial.
I don't understand why the Hilbert polynomial is unique.
In the theorem I.7.5 we find a polynomial $P_M(z)$ such that $\varphi_M(l)=P_M(l)$ for all $l\gg0$ ($\varphi_M(l)=dim_K M_l$ as discribed in ...
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When existence of transfer map of algebraic $K$-theory implies rational injection.
Given a map of smooth projective varieties $f:X\rightarrow Y$ over fields, there is a projection formula in algebraic $K$-theory given by $f_*(\alpha.f^*(\beta))=\beta.f_*(\alpha)$. I was wondering ...
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Milne's Intersection theory simple example
In Milne's Divisors and intersection theory, Example 12.3a) computes the intersection number of the curves $Z_1: Y=X^2$ and $Z_2: Y^2=X^3$ at the intersection point $P=(0,0)$ in the affine plane over ...
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Chern classes of tangent bundle over the Grassmannian G(2,4)
What are the Chern classes of the tangent bundle $\tau_G$ of the Grassmannian $G=G(2,4)$ of lines in $\mathbb{P}^3$? This is Exercise 5.37 on page 191 of 3264 & All That by Eisenbud and Harris.
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For complex subspace $S$, there are vectors $u,w ∈ S$ such that $u\neq zw,\ \forall z\in \mathbb{C}$
The official question reads:
Suppose U and W are each 5-dimensional subspaces of $\mathbb{C}^8$.
Prove that there are two vectors $u,w ∈ U∩W$ $\hspace{1mm}$ s.t. $\hspace{1mm}$ $u≠zw$ $\hspace{...
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The complement of a neighborhood of a divisor.
Given a projective variety $X$ and a hyperplane section $H$ (intersection of some hyperplane in the ambient projective space with $X$). Is it true that complement of any neighborhood of $H$ is a ...
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How can I use Grothendieck--Riemann--Roch theorem?
For my research, I'm trying to understand how the Grothendieck--Riemann--Roch theorem is used in the paper The Birational Geometry of the Hilbert Scheme of Points on Surfaces by Aaron Bertram & ...
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Intersection form of $\mathbb P^1 \times \mathbb P^1$
This comes from Vakil 20.2 C, self-study.
We have to show the intersection form of $X = \mathbb P^1 \times \mathbb P^1$ is given by $l \cdot l = m \cdot m = 0$ and $l \cdot m = 1$, where $l$ is the ...
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Formula of intersection of 2 points with the x axis
I'm trying to figure out how to get the point x = 3 :
What's given here are the points S and G .
(Assuming the 2 angles are equal)
Apparently, we can assume that the ball does not bounce off the x-...
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Kleiman's theorem on intersection theory
I study the book 3264 & All That
Intersection Theory in Algebraic Geometry by Eisenbud & Harris and I'm a little bit confused on the proof of Kleiman’s theorem on pages 21:
Theorem 1.7 (...
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Why one cannot equate a sphere and a plane but a sphere with a sphere? [closed]
This is a general question about the intersection of a sphere with a plane or sphere which is confusing me.
To find the intersection between two spheres K1 and K2, you can equate them, solve the ...
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How to count constraints in algebraic geometry?
I am interested in where to look to find techniques to approach the following problem:
Let $M$ be a smooth projective manifold, and let $S \subset M$ be some submanifold.
Consider on $M$ a complete ...
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Scheme theoretic intersection vs intersection of cycles
Let $X$ be a scheme for which we have an appropriate moving lemma to define an intersection product of cycles. For example a smooth projective variety. Then if $V$ and $W$ are properly intersecting ...
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Local geometry near a transverse intersection of two Riemann surfaces
Suppose $C_1$ and $C_2$ are two Riemann surfaces (embedded as submanifolds in some ambient space, say, $M$) which intersect transversally at a point $p_1 \in C_1$. How does one see that near $C_1$, ...
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What is the difference between intersection theory and enumerative geometry?
I am about to finish the book : $3264$ & All that. Intersection theory in Algebraic geometry, by David Eisenbud and Joe Harris, available freely on the net.
I would like to know, what is the ...
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Chow ring of projetive space
The Chow ring of $\mathbb{P}^n$ is $\mathbb{Z}[H]/H^{n+1}$, where $H$ is the linear equivalence lass determined by the zero-set of a linear functional. I have a few (very basic) points of confusion:
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Self-intersection of a cubic Bezier, interpretation of the solution
I am trying to understand the form of the determinants that I get when I try to calculate the self-intersection. Since I might have made some mistakes, I'm including the whole derivation:
Let $\vec{P}(...
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Proving that intersection number of divisor is 0
My question arises from proposition 8.3 in the algebraic geometry text by Shigeru Iitaka:
Let $f:W\rightarrow S$ be a surjective morphism between nonsingular projective surfaces. Let $E$ be a divisor ...
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Polynomial equivalence
Given a variety $X$, rational equivalence gives an equivalence relation on $Z_i(X)$ leading to the Chow groups and used in the Grothendieck-Riemann-Roch theorem.
I was wondering about replacing $\...
CommunityBot
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Lines on a singular cubic surface
How many lines the cubic surfaces $xyz=w^3 \in \mathbb P^3$ has? I found only three: $x=w=0$, $y=w=0$ and $z=w=0$. How to prove that there are no other lines? Also, this surface is singular, is it ...
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Negative intersection numbers
Let $X$ be a complex variety of dimension $m$ and $D_1,D_2,\dots, D_m \subset X$ irreducible divisors with $D_i \neq D_j$. Can we have $D_1 \cdot D_2 \dots \cdot D_m < 0 $ ?
The only concrete ...
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Néron-Severi groups
Let $X$ be a smooth projective variety over the complex numbers.
Let $$c_1 : \text{Pic}(X)\to H^2(X,\mathbf{Z}(1))$$
Its kernel is the subgroup of homologically trivial divisor classes on $X$.
How ...
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definition of cycle theoretic fibre
I am studying the definition of Chow variety on Kollar's Rational Curves on Algebraic Varieties, and I am having some trouble in understanding Definition 3.9.
Here we have a proper morphism of ...
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Versions of Bertini's theorem?
I read the following statement:
Let $X$ be a Del Pezzo surface (i.e. smooth surface with $-K_X$ ample) over an algebraically closed field $k$. Then the general member of $|-K_X|$ is irreducible and ...
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Segre classes of subvarieties
I am following 13.2 of "3264 and All That".
I read the definition of "Segre classes of subvarieties" but I couldn't calculate with describing specific examples.
The definition is ...
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Conics meeting 8 general lines
I am trying to show that the number of plane conics in $\mathbb{P}^3$ meeting $8$ general lines is $92$, using what I know about intersection theory.
I started considering the tautological bundle $S$ ...
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Hartshorne Chapter V Proposition 1.5
This proposition states as follows,
What confuses me is the last sentence. Notice that the Lemma 1.3 states as follows.
We also have the fact that,
The pairing Div $X \times \operatorname{Div} X \...
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Self intersection of zero section of decomposable ruled surface
Let $L_1, L_2$ be two line bundles over a curve $C$. Then we can consider the ruled surface $S=\Bbb P (L_1\oplus L_2)$ over $C$. Let $s_0$ and $s_\infty$ denote the zero section and the infinity ...
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When do these functions equal each-other?
I'm looking for the $x$ such that
$$ -\frac{1}{q^x}+1=a^x -1$$
For any $q$ and $a$
A.K.A the $x$ at the intersection of the two functions. Am I asking the wrong question here?
One obvious solution is $...
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How to find the directrix in order to get intersection between two parabolas and a segment?
So I am trying to implement the Fortune's algorithm in order to have a Voronoi pattern (wonderful post about it here : http://www.ams.org/publicoutreach/feature-column/fcarc-voronoi)
For this, I need ...
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Calculate Seshadri constant
I understand the definition of Seshadri constant
$$\epsilon_{D}(E)=\sup\{k \in \mathbb{R}_{\geq 0}|E-kD\text{ is nef}\}.$$
I am not sure how to apply this definition to practice calculation, or maybe ...
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How should I calculate the intersection of subspaces? [closed]
How should I calculate the intersection of subspaces? In Z5.
span{(1, 4, 4), (2, 3, 4)} ∩ span{(1, 1, 4), (2, 4, 0)}
And how many vectors it includes?
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27 lines on a cubic surface - which proof is better?
There is a famous fact that any smooth cubic surface has exactly 27 lines.
I am a undergraduate students who want to view some detailed proofs of this result. I know there are several different ...
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Existence of a transversal map prevents density?
Let $S$ be a $C^{\infty}$-submanifold of $N$ and suppose that $N-S$ is dense in $N$, where $M,N$ are $m$ and $n$ dimensional $C^{\infty}$-manifolds, respectively.
In this post the answering poster ...
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