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)

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Knowing two Vectors, and the distance to a 3rd, how to get the 3rd

If i know the two Vectors v1 and v2, which discripe points in a 2D space, and i also know that a vector v3 is on the line segment between v1 and v2, how can i get the x and y coodinates of v3 if the ...
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Calculating intersection with diagonal in $\mathbb{P}^2 \times \mathbb{P}^2$

The following is example 6.1.2 from Fulton, Intersection Theory. Denote projective coordinates on $\mathbb{P}^2$ by $[x,y,z]$, and on $Y=\mathbb{P}^2 \times \mathbb{P}^2$ by $([x,y,z],[u,v,w])$. ...
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Geometry of cubic 3-fold

I'm having some questions about the geometry of the cubic 3-fold. Every variety is over $\mathbb C$. Take $Y$ a smooth cubic 3-fold in $\mathbb P^4$ and $E$ a curve of degree 6 and genus 1 contained ...
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Ideal sheaf of intersection of two surfaces in $\mathbb P^3$

Let X be an intersection of 2 surfaces of degree $d_1,d_2$ in $\mathbb P^3$. Is it true that there is a short exact sequence $$ ...
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When does a homogeneous morphism have only finite fibers?

Suppose that we have a map ${\bf f}:=(f_1,f_2,\cdots ,f_n):\mathbb{C}^n\rightarrow \mathbb{C}^n$ given by $$ \mathbb{C}^n\ni {\bf z}:=(z_1,z_1,\cdots,z_n)\rightarrow \big(f_1({\bf z}),f_2({\bf ...
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Line intersection with Sphere

I'm trying to get a formula for calculating intersection points of a line with a sphere (3d space). I've been following this one: Wiki Line-sphere intersection But I'm 99% sure that this one is ...
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27 views

Intersection of two convex lattices polygons

A convex lattice polygon is a polygon whose vertices are points on the integer lattice. Let P and Q two convex lattice polygons with n ,(resp. m) vertices. Let R be the convex lattice polygon ...
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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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19 views

Intersection numbers on blow ups of toric varieties

Suppose we have a smooth, complete toric varietiy $X_{\Sigma}$ of dimension $n$. Let $\sigma_k \in \Sigma(k)$ a smooth $k$-dimensional cone in $\Sigma$ and suppose we make the toric blow up at the ...
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60 views

finding discrète coordinate of Intersection of two convex polygon?

I seek for cartésien coordinate of vertex's of the intersection area between two polygons ? We have two convex polygon's P & Q such that : all vertex of P (resp. Q) are in 2D cartésien plane. I ...
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Are transverse maps in intersection theory local diffeomorphisms?

Suppose that $f: X \rightarrow Y$ and $Z$ is a submanifold of $Y$, all boundaryless. Suppose that $f$ is transverse to $Z$, so that: $$df_x T_xX + T_{f(x)}Z = T_{f(x)}Y$$ for every $x \in f^{-1}(Z)$ ...
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53 views

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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122 views

Tangential space to the rational normal curve

Exercise 15.5 (Harris, Algebraic Geometry: A First Course): Describe the tangential surface to the twisted cubic curve $C \subset \mathbb P^3$. In particular, show that it is a quartic surface. What ...
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28 views

Explicit value of $\overline{\mathcal{O}}_{\mathbb{P}^1(\mathbb{Z})}(1).\overline{\mathcal{O}}_{\mathbb{P}^1(\mathbb{Z})}(1) $

Let us consider the Fubini-Study metric on the part at infinity of the line bundle $\mathcal{O}_{\mathbb{P}^1(\mathbb{Z})}(1)$ to obtain the Hermitian line bundle ...
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27 views

Find the intersection between two lines in a polar notation

I've a polar chart in an application, which displays a curve: I would like to add a functionality when I click on the plot. When I click(at the point M here), I know the orientation and amplitude ...
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26 views

Orthogonal spaces, span-formula

I read in a book this formula: ($v_i$ are vectors of an euclidean vector space, each one $\neq$ 0) $(\cap v_i ^\bot )^\bot = \sum v_i^{\bot \bot}$, The intersection and the sum are build over a ...
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Intersection theory proof of the poincare hopf theorem.

Suppose that $M$ is a connected compact oriented smooth manifold, and $X:M\rightarrow TM$ a vector field with isolated zeros. Then if $Z$ is the zero set of $X$ (a $0$-dimensional oriented manifold), ...
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Point inside the area of two overlapped triangles

The question is as simple as that, but I have been trying to figure out an answer (and searching for it) with 0 results. I mean, given two triangles (in 2D) I want to find just a single point which ...
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17 views

Proving that the intersection over a general index is the null set

Prove that the generalized intersection of the interval $[n,n+2]$ is the empty set. I can prove this by contradiction by assuming that there there is an $x$ in the intersection such that $n+2<x.$ ...
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114 views

Intersection multiplicity , Hilbert samuel polynomials and normal cones

I'm in the process of reading some parts of Fulton's "Introduction to intersection theory" and there's a short part there which I don't quite understand where I think I'm missing something obvious. ...
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34 views

Self-intersection of the canonical sheaf of product of varieties

Let $X$ and $Y$ to be two smooth projective algebraic varieties of dimension $n$ and $m$, respectively. Then, $K_{X\times Y}=\operatorname{pr}_1^*K_X+\operatorname{pr}_2^*K_Y$. Can we can compute ...
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39 views

Size of the product of two subgroups

Let $(G, \ast)$ be a group and let $H\le G$ and $K\le G$ be subgroups of $G$. Prove that $|HK|$=$\frac{|H|\cdot|K|}{|H\cap K|}$. Intuitively this is quite obviously true, as otherwise the products of ...
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Restriction and intersection

Say $X$ is a smooth threefold, and $A$, $B$, and $C$ are three smooth divisors on $X$. Is it true that the three-way intersection $(A \cdot B \cdot C)_X$ is equal to the intersection $(A\vert_C \cdot ...
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Geometric proof: Legs intersect on CM (median of triangle)

$M$ is the midpoint of $AB$ in the triangle $\triangle ABC$. The angle $\angle ACM$ is copied and drawn on the leg $AB$ in $A$. The angle $\angle MCB$ is copied and drawn on the leg $BA$ in $B$. The ...
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48 views

Find the intersection point of a great circle arc and latitude line

In spherical geometry, I need to know at what longitude λ a great circle arc φ1,λ1-φ2,λ2 has intersected a line of latitude φ. I have found the equivalent equation for solving latitude φ for an ...
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26 views

If $f\colon X\to Y$ is nullhomotopic, then $I_2(f,Z)=0$ for closed $Z\subset Y$ of complementary dimension?

I think I have a near solution, but one doubt. Problem 2.4.4 on page 83 of Guillemin and Pollack asks If $f\colon X\to Y$ is homotopic to a constant map, show that $I_2(f,Z)=0$ for all ...
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50 views

Given the equation of a cylinder $ x^2+z^2=1,$ find the parametric and locus form of the curve of intersection with plane

Given the equation of a cylinder $x^2+z^2=1,$ describe the curve of intersection between the cylinder & the planes z=x & y=x in the parametric form & the form F(x,y,z)=0. I am so lost ...
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79 views

Calculate the intersection numbers by a plane section

This question is from the chapter A of Reid's note: Chapters on algebraic surfaces Let X = X$_d$ $\subset$ P$^3$ be a nonsingular surface of degree d and suppose that X has a plane section P ...
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Self-intersection of a cycle

If $X$ is a smooth projective variety of dimension $2n$, and $V \subset X$ is a smooth subvariety of dimension $\geq n$, then $V \cdot V$ makes sense as a class as an element of the Chow ring ...
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48 views

Trouble doing polynomial interpolation

I need to do a polynomial interpolation of a set $N$ of experimental points; the functional form I have to use to interpolate is this: $$ f(x) = a + bx^2 + cx^4,$$ as you can see the coefficient that ...
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A question about intersection number on surfaces

This question is from the Qing Liu's book: Algebraic Geometry and Arithmetic Curves, Exercise 9.1.6. Let $X\to S$ be an arithmetic surface and $X_s$ a closed fiber. Let $C_1,...,C_m$ denote the ...
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Meaning of “local equation” of a divisor.

Let $X$ be a smooth surface and moreover let $C,D$ be two effective divisors of $X$. Hartshorne says (page 357) that $C$ and $D$ meet transversally if the local equations $f,g$ of $C,D$ at $P$ ...
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Calculating canonical divisor in product of projective spaces.

Let $X$ be an intersection of two divisors of bidegree $(a,b)$ and $(c,d)$ in $\mathbb{P^2}\times \mathbb{P^2}$. Then how can I find the canonical divisor $K_X$? I'm asking because I have no ...
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67 views

Basic question: degree of normal bundle is not self-intersection number

For $C$ a (possibly singular) curve on a nonsingular projective surface $X$, let's define $C^2=deg_C(\mathcal{O}_X(C))$. Why is it not the same as $deg_C(N_{X|C})$ when $C$ is singular? Why do ...
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Questions about intersection of linear varieties in a projective space

Let $X, Y, Z$ be linear varieties of dimension $r, s, t$ respectively in $\mathbb{P}^n$. If $r+s\ge n$, then $X\cap Y\neq \varnothing$. Furthermore, if $X\cap Y\neq \varnothing$, then $X\cap Y$ is a ...
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Blowing-up of a linear sub-space is Fano?

Consider $P=\mathbb{P}^3\times \mathbb{P}^1$ with coordinates $([x_0:x_1:x_2:x_3],[u:v])$ and let $\varepsilon:X\to P$ be the blow-up of $\mathbb{P}^2\cong A=(x_3=v=0)\subseteq P$, of pure codimension ...
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Possible Intersection of Intervals

Suppose there are two intervals, where one of them is fixed. Is there a way to calculate all possible intersections of the intervals as shown in the figure? ? Notice that because $a,b$ and $c$ are ...
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antiholomorphic involution action on resolved orbifold of tori

Consider the space $\tilde X= T^2 \times T^2 \times T^2 / G$ where $T^2$ is a torus and $G=Z_2 \times Z_2$ acts as $\theta_1:(z_1,z_2) \mapsto (-z_1,-z_2)$, $\theta_2:(z_2,z_3) \mapsto (-z_2,-z_3)$ ...
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rational quartic in $\mathbb{P}^3$

According to Hartshorne (exercise IV.6.1), a rational curve of degree 4 in $\mathbb{P}^3$ is contained in a unique smooth quadric surface. If this is the case, then it must define a divisor on it. My ...
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how to calculate these intersections without having to count all combinations

We have the following sets: $X= {(a,b,c,d) ∈S: b< c < d},$ $Y= {(a,b,c,d) ∈S: a< c < d},$ $Z= {(a,b,c,d) ∈S: a< b < d},$ $F= {(a,b,c,d) ∈S: a< b < c},$ Where each of ...
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Generalization of Bezout Theorem to many-hypersurface case in Hartshorne's setting

I try to follow the ideas in Hartshorne's Chapter 1, Section 7. Suppose we have algebraic sets $Y_1,...,Y_l$, I try to define their intersection number $I(Y_1,...,Y_l)$ to be the leading term of the ...
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Dimension of moduli of lines on quadric

What is the dimension of the moduli space of lines on a general quadric hypersurface in $\mathbb{P}^n$? Maybe the question is quite trivial, but different intuitive approaches (à la Italian algebraic ...
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Intersection product of submanifolds of complex manifolds - Selfintersection

I do not understand something about the intersection product. I'm kind of new to this topic, so please consider that. I write down everything we discussed in a lecture. We defined the intersection ...
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Definition of intersection multiplicity in Hartshorne VS Fulton for plane curves

In exercise I.5.2 in Harshorne there is the following definition of intersection multiplicity for two curves in $\mathbb{A}^2$: \begin{equation} \mathrm{length}_{\mathcal{O}_P}\mathcal{O}_P / (f,g) ...
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Containment of two varieties with a lot of intersection

Given a projective variety $X\subset \mathbb P^n$ and a curve $C\subset \mathbb P^n$, when can I conclude that $C\subset X$, from the fact that $C$ and $X$ have 'many' points in common. I.e., is there ...
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Number of intersections of two closed loops on a genus zero surface

I have stumbled onto the following fact and I am quite helpless in seeing why this is true (although I can agree intuitively). Let $M$ be a surface of genus zero (open or closed, with or without ...
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Probability of infinite intersections

While I was studying Probability and random processes I came across the following question. Say I have $A_1,A_2, \ldots, A_n$ events such that $A_i$ is in $E$ but not equal to $E$. What is: ...
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Why is this theorem important in fulton's book?

In Fulton algebraic curves book, we have the following proprieties which help us to find the intersection number of a pair of function at a given point. afterwards he states this theorem: ...
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348 views

What are the prerequisites for Fulton's “Intersection Theory”?

Is it necessary to read SGA VI to understand "Intersection Theory" by William Fulton?
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Calculating the intersection product in CH(X)

Let CH$(X)$ be the Chow-Ring of a projective,smooth variety with cycles modulo rational equivalence. Lets assume Kunneth-Formula holds. There is an intersection product CH$^a(X) \otimes $ CH$^b(X) ...