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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What kind of morphisms should I expect to be proper?

I'm trying to learn more about proper pushforward, but I'm stuck at coming up with interesting examples of proper morphisms of schemes. The only examples I can think of are inclusions of projective ...
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84 views

circle tangent to three circles

To-day I want to look at CCC - one circle tangent to three circles whose radii and positions of their centers are known. How does one solve this.. old fashioned ways like ruler and compass, or ...
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16 views

Peanut Shape (Intersection of two circles)

I have two circles with centres (xo,yo) and (x1,y1) respectively. The two circles have radius R1 and R2. The circles are already plotted with help of a series of cumulative small angles (theta) and ...
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Tropical top self-intersection numbers of boundary divisors in toroidal embeddings

Let $X_0 \subset X$ a toroidal embedding without self intersections and denote by $\overline{\Sigma}$ its corresponding (weakly embedded) extended conical simplicial complex. Let $D$ be a divisor on $...
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1answer
25 views

Intersection of a nested interval of $A_n=\left[3-{\frac{1}{\sqrt{n}},3+\frac{1}{3^n}}\right]$

$A_n=\left[3-{\frac{1}{\sqrt{n}},3+\frac{1}{3^n}}\right]$ What is $\bigcap_{n=1}^{\infty}A_n$ Since every set becomes a subset of the next set, is it correct to say that the intersection of all ...
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1answer
58 views

How to prove that the Lefschetz number is invariante under homotopy?

How to prove that the Lefschetz number is invariante under homotopy? We define the Lefschetz number as the number of $f : M \to M$ as the number of intersection of the map $g(x) = (x,f(x))$ with the ...
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1answer
30 views

Calculating a composition of finite correspondences

This is part of exercise 1.11 in Mazza, Voevodsky, Weibel's Lecture Notes on Motivic Cohomology. Here's some notational stuff: we're working in the category $Cor_k$ whose objects are the smooth ...
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25 views

Calculating the intersection multiplicities of algebraic curves using Gröbner Basis

In my class, the lecturer told me that in order to calculate the intersection multiplicities for multiple space curves, sometimes I may have to calculate the Gröbner Basis. I just can't see how ...
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28 views

Self-intersection of an axis

Let $X$ be a projective, smooth curve over an algebraically closed field, $Y = X\times X$ and $\mathcal{l}= pt\times X$ where $pt$ is a closed point of $X$. Can one say that the intersection number $\...
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1answer
18 views

Let P(x,y,z) be an irreducible homogeneous second degree polynomial. Show that the intersection multiplicity of V(P) with any line l is at most 2.

I came across this question in Algebraic Geometry: A Problem Solving Approach: Let P(x,y,z) be an irreducible homogeneous second degree polynomial. Show that the intersection multiplicity of V(P) with ...
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40 views

How many different ways can a circle intersect a triangle N ways?

Consider a circle intersecting a triangle. The circle and triangle can have between 0-6 total intersection points. Is there a mathematical formula for the number of possible ways they can intersect ...
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51 views

Intersection of hypersurfaces in the projective space

Fix an integer $n>0$. Is it true that for any $k>0$ and a closed point $x \in \mathbb{P}^n$, there exist hypersurface sections of degree $k$ (i.e., global sections of $\mathcal{O}_{\mathbb{P}^n}(...
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38 views

Optimising 2D acceleration to intercept a moving target

Related to this question: Accelerating one moving body to intercept another body in 2d I have a spaceship that needs to intercept a moving target in 2D. They have a relative velocity and I need to ...
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30 views

Intersection of subgroup cosets

Let H and K both be subgroups of G. Show that for all a ∈ G we have Ha ∩ Ka = (H ∩ K)a? Considering H and K are subgroups I thought there could be two general cases; H ∩ K = {1} or H ∩ K not equal to ...
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1answer
30 views

Accelerating one moving body to intercept another body in 2d

I have a spaceship moving in a 2D plane and wish to intercept another body which is also moving in the same plane. To do this I need to find a series of accelerations over a series of times that will ...
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2answers
52 views

Trouble with definition of signature of a compact manifold

The signature of a manifold, as I understand it, is defined as follows: Given a connected, compact, and oriented manifold $M$ of dimension $4n$, we may define a quadratic form on the cohomology group ...
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1answer
45 views

Smoothness of the intersection of smooth varieties

Let $(L_i)_{i \in I}$ be a finite family of (distinct) smooth irreducible embedded algebraic varieties of codimension one in some complex projective space of dimension $n$. Consider a finite ...
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2answers
70 views

Modelling the difference between intersections of two lines on the circumference of a circle

I have a line which is divided into small segments. In the following diagram we have the first segment defined by two points $P_1$ and $P_2$. However, imagine the line having other segments evenly ...
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45 views

Equivalence of two definitions of rational equivalence

Let $X$ be a variety. In order to define the Chow ring of $X$, there seem to be two different ways of defining rational equivalence of $i$-cycles $Z \sim Z'$. Definition 1: $Z \sim Z'$ iff there is ...
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3answers
67 views

Intersection point of two functions - one linear, the other with logarithmic and sqrt terms

I would like first to appreciate everything that is being done on this forum and to greet you all! I have namely two functions and the goal is to find the intersection point of them. $y_1 = a + \...
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2answers
39 views

How to find the intersection of level curves?

For two functions $f,g:\mathbb{R}^2\rightarrow\mathbb{R}$, how would I show that the level curves of these two different functions intersect at right angles? I can give the specific functions, but I ...
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71 views

The product of (first) chern classes

I was reading Fulton's Intersection Theory and came across this useful formula for the chern class of the tensor product: $$ c_r(E\otimes L) = \sum_{i=0}^r c_1(L)^ic_{r-i}(E). $$ However, I couldn't ...
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34 views

Intersection multiplicity for two surfaces definded by $f=0,g=0$

I want to understand how I can find the intersection multiplicity $I_p$ at a point $p$ for two curves $f,g$. I have the example where $$ f(x,y) = y^2-x^3, \,\,\,\, g(x,y)=y^2-x^2(x+1) $$ Then I am ...
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2answers
48 views

Finding points of intersection of two curves

I have a problem where I needed to find the intersection of two curves, with $r=3+\sin(\theta)$ and $r=7\sin(\theta)$ for $[0<\theta<2\pi]$ I have that it's supposed to be $(7/6,\pi/6)$ and $(7/...
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1answer
72 views

Definition of the degree of a sheaf

Let $C$ and $D$ be smooth closed curves on a projective smooth surface $X$ of finite type over an algebraically closed field. I'm looking for a definition of the term $\deg_C(\mathcal{O}(D)_{|C})$. ...
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36 views

Intersection/overlap of two sectors

how are you all? Can anybody help me with a rather straight forward approach of detecting or, even better, computing the area of overlap of two (or more - if possible ) sectors whose radii and arc ...
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39 views

Common Intersection Point of ellipsoids

Suppose we have two ellipses in $2$-dimensions centered at the origin. It is easy to visualize that (unless one is contained in the other) they will have $4$ points of intersection. Is it known ...
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2answers
33 views

Find Length of line which has rotating object.

I have 3 Images. A, B and C. if I place it on graph, its look something like this. Now main image is A and I place B and C on that image's (A) center point. For easy understanding, let's consider ...
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1answer
76 views

Finding a 3rd point in a 3D triangle with known plane, two points and lengths of each side

I have a very similar problem to the below question. right triangle in 3D space, vectors, line intersection? Rather than having the unit vector $A$ I have the lengths $i_2$ to $i_3$ and $i_1$ to $...
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2answers
58 views

General form of intersection line between 2D plane and 3D hyperboloid when offset from symmetry axis?

General Background Suppose I have one side of a two-sheet hyperboloid as a general three dimensional shape, where the symmetry axis is along, say, my x-axis ($\hat{\mathbf{x}}$) in my chosen ...
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1answer
68 views

How do you find the angle of intersection between two given polar curves?

How does one find the angle of intersection between two given polar curves? For example, between $a^2=r^2\sin(2\theta)$ & $b^2=r^2\cos(2\theta)$
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2answers
85 views

Bertini's Theorem and singular divisors on a surface

I'm trying to understand the following: Let $X$ be a projective, smooth surface over an algebraically closed field and $D$ a divisor on $X$. How can I see that $D$ is linear equivalent to the ...
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1answer
55 views

right triangle in 3D space, vectors, line intersection?

I'm having way to much issue with this, I would think it's not super hard, but I'm getting no where with it, and I need to slove it to progress with the thing I'm making. Anyways here is my problem, ...
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2answers
74 views

collision point of circle and line

I'm trying to figure out the collision point of circle and a line, utimatly it should work in 3D but for now just in 2D to simplify the problem as much as possible. I've created 2 examples here on ...
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0answers
31 views

Explicit form of certain polynomials and intersection of curves

Let $X$ be a smooth degree $d$ hypersurface in $\mathbb{P}^3$ and $C, D$ two effective divisors on $X$ intersecting at finitely many points. Is it true that if $C$ and $D$ intersect in ''low'' number ...
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0answers
32 views

Intersection product of pull back under projection

Let $X$ be a surface and $Y$ be a curve over $\mathbb{C}$. Let $L$ and $L'$ be ample line bundles on $X$ and $Y$ respectively. Consider the product $X\times Y$. Let $p$ and $q$ be the projection from $...
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18 views

intersection multiplicity in the special fibre

Let $A$ be a (complete) DVR with maximal ideal $m$ and residue field $A/m=k$ of positive characteristic, and let $K$ be its fraction field. Let $X, Y$ be curves (or more generally cycles of ...
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85 views

Intersection theory question from Vakil's notes on Algebraic Geometry

I cannot seem to solve what I think should be a straightforward intersection theory question in Ravi Vakil's algebraic geometry notes. Let $X$ be a scheme and $Y$ a closed subscheme of dimension less ...
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1answer
27 views

Points of intersection of vector with cone.

I have a Vector $\vec A$ defined as : $(A_o+t*A_d)$ I also have a Cone with vertex (cone tip) V and axis direction $\vec D$, base radius R and height H. The cone angle can be computed via $θ=2{tan^-}^...
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1answer
19 views

avg # of maximum intersections for m 1-dimensional segments with length L in a range [0,t]?

I have a discrete range, let's say $[0,T]$. I also have $m$ segments of length $L\leq T$. A segment is $seg=(a, a+L)$, with $0 \leq a \leq (t-L)$. The total number of possible configurations of ...
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63 views

Geometric intuition under a negative intersection

I am studying Intersection Theory in algebraic geometry and in the following when I say intersection I mean the intersection of a dimension k cycle with a codimension k cycle. I was reading this ...
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41 views

Looking for an elegant proof of why there is only ever a single intersection point of the nullclines for a 2D ODE system

I have the following ODE system $$ \dfrac{d \ell}{dt}=\sigma_{\ell} - \mu_{\ell} \dfrac{M \ell}{1+\ell}-d_{\ell}\ell \\ \dfrac{dM}{dt}=\dfrac{\ell}{\beta+\ell}+\sigma_M \dfrac{M \ell}{1+\ell}-M $$ ...
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1answer
27 views

How to do this simple set operation?

Suppose A and B are events with P(A) 0.4 , P(B) 0.6 and P(A and B) 0.25 . Calculate the probability P(A complement union B). A 0.25 B 0.65 C 0.75 D 0.85 What I tried?- P(A ...
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36 views

Modify Equation of sphere intersection

I got to know how to modify formula for radius of circle formed after intersection of two spheres when centres of both spheres where at origin or at x axis to centres at any arbitary positions $(x_1,...
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1answer
52 views

Is a point on a plane part of a face on the plane?

There is a line and a face in $\Bbb R^3$, does the line inersect the face? I have a plane (infinite area) in $\Bbb R^3$ defined by a point $(x_0,y_0,z_0)$ and its normal $n$. The plane contains a ...
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33 views

Evaluate this statement / find the set

$X = (\{1\}\cup\{2,3\})\cap(0,4)$ I think the solution is an empty set but I would like confirmation or refutation from you. I think it is an empty set because we take the union of 2 sets containing ...
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1answer
21 views

Computing closest edge of a point within a polygon that is constructed with lat/longs?

So here's what I'm trying to do: Given a collection of lat/long coordinates that form a polygon like below, I want to be able to select a point inside the polygon and determine which side of the ...
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17 views

How to efficiently calculate points of intersection of a straight line and a contour?

Ex. I have an image, let's say 100x100 pixels with some shape on it. And a set of straight lines that are passing through origin located at some point of image, for example origin located at point (50,...
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2answers
53 views

Intercept moving target

Here's a problem that has bugging me for a while: Say I have a friend that is passing my house. My friend is moving at a constant speed in a perfectly straight line. If she already hasn't done so, ...
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38 views

Why is the dimension of this Grassmann manifold $G_{d, n}$ equal to $(d+1)(n-d)$ formed by the Plucker coordinates of a $d$-plane?

A $d$-plane $L \subset \mathbb{P}^{n}$ is defined as the set of points $P=(p(0), p(1), \ldots, p(n)) \in \mathbb{P}^{n}$ that satisfy equations $\sum_{j=0}^{n} b_{\alpha j}p(j) = 0$, where $\alpha = 1,...