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)

learn more… | top users | synonyms

2
votes
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 ...
0
votes
1answer
52 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 ...
0
votes
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 ...
1
vote
0answers
24 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 ...
0
votes
0answers
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 $\...
0
votes
1answer
17 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 ...
2
votes
0answers
36 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 ...
2
votes
0answers
48 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}(...
0
votes
0answers
34 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 ...
0
votes
0answers
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 ...
0
votes
1answer
28 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 ...
0
votes
2answers
48 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 ...
2
votes
1answer
44 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 ...
0
votes
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 ...
0
votes
0answers
41 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 ...
2
votes
3answers
65 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 + \...
4
votes
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 ...
1
vote
0answers
69 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 ...
0
votes
0answers
33 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 ...
0
votes
2answers
45 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/...
2
votes
1answer
70 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 $\deg_C(\mathcal{O}(D)_{|C})$. ...
0
votes
0answers
35 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 ...
1
vote
0answers
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 ...
2
votes
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 ...
0
votes
1answer
67 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 $...
1
vote
2answers
54 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 ...
0
votes
1answer
63 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 $a^2=r^2\sin(2\theta)$ & $b^2=r^2\cos(2\theta)$
1
vote
2answers
82 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 ...
1
vote
1answer
52 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, ...
0
votes
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 ...
2
votes
0answers
30 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 ...
1
vote
0answers
30 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 $...
1
vote
0answers
17 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 ...
3
votes
0answers
78 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 ...
2
votes
1answer
26 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^-}^...
0
votes
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 ...
2
votes
0answers
59 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 ...
1
vote
0answers
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 $$ \dfrac{d \ell}{dt}=\sigma_{\ell} - \mu_{\ell} \dfrac{M \ell}{1+\ell}-d_{\ell}\ell \\ \dfrac{dM}{dt}=\dfrac{\ell}{\beta+\ell}+\sigma_M \dfrac{M \ell}{1+\ell}-M $$ ...
1
vote
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 ...
0
votes
0answers
35 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,...
0
votes
1answer
51 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 ...
0
votes
0answers
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 ...
0
votes
1answer
17 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 ...
0
votes
0answers
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,...
1
vote
2answers
52 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, ...
2
votes
0answers
35 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,...
0
votes
0answers
32 views

Self Intersection Formula

Let $\pi$: $X\to \Delta$ be a good degeneration of surfaces with degenerate fiber $X_0 = V_1 + ... + V_n$. Let $C$ be a component of the double curve $V_1 \cap V_2$ I am trying to understand why the ...
4
votes
0answers
109 views

Question about Gysin map (pushforward in cohomology)

The setting is the following: $X \subset \mathbb{CP}^r$ is an algebraic subvariety of dimension $n$, codimension $e=r-n$, and degree $d$. Call $j$ the inclusion. Then, Poincaré duality induces a map $...
7
votes
1answer
128 views

Intersection of curves on surfaces and the sheaf $\mathcal O_{C\cap C'}$

Let $S$ be a (complex) projective surface and let $C,C'\subset S$ two closed irreducible curves (namely two prime Weil divisors). It is well defined the scheme $C\cap C'$ (as fibered product of $C\...
2
votes
0answers
95 views

How to compute intersection numbers in practice?

When speaking about plane curves, one of the most fundamental and important results is Bézout's Theorem, which states that over an algebraically closed field $k$, two plane projective curves of ...