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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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
25 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
31 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
30 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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1answer
40 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
40 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
65 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
25 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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17 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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0answers
12 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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63 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
22 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 ...
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1answer
17 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
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0answers
49 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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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 $$ \dfrac{d \ell}{dt}=\sigma_{\ell} - \mu_{\ell} \dfrac{M \ell}{1+\ell}-d_{\ell}\ell \\ \dfrac{dM}{dt}=\dfrac{\ell}{\beta+\ell}+\sigma_M \dfrac{M \ell}{1+\ell}-M $$ ...
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1answer
24 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?- ...
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0answers
27 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 ...
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0answers
15 views

Given the Poincarre map, find the points where the limit cycle intersects the hyperplane S

The system $f(x(t))$ is known to have a limit cycle that intersects the hyperplane $S = ${$(x_1,x_2)|x_2=0$}. The Poincarre map is given by $P\binom{x_1}{0}=\binom{x_1^2-x_1-3}{0}$ 1) Find the ...
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1answer
46 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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32 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
13 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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0answers
13 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 ...
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2answers
40 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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34 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 = ...
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0answers
28 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 ...
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0answers
68 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
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1answer
114 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 ...
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0answers
64 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 ...
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2answers
63 views

Everyone knows that there is only one nonsingular conic tangent to five general lines in $\mathbb{P}^2$

This is a statement in Fulton's Introduction to intersection theory: ''there is only one nonsingular conic tangent to five general lines in $\mathbb{P}^2$ '' He implied that this is clear by duality, ...
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19 views

What is the definition of quasi-transversally intersection?

I'm considering a reduced curve $Y\subseteq \mathbb{P}^3$ and a multisecant line $L$. What does it mean that $L$ intersect $Y$ quasi-transversally?
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14 views

Detecting intersected cells of a 3D grid by a 3D closed triangulated surface

I have a Cartesian rectangular grid defined as: $x_i = x_0, x_1, \cdots, x_{NI-1}$ $y_j = y_0, y_1, \cdots, y_{NJ-1}$ $z_k = z_0, z_1, \cdots, z_{NK-1}$ In this 3D grid, a closed triangulated ...
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1answer
23 views

Single Plane Single Line Intersection using points

Background: This is for a programming assignment. The actual programming work is no problem, but I'm just having trouble understanding the equations/formulas. Given Data: I'm only given 3 points for ...
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0answers
28 views

Plane-ellipsoid intersection with rational coordinates

Consider an axis-parallel ellipsoid $E_d$ in ${\mathbb R}^d$ that is defined by an equation of the form $a_1x_1^2+\cdots+a_dx_d^2 = r^2$. Consider a hyperplane $h$ that is defined by the equation ...
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1answer
22 views

Calculating intersection numbers of surfaces in example from knot surgy

There is a section in an explanation about knot surgery which I do not understand in "Knot surgery revisited" by Fintushel, p.203. Let $X$ be a simply-connected compact $4$-manifold. Let $K$ be a ...
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0answers
11 views

Intersection of globally generated line bundles is nonnegative.

I'm reading Fulton's Introduction to Toric Varieties and in section 5.4 he states without further comment the following result from intersection theory: "This follows from the fact that the ...
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1answer
93 views

Minimum intersection of n lines

I saw many questions about intersections on here but didn't find an answer to my question. My question: Imagine you have n points which are randomly spread over a table or a sheet of paper or ...
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0answers
61 views

Two simple counterexamples in algebraic geometry

Suppose we have a smooth complex algebraic variety $X$. Then in general, $K^a(X)\to K(X^{an})$ is not surjective. Could someone give an example of a topological vector bundle class which contains no ...
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1answer
36 views

Bezier bicubic surface intersection genus

I have two bicubic Bezier surfaces that will intersect. According to this paper: http://nishitalab.org/user/nis/cdrom/cad/CAGD91geometric.pdf At the end of page 1. The general genus of intersection ...
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0answers
12 views

Intersection of 2 continua with an arbitrary number of dimensions using matrices

If I have 2 continua (i.e. a line, plane, space, hyperspace etc.) of an arbitrary number of dimensions, each described parametrically as: x = xs + x1*t + x2*u + x3*v + ... + xn*k y = ys + y1*t + ...
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2answers
58 views

Spacecurve of intersection between surfaces

Problem The spacecurve of intersection between the surfaces $opp$ & $\alpha$ (above z=3) has to be found, i.e. the intersection of the blue and green surface, above the red pane (z=3). Plots ...
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2answers
55 views

Polygon and line intersection

Does anyone help me with the fast algorithm to determine the intersection of a polygon (rotated rectangle) and a line (definite by 2 points)? The only true/false result is needed.
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34 views

Self-intersection number of fibered surface

Let $R$ be a complete discrete valuation ring with residue field $k$ (can assume algebraically closed), $f:X \to \mathrm{Spec}(R)$ a flat, proper family of projective curves (i.e., $f$ is also ...
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1answer
52 views

A fourth degree curve with four singular points is reducible

I'm trying to learn some algebraic geometry, and I would like to know if the following approach is correct. I would like to show that a fourth degree curve in $P\mathbb C^2$ with four singular ...
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1answer
44 views

Intersection product of line bundles with $\mathcal{F}$.

I am having trouble with computing the intersection number using the following definition [Ref Vakil Chapter 20]: Suppose $\mathcal{F}$ is a coherent sheaf on $X$ with proper support of ...
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1answer
20 views

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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0answers
80 views

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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77 views

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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1answer
88 views

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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89 views

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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130 views

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 ...