# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $$\dfrac{d \ell}{dt}=\sigma_{\ell} - \mu_{\ell} \dfrac{M \ell}{1+\ell}-d_{\ell}\ell \\ \dfrac{dM}{dt}=\dfrac{\ell}{\beta+\ell}+\sigma_M \dfrac{M \ell}{1+\ell}-M$$ ...