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Questions tagged [intersection-theory]

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). Do not use this tag for elementary problems in linear algebra or geometry. (e.g. determining whether two lines intersect)

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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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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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circle circle intersection + cordinates + 3d + normal plane

good day. Actually I'm stuck with this problem I want to get the 2 points (vertex coordinates) in a 3d circle circle intersection actually I know a lot of data, ...
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Intersection paring on homology and cup product cohomology?

One advantage of working with cohomology rather than homology is that cohomology is naturally endowed with a ring structure via cup product. However the cup product is often understood via ...
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Chern numbers of Projective Space

Consider the $k$-th chern class $c_k:=c_k(\mathcal{T}_{\mathbb{P}^n})$ of the tangent sheaf of projective space $\mathbb{P}^n=\mathbb{P}^n_\Bbbk$ over some (algebraically closed, if you want) field $\...
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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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Dimension of the image of the morphism associated to a Divisor

Let $S$ be an algebraic smooth surface over $\Bbb{C}$. Let $D\in\mathrm{Div}(S)$ be such that the complete linear system $|D|$ is base-point free and suppose $h^0(D)=N+1$ with $N>0$. To $D$ is ...
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Use Quadrilateration to locate a point

I'm having lots of trouble with this. At first I thought the problem was a matter of equating four spheres to find their one common point, i.e. point of intersection. I've looked up lots of things on ...
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“intersection number” of surface with boundary

Consider in an $2n$ dimensional manifold $M$(compact, smooth), for embedded $n$ dimensional surfaces with boundary, denoted by $\Sigma$ and $\Sigma'$, in M, we can consider "intersection number"(...
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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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intersection multiplicity with a hyperplane

Let $V$ be an algebraic variety embedded in $\mathbb P^n$. Let $H$ be a hyperplane (that is to say, a variety defined by a single equation of degree $1$) such that for every $x \in H \cap V$, $x$ is a ...
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Codimension 1 points

Today I was reading a proof of the following Lemma from Liu's "Algebraic Geometry and Arithmetic Curves" Recall: A a point $x \in X$ is called a codimension 1 point if $ \overline{ \{x \}}$ has ...
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Intersection Between a Cylinder and a Given Line

I have a given point in space (Point C in Figure) and a cylinder, defined by its start/end center points and radius. I have a point on the axis (Point D), and I need to find the intersection between ...
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Intersection of half spaces

Let $X \neq \{0\}$ be a normed vector space, $C \subset X $ a convex subset. Is it true that $C$ is the intersection of some half-spaces in $X$? So I think that the statement is true, but I find it ...
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Intersection number and cup product

It is known that for a closed oriented smooth manifold $M$, if A and B are oriented submanifolds of $M$, and if A and B intersect transversely, then the Poincare dual of A ∩ B is the cup product of ...
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Divisor with positive Selfintersection Number Semi Ample

Let $X$ be a surface (therefore $2$-dimensional, proper $k$-scheme) and $D$ a divisor with positive self intersection number $(D \cdot D) >0$. Futhermore it is nef therefore for each irreducible ...
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Section of Hirzebruch Surface Semi Ample

We consider the Hirzebruch surface $S = \mathbb{P}(\mathcal{E})$ with locally free sheaf $\mathcal{E} = \mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(2)$. By definition of ...
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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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collision point of circle and line

I'm trying to figure out the collision point of the circle and a line, ultimately 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 ...
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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 $...