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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)

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singular intersection only comes from tangent?

Let $X$ be a smooth hypersurface in $\mathbb {CP}^n$, $H$ be a hyperplane. If $X\cap H$ is singular, is it true that $H$ is a tangent plane at some point of $X$? Note the converse is always true, if ...
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Intersection Theory in the projective space

Let $X$ be an irreducible projective variety embedded in $\mathbb{P}^N$. Let $V,W\subseteq X$ be two irreducible subprojective varieties intersecting properly in $X$, that is $$\mathsf{codim}_X(V)+\...
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Find intersection area of a diamond and movable/resizable square or rectangle.

I am creating a program in which I have to crop the area intersected by movable and resizable rectangle in a diamond. to achieve this I need a formula which will give me all the vertices of the ...
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Singular locus of dual hypersurfaces

Everything is over field $\mathbb C$. Let $X$ be a hypersurface of degree $d$ in $\mathbb P^n$. We know that if $X$ is smooth, then its dual $X^\vee$ is still a hypersurface in $(\mathbb P^n)^\vee$, ...
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How to find intersection points of circle and line given only a radius and end points

I need to find the point of intersection between a line and a circle. here is the information i have ...
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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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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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Automorphisms of generic hyperplane sections

Let $X\subset \mathbb {P}^n=\mathbb {CP}^n$ be a smooth hypersurface of degree $d$, $\{H_\lambda\}_{\lambda\in {\mathbb P^n}}$ be the set of hyperplane sections of $X$. We exclude the case $(d,n-2)=(4,...
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Calculate the intersection of a line segment on the radius of a circle

Given the length and one endpoint of the line segment, how can we find the other endpoint so that it is on the radius of a circle (known coordinates and radius)? Assume that there is at least one ...
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Non-trivial intersections of row spaces in Matlab

Let $W_p$ and $W_f$ be two subspaces (past and future data in matrix form: rows as basis vectors). Let x be a vector that lies in intersection of these two subspaces. Then $∃$ two coefficient vectors $...
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rational points on the quadrifolium $(x^2 + y^2)^3 = (x^2 - y^2)^2$

I have been reading the Wikipedia page on the Quadrifolium there are two of them: \begin{eqnarray*} r &=& \sin 2\theta \\ (x^2 + y^2)^3 &=& 4 x^2 y^2 \end{eqnarray*} and it's $45^\...
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Prove that a function from R to the unit circle is a local diffeomorphism.(2.4.8 G&P)

In order to prove the existence of the function $g$ in the question I want to proof that the following function is a diffeomorphism (I was told a hint that it is a diffeomorphism): $$p(t) = (\cos t, ...
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A difficulty in understanding the proof of boundary theorem in G&P.

The theorem and its proof is given below: But I could not understand the last line in the proof in particular: Why $F^{-1}(Z)$ is a compact one dimensional manifold with boundary? And why this leads ...
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A difficulty in understanding a case in the intersection theory mod 2(p.80 Guillemin and Pollack)

The following is written just before the boundary theorem in Guillemin & Pollack : But I see that if Z is not transversal to X this is not true,why the book did not consider this case? why the ...
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Euler characteristic of matrix manifolds

I'm reading through examples of computing Euler characteristic of manifolds. I know how to compute it for generic manifolds like sphere and torus. But what about matrix manifolds? I'd like to know how ...
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Intersection number for projective plane curves

Context: In our lecture about algebraic geometry we defined the intersection number of two algebraic curves $F,G$ to be $dim_k (\mathcal{O}_P (\mathbb{A}^2)/(F,G))$. Then we proved that it satisfies ...
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intersecting lines in the projective plane

Consider three points $A, B$ and $C$ in the projective plane [the white points in the picture below] not all on one line. Next, choose a point $O$ [the blue point] and draw the lines that connect $O$ ...
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Any hyperplanes is covered by non-Lefschetz pencils?

Let $X\subset \mathbb P^n$ be a smooth hypersurface over base field $\mathbb C$. A pencil of hyperplanes is just a projective line $(X_t)$ in $\mathbb P^{n*}$. It is called a Lefschetz pencil if it ...
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Find the equation of the plane that passes through the line of intersection of two planes and a point

Find the equation of the plane that passes through the line of intersection of the planes $2x-3y-z +1 =0$ and $3x+5y-4z+2=0$, and that also passes through the point $(3,-1,2)$ $\vec n_1 = [2,-3,-1]$ ...
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How can one show that an elliptic curve has a point whose tangent line also meet another point of the curve

I'm coming from a projective setting of a smooth cubic plane curve over a field $K$ and want to show that I can bring it to the Weierstrass long form. The usual method is to start with a point and ...
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Area of the intersection between a sphere and a cone (located in the center of the sphere)

Please, how do I calculate the area of the intersection between a sphere and a cone, as shown in the image below? The beginning of the cone is located in the center of the sphere, and both geometric ...
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General statement about how many lines in Euclidean space will determine a line

It is easy to see that in $3$-dimensional Euclidean space, given $4$ lines in general position, there exists precisely one line who intersects with each of the $4$ lines. We call the $4$ lines ...
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Question about adjunction formula $K_Y = (K_X + Y)|_Y$

I've learnt that if $Y\subset X$ is a smooth subvariety of codimension 1 then the degree of canonical divisors of $Y$ and $X$ are related by \begin{equation} \deg K_Y = \deg((K_X + Y)\cap Y). \end{...
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Intersection of the two planes

I need help for my vector's assignment!!! Let L be the line of intersection of the two planes x+y+z-1=0 and 2x+3y-z+2=0. Find the scalar equation of the plane that contains the line L and passes ...
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Intersection of a plane

I need help for my grade 12 Vector's homework. Can a plane be perpendicular to the x-axis and contain the line x=z, y=0? Explain. I really hope someone can answer this question
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Intersection of span

Let $a_1,a_2,a_3,a_4$ be vectors of a vector space. If $\operatorname{span}\langle a_i\rangle \cap\operatorname{span}\langle a_j\rangle = \overline 0 $ for all $i\ne j$. Does $\operatorname{span}\...
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Average percentage overlap between two/more datasets

I am analyzing different schemes of mutual funds. Each scheme has many funds in its portfolio. I wanted to analyse the overlap (of funds) between these schemes. I can find the overlapping of scheme1 ...
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1answer
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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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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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Calculating intersection line between two semi-cicles based on proportional distance to their origins

I hope you can help me here. We are trying to create the curve or line that gets draw between two semi-circle polygons that have each of them an origin point and a given radious. As on below figure: ...
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Intersection Theory on a Surface

I have some problems to prove the exercise 20.2.A part (b) in Ravi Vakil's "Fondation of Algebraic Geometry". Here the excerpt: The setting is: We have a surface $X$ (therefore 2-dimensional, proper $...
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Why is : $ g \circ f = ( \mathrm{pr}_{XZ}^{XYZ} )_* \big( ( \mathrm{pr}_{XZ}^{XYZ} )^* . \mathrm{pr}_{YZ}^{XYZ} )^* g \big) $?

Let $ \mathcal{E} (k) $ be the enumerative category of smooth projective varieties over a field $k$. This is a $ \mathbb{Q} $ - linear category which has for objects : smooth projective varieties over ...
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Every Schubert cycle a Chern class?

Consider the Grassmann variety $\mathbb{G}(k,n)$ and its Chow ring $A$. It is known that the classes of Schubert cycles form a $\mathbb{Z}$ basis of $A$. Is it known which of these Schubert cycles can ...
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1answer
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Every cycle class a Chern class?

I am currently learning intersection theory of smooth algebraic varieties and I have the following question. Let $X$ be a smooth projective variety and $\mathcal{F}$ a vector bundle on $X$. Then the $...
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Do these lines in 3D space intersect?

Th lines formed by $(0,0,0)+ \lambda(1,1,1)$ and $(0,6,0)+ \lambda(0,-3,2)$ ever intersect? It seems like the do but they don't. How do I show this algebraically?
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Proving non-centered at the origin ellipsoid A is a subset of ellipsoid B

I would like to prove an ellipsoid $E_a$ center at $q_a$ with shape matrix $A$ and radius $r_a$ defined as: $(x-q_a)^T A^{-1} (x-qa) <= r_a$ is included into another ellipsoid $E_b$ -i.e. $E_a$ ...
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Intersection of Two Polyhedrons Linear Programming

I am stuck on the following linear programming problem: If P and Q are two n-dimensional polyhedra Devise a linear programming such that: If P ∩ Q is nonempty, return a point in P ∩ Q Else: LP is ...
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Line intersecting spheroid

I have two planes $(A): u_{1}x + v_{1}y + w_{1}z = d_{1}$ and $(B): u_{2}x + v_{2}y + w_{2}z = d_{2}$. They intersect together, then they yield a line $(L)$ that has a direction vector $M (x_{M},y_{M}...
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Schubert class in the Grassmannian G(3,6)

How to compute the Schubert class $\sigma$$^2$$_2$$_1$ in the Grassmannian G(3,6)? I remember the result is $\sigma$$_3$$_3$ + 2$\sigma$$_3$$_2$$_1$ + $\sigma$$_2$$_2$$_2$.
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Can two projective surfaces intersect in points only?

Let $S_1,S_2\subset \mathbb P^n$ be two algebraic surfaces (we may assume that $n=3$). Is it possible that $S_1\cap S_2=\{x_1,\dots,x_N\}$ a finite set of points? I can imagine the surfaces two be ...
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Modified homotopy and relation with intersection theory.

Denote by $\Gamma$ a hypersurface in $\mathbb{C}^2$, i.e. the zero locus of a polynomial of two complex variables. Denote by $X$ the complement of $\Gamma$ in $\mathbb{C}^2$. I am trying to define a ...
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Degree of curve where matrix of polynomials has rank 1

My question is about a step in Exercise 12.8 on page 442 of 3264 & All That by Eisenbud and Harris. Chapter 12 is about Porteous' formula. The exercise reads: Let $A=(P_{i,j})$ be a $2 \times 3$ ...
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Fulton, example 3.2.16

In his "Intersection theory" book Fulton proves in lemma 3.2 the following fact: if a $E$ is a filtered vector bundle of rank $r$ over $X$ with quotients line bundles $L_i$, $s$ is a section of $E$ ...
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1answer
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Plane intesection with vertical vector given plane normal

There is a plane defined by a normal and an origin. For simplicity's sake, the origin is $(0,0,0)$. And then, there are two coordinates ($x$ and $z$) of a point on this plane. How can I find the ...
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Can a curve intersect 0 to inf without crossing inself?

Can you draw a curve such that it intersects every x at y=0 without ever crossing over itself (that is to say, without hitting the same value twice)? Wouldn't accomplishing this feat mean infinitely ...
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Massey product used to show that Borromean rings are linked

I'm trying to understand an example in "Elements of Homology Theory" from V.V. Prasolov (p. 85-88) where he shows that the Borromean rings represented by three spheres $S_1, S_2, S_3$ in $S^3$ are ...
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1answer
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Chern classes of tangent bundle over the Grassmannian G(2,4)

What are the Chern classes of the tangent bundle $\tau_G$ of the Grassmannian $G=G(2,4)$ of lines in $\mathbb{P}^3$? This is Exercise 5.37 on page 191 of 3264 & All That by Eisenbud and Harris. ...
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How to find intersection of two and three sine waves on x axis intercept? (Biorhythms)

I'm currently studying the trigonometry behind biorhythms. I was reading through the Wikipedia article on the topic (https://en.wikipedia.org/wiki/Biorhythm) which states that: Basic arithmetic ...
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Why do vertical divisor not contribute to the “intersection pairing”?

Let $X\to S=\operatorname{Spec}(O_K)$ be an arithmetic surface. We denote with $X_s$ the fiber over $s\in S$ and let $\operatorname{Div}_s(X)$ be the set of divisors on $X$ with support contained in $...