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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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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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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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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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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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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 $...
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Alternative way of calculating the intersection number on a surface

Let $X$ be an algebraic surface over a field $k$ and let $D,E$ two smooth prime divisors on $X$. Assume that $e_x,f_x\in\mathcal O_{X,x}$ are the local equations at $x$ of $D$ and $E$ respectively. ...
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Exact sequence on the level of Chow group

I am just reading Fulton's book. I don't understand something in the proof. I understand that the image is subset of the kernel. I don't understand how does the proof get the reverse inclusion? ...
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How do we get empty fiber over $\infty$?

I am trying to understand a proof in 3264 and all that. I don't fully understand how do we get empty fiber over $\infty$ ?
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Notation in 3264 and all that algebraic geometry

I don't understand the notation $\mathcal{O}_{Y,Y_i}$. I know the notation $\mathcal{O}_{Y,s}$ that is the stalk with respect to a point. Can someone explain this ? This is in page 15 in the book ...
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Generic hyperplane and transversal intersection

I'm trying to solve an exercise I.7.7 in Hartshorne's Algebraic Geometry: 7.7. Let $Y$ be a variety of dimension $r$ and degree $d>1$ in $\mathbb{P}^n$. Let $P \in Y$ be a nonsingular point. ...
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Kahler Form Corresponding to Hyperplane Line Bundle?

Let $S\subset \mathbb P_\mathbb C^3$ be a cubic hypersurface. By exponential sequence we know the natrual map $$Pic(S)\cong H^1(S,\mathcal O_S^*)\to H^2(S,\mathbb Z)$$ is an isomorphism. We denote ...
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Intersection spheroid-polar plane in parametric or spherical coordinates

The Earth is assumed to be a WGS84 ellipsoid (oblate spheroid) $E$. $E: (x^2+y^2)/R_{eq}^2+z^2/R_{pol}^2=1$ With $R_{eq} > R_{pol}$, a point $M$ is outside $E$, and $P$ is its polar plane, that ...
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Smooth quartic surface in $\mathbb{P}^{3}$ that contains a smooth curve of genus 2 and degree 6.

I am reading through an article by Matsumura and Monsky on Automorphisms of Hypersurfaces in which they state that there exist quartic surfaces is $\mathbb{P}^{3}$ which have infinite automorphism ...
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Points of tangency from a point M to an ellipsoid-plane intersection

Having an spheroid $S$ of semi-major axis $a$ in the equatorial plane in direction $x-axis$ and $y-axis$, and of semi-minor axis $b$ in direction $z-axis$. $(S): (\frac{x}{a})^2+(\frac{y}{a})^2+(\...
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Calabi Yau: complete intersection condition

When you express a Calabi Yau manifold as a member of the configuration matrix defined as equation (2.1) of this paper https://arxiv.org/abs/1303.1832 how do you know a priori that every choice of the ...
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Coordinates of tangent points from a point outside a spheroid (ellispoid of revolution)

The blue conics in the figure is an ellipse (it would be in a case a circle), that represents the tangency points, from the line drawn from the point $P$ to the ellipsoid surface in all direction, ...
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Condition to ensure $n$ equations in $\mathbb P^n$ cut out maximal points

Let $\mathbb P^n$ be the projective space over $\mathbb C$, and let $F_i$ be $n$ homogenous polynomial equations with ${\rm deg} F_i=d_i$. My question is: What is the right condition to ensure the ...
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How to check if this non-proper intersection is reduced?

Given a smooth and projective variety Z over the complex numbers. Then we have $X:=Z\times Z$ and in $X$ we have the diagonal $Y=\Delta\cong Z$. Now let $p: Bl_Y(X) \rightarrow X$ be the blow up. In ...
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Find point of intersection of two gradients

I have four points $A$, $B$, $C$, $D$ on the Cartesian plane, with coordinates $(x_a,y_a)$, $(x_b,y_b)$, $(x_c,y_c)$, $(x_d,y_d)$. The lines $AB$, $BC$, $CD$ all have a gradient greater than or equal ...
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Chern class of the tensor product of a torsion free sheaf with a line bundle

I'am try to work with Chern class of the coherent sheaves, in this sense. If I have a vector bundle $E$ of rank $r$ and $L$ a line bundle we have the Chern class property $$c_{r}(E\otimes L) = \sum_{...
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Intersection spheroid -plane

I have a spheroid $S$ with $a$ is the equatorial radius, and $b$ is the polar radius, and ($a>b$) I would get the intersection between $S$ and a plane $P: ux+vy+wz+d=0$ Then calculating the semi-...
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Intersection Index invariant under homotopy, counterexample?

I am looking specifically at the theorem in Guillemin and Pollack Page 78, where the intersection number modulo 2 is invariant under homotopy given that X is compact, Z and X are in complementary ...
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Parametric curves intersection

I would like to ask how to solve following problem: We have 2 "super heroes". Their path are given by parameterized function where t means time: $r_{1}(t)=$$\begin{bmatrix}t\\t^2\\t^3\end{bmatrix}$ ...
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self intersection of divisor

My question concernes following former thread in MO: https://mathoverflow.net/questions/117808/the-intersection-multiplicity-of-the-canonical-divisor-of-a-surface-with-a-fibre here we have a (minimal)...
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Algebraically Equivalent Fibers

Let consider the ruled surface $X$ with morphism $\pi: X \to C$ to a curve. Why then all fibers (of closed points) are as divisors algebraically equivalent? Hartshorne explains it using an argument ...