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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Boundary of smooth domain difference

Let $\Omega_{1},\Omega_{2} \subset \mathbb{R^2}$ be open, connected and bounded domain with piecewise smooth boundary. Let suppose that $\Omega_{1}\cap\Omega_{2}\neq \emptyset$ has a piecewise smooth ...
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how we can compute $\hat{i}(\alpha,\beta)$ and ${i}(\alpha,\beta)$ for following curve?

in the Farb and Margalit: A primer on MCGs. on page 28 we have : There are two natural ways to count the number of intersection points between two simple closed curves in a surface: signed and ...
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What is a generically reduced scheme?

I am reading the book "3264 & All That Intersection Theory in Algebraic Geometry". In the following definition (see page 30) Definition 1.22. Let $f:Y\rightarrow X$ be a morphism of ...
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Calculate Intersection point of 2 Lines with angle [closed]

I would like to calculate an intersection point of two lines in a 2D area. I think it should be really simple but i cannot figure it out. I have two points P1(x,y), P2(x,y) and 2 angles alpha and beta....
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Finding Multiplicities of Intersections Between cubic and quartic functions f(x,y)=0 and g(x,y)=0.

I fear this may be a naive question, as I have very little experience with algebraic geometry... While working on a problem in Linear Algebra, I find that my problem reduces to finding the number of ...
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Determining A Vector Through the Center of Multiple Points on a Sphere

I am working on a machine vision task that requires me to determine spin rate and spin axis of a moving ball. I have had some luck, and actually do have a solution but am looking for a more efficient ...
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I need to find the points in which the exterior and interior lines of a wall are supposed to intersect.

I am trying to produce the same result of that of Floor planner software in making walls in 2D.Here while drawing these walls, I need to join these walls, for that I need to find the exact location ...
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find the point two lines intersect in 3d with a mixture of cartesian and spherical coordinate system knowns

I have line 1: originating from origin (0,0,0) and magnitude (length) of 7 I have line 2: originating from a shifted position (0, 0, 0.4) and the polar and azimuth are also known (see picture) I ...
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Equivalence of two notions of intersection multiplicity for curves.

Let $C,D\subset\mathbb{C}^{2}$ be algebraic curves with no common component, and suppose that $P=(0,0)$ is in $C,D$. Let $f,g$ be the equations of $C,D$ respectively and let $i_{P}(C,D)$ be any ...
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Intersection multiplicity with a line of projective surface

In Shafarevich´s Example 4.7 in page 244 (Basic Algebraic Geometry), one finds the following. I have two questions: 1.- Why $\sum k_i deg C_i=m-1$? He defines the degree of a of a projective variety ...
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Multiplicity of intersection in terms of degree of the quotient of ideal by its saturation

In this guide to Macaulay2 I found an interesting way to compute intersection multiplicity. On page 61 the authors give an explicit case of the following argument: We want to compute the intersection ...
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Intersection between a Plane and a Line (2 points)

I have to calculate the intersection between a line and a plane. Of the line I know two points $P_1=(x_1, y_1, z_1)$ and $P_2=(x_2, y_2, z_2)$ while of the plane I know the equation $Ax + By + Cz + D =...
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Rigorously working with flat limits: lines meeting a curve by specialization

I am trying to get comfortable with flat limits. This question is motivated by Section 3.5.3 of Eisenbud and Harris's '3264 And All That' and Exercises 3.35 and 3.36. This section and the surrounding ...
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$30\%$ students have glasses. $20\%$ of students with glasses play. $60\%$ of students without glasses play. Probability, student without glass plays.

Problem: In a school, $30\%$ of students have glasses. $20\%$ of students with glasses play sports. $60\%$ of students without glasses play sports. If we randomly choose a student, find probability ...
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Probability of one discrete random variable having an higher value than another

I am trying to understand a solved question about statistics. I have two identical independent binomial random variables X and Y with identical pmfs $P[X=x]={100 \choose x}0.05^x0.95^{100-x}$ and $P[Y=...
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Euler characteristic of union of oriented spaces with non matching orientation

I'm studying some oriented geometries in the Grassmannian. These geometries, even if their interior is connected and oriented, have some funny behaviour on their boundaries. For example, some one-...
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Computing the degree of the locus of unions of a double line and a line in the space of plane cubics

This question is inspired by exercises 2.51-2.52 of Eisenbud's "3264". Here $\mathbb{P}^n$ denotes the $n$-dimensional projective space over an algebraically closed field of (for safety) ...
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Nef divisor on a surface and intersection number

Let $X$ be a complex surface and $D$ a nef divisor that is not numerically trivial. Then for any $n\in \Bbb N$, can we choose a smooth curve $C$ such that $D\cdot C\geq n$? Certainly there is a ...
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Find where two functions intersect

I have the functions $f(\lambda) = e^{\frac{\ln(\frac{-\lambda\theta^k}{3\lambda-2T})}{k}}$ and $g(\lambda) = \frac{\ln(\frac{-\lambda}{3\lambda-2T})}{k}+\theta$ where the constants $\theta, k, T$ are ...
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Intersection of two parametric surfaces (governing equation)

in order to solve the interaction between two parametric surfaces (represented as Bezier oder B-Splines) i need to "solve" the non linear equation system. As both surfaces are depended in ...
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Dimension of a quasivariety

Let $f_1, ...f_n$ be $n$ homogenuous polynomials, on variable $z_0, z_1, ..., z_n$. Let $V=V(f_1, ..., f_n)$ be the projective variety in $P^n$. Let $V_1=V\bigcap \{ z_0=1\}$ and $V_2=V\bigcap \{ z_0=...
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how to compute the probability of intersection of two subsets of the sample space? (in difficult problems)

let A and B be 2 events (i.e subsets of the sample space). how can I compute the probability of the intersection (theoretically)? I know that in simple exercises, I have to draw each set, in order to ...
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Difficulty with Intersection Theory on a Manifold with Boundary

I'm having trouble understanding a certain situation having to do with (topological) intersection numbers of submanifolds of a manifold with boundary. This whole question will be based on theorem 11.9 ...
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degree computation in the proof of Theorem V.1.10 in Hartshorne

Here is our setting: $X$ is a surface (in the sense of chapter V in Hartshorne), $D$ is a reduced curve on $X$ (possibly singular), $C_i$ is an irreducible component of $D$, $f: \tilde{C}_i \...
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nonsingularity assumption in Lemma V.1.3 in Hartshorne

Lemma V.1.3 in Hartshorne states that if $C$ is an irreducible nonsingular curve on a surface $X$, and $D$ any curve meeting transversally with $C$, then $\#(C \cap D) = \deg_C (\mathscr{L}(D) \otimes ...
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Intersection of cone and sphere

I have the following problem: There is a sphere (Earth) and a cone (the FOV of a satellite orbiting Earth). So the tip of the cone is at the satellite's center orbiting Earth, and the wide part of the ...
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Possible Intersection Multiplicities of a curve and hyperplanes

Let $k$ be an algebraically closed field with positive characteristic $p>0$ and let $1<q_1<q_2<q_3$ be its powers. Let $X\subset \mathbb{P}^4$ be given by $(1:t:t^{q_1}:t^{q_2}:t^{q_2+1}+t^...
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Vanishing Cohomology of Cubic Surface

I recently read somewhere that for a cubic surface $S$ in $\mathbb{P}^3$, the classes $[\Delta] - [\Delta']$ generate the vanishing cohomology $H^2(S,\mathbb{Q})_{van}$, taken over all pairs of lines $...
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How to determine the properties of a self intersecting curve (parametric functions) : number of loops and of passages through a point?

Let's define a parametric vector $(f(t),g(t))$. If there a way to determinate how many times this function will come back to a certain point ? For instance here is the plot of $(f(x),g(x))=\left(-2 ...
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Example of a normal cone, that is not irreducible

For a pair of (irreducible) varieties $W \subset V$ over an algebraically closed field $k$, let $C = C_{W} V$ be the normal cone, i.e. $$C = \operatorname{Spec}_W \bigoplus_{n\geq 0} I^n / I^{n+1},$$ ...
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Definition of the Polynomial $P(D, E)$ in Riemann-Roch Without Denominators.

In the proof of Lemma 15.3 in Fulton's Intersection Theory, there appears the formula $$c(\Lambda^\bullet D^\vee \otimes E) = \prod_{p=0}^d \prod_{j=1}^e \prod_{i_1 < \dotsb < i_p} (1 + y_j - ...
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Calculating coordinates of interception from field of view to target

Say we are working in a 2D plane, with a camera and a ball flying past as shown. Camera at bottom, ball flying from left to right Given that I have the X/Y coordinates of the camera, as well as the ...
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Any two closed curves in $S^2$ has intersection $0$ mod $2$.

I know that $S^2$ is simply connected since by Sard's theorem any curve in $S^2$ ($1$ dimensional manifold) is a measure zero set hence it should be contained in $S^2-\{p\} $ for some $p\in S^2$ and $...
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Find projection source given projected points on 2D lines.

Assume we have M points in 2d Cartesian space. There are also $N$ lines crossing point $(0,0)$. Let $(x_{ij},y_{ij})$ be the projection of the $i$ th point of $j$ th line. Given These projected ...
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Intersection number of very ample divisor and curve on a surface equals the degree

I'd like to show that if $X$ is a (nonsingular, projective, algebraic) surface, $H$ a very ample divisor on $X$, and $C$ an effective divisor (curve) on $X$, then the intersection number $C.H$ equals ...
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A $k$-dimensional reduced subscheme $Y \subset \mathbb{P}^n$ of degree $1$ is a linearly embedded $k$-plane

In following all schemes $X$ will be considered as separated, of finite type over an algebraically closed field $K$ of characteristic $0$. Recall that a Hilbert function $HF_S: \mathbb{N}_0 \to \...
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Number of intersection points of plane and algebraic curve

Suppose $C\subseteq \mathbb{R}^3$ is an irreducible real-algebraic curve of degree $k$ and $P\subset \mathbb{R}^3$ is a plane. Suppose that $C$ intersects $P$ finitely many times. What is the best way ...
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Algebraic curve contained in plane or only intersects it finitely many times

Suppose that $C\subset \mathbb{R}^3$ is an irreducible real-algebraic curve that meets a plane infinitely many times. Is it true that $C$ must be contained in this plane? I'm working on a paper that's ...
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Ray Tracing from Scratch. Reflections not quite right.

I am currently writing a simple demo for a 3D raytracing engine. The program basically has the following structure: An array stores all the planes using three coordinates. For every (2D) pixel on the ...
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Question on P((A|C)∩(B|C)) = P(A∩B|C)

I've been exploring conditional probability with three events. In particular, I've tried to gain a better understanding of probability conditioned on a third event. In doing so, I'm struggling to ...
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Find max distance between an intersection and a plane.

Determine the point of intersection between the plane $x/2 + y/2 -z = -1$ and the cone $z^2 = x^2 + y^2$ that is farthest from the plane $y = 0$
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How to calculate the Euler class, Euler characteristic and top Chern class of $End(E)$?

It's me again. Could someone please ilustrate the relationships between these concepts through the following example: Let $E$ be a rank 2 holomorphic vector bundle on $\mathbb{CP}^2$. Find the Euler ...
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3 votes
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A formula for the second Chern class of the tensor product of a line bundle and vector bundle

If possible I would like someone to prove or suggest a place to see the proof of this relation: $$c_2(V \otimes L)=c_2(V)+(r−1)c_1(V)c_1(L)+ {r \choose 2} c_1(L)^2$$ Here $L$ is the line bundle and $r$...
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When are calculations up to numerical equivalence appropriate?

William E. Lang writes in Examples of Surfaves of General Type with Vector Fields In the next two lemmas, we show that $K_X$ and $\mathcal O_X(D)$ are linear compinations of [some other bundles] . ...
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Intersection pairing of weighted projective plane

Let $a,b,c$ be mutually relatively prime positive integers. The weighted projective plane $X:=\Bbb CP^2(a,b,c)$ is the quotient space $\Bbb C^3-\{0\}/(z_1,z_2,z_3)\sim (\lambda^a z_1,\lambda^b z_2,\...
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Question about Push-forward

In Fulton's book "Intersection theory", Theorem 1.4. It has the following argument: Let $X$ and $Y$ are normal varieties, $f : X \to Y$. If $A$ is the local ring of $W$ on $Y$, $W$ a ...
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Intersecting three quadrics in $3D$

A quadric in $3D$ can be expressed as $ r^T Q r = 0 $ where $ r = [x, y, z, 1] $ , and $Q $ is a symmetric $4 \times 4 $ matrix. Suppose I have three quadrics and want to find their intersection ...
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Intersection between a horizontal plane and a torus. Problem with complex values

[EDIT: I have replaced $a$ by $R_1$ in my previous post to make your reading and understanding easier, as this is a more general expression.] I think I have a basic question but I cannot figure out an ...
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Chow group of disjoint union

I have a disjoint union of open sets $U_1,..., U_k$ on a variety $X$. In Fultons "Introduction to toric varieties", he used $X=X_\Sigma$ a toric variety and $U_i=\mathcal{O}(\sigma_i)$ ...
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Is there a relation between self-intersection and covering map?

Let $X$ and $Y$ be compact oriented smooth 4-manifolds. There is a well-defined intersection form $H_2(X)\times H_2(X)\to \Bbb Z$, $(\alpha,\beta)\mapsto \alpha \cdot \beta$, and similarly for $Y$ (...
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