# 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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### 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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### 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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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 - ... 0 votes 0 answers 12 views ### 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 ... 0 votes 0 answers 43 views ### 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 ... 0 votes 0 answers 12 views ### 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 ... 1 vote 1 answer 69 views ### 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 ... 0 votes 0 answers 65 views ### 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 \... 0 votes 0 answers 33 views ### 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 ... 0 votes 1 answer 98 views ### 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 ... 1 vote 0 answers 66 views ### 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 ... 0 votes 0 answers 48 views ### 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 ... 0 votes 0 answers 23 views ### 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 2 votes 1 answer 156 views ### 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 ... 3 votes 2 answers 217 views ### 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 $r$... 1 vote
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] . ...