# 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)

622 questions
Filter by
Sorted by
Tagged with
1 vote
0 answers
21 views

### 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 ...
• 441
0 votes
0 answers
19 views

### 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 ...
• 4,083
0 votes
1 answer
42 views

### 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 ...
• 161
0 votes
2 answers
37 views

### 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....
0 votes
0 answers
31 views

### 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 ...
0 votes
0 answers
18 views

### 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 ...
-1 votes
0 answers
15 views

### 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 ...
0 votes
0 answers
20 views

### 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 ...
0 votes
0 answers
21 views

### 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 ...
• 1,269
0 votes
0 answers
29 views

### 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 ...
• 148
1 vote
1 answer
43 views

### 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 ...
• 2,702
-1 votes
1 answer
48 views

1 vote
0 answers
49 views

### 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-...
0 votes
0 answers
22 views

### 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) ...
• 2,702
0 votes
0 answers
41 views

### 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 ...
• 2,034
0 votes
1 answer
53 views

### 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 ...
0 votes
1 answer
37 views

### 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 ...
• 11
0 votes
0 answers
25 views

• 24.2k
3 votes
1 answer
68 views

• 413
4 votes
1 answer
156 views

• 323
1 vote
1 answer
52 views

### 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. $$C = \operatorname{Spec}_W \bigoplus_{n\geq 0} I^n / I^{n+1},$$ ...
• 4,432
0 votes
1 answer
21 views

1 vote
0 answers
35 views

### 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] . ...
• 4,432
2 votes
1 answer
155 views

• 1,720