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)

Filter by
Sorted by
Tagged with
0
votes
0answers
21 views

Nearest intersection point to the center of multiple spheres

I have multiple spheres with the same radius r, and I need to find the nearest intersection point to the center of all the spheres. I also know that all the spheres intersect together. I'm using ...
0
votes
0answers
15 views

Ray intersection with cylinder of arbitrary rotation

I'm working on writing an algorithm for a the distance of a ray intersection with a cylinder (t), where the cylinder is of arbitrary rotation. Using this website as inspiration, I know I can find the ...
0
votes
0answers
10 views

Tangent with intersection multiplicity greater that multiplicity of the point

$\mathcal C$ is an affine or projective curve, $P\in \mathcal C$ is a singular point (an $m$-uple point, so its multiplicity is $m$) and the line $\tau$ is tangent to $\mathcal C$ in the point $P$. ...
1
vote
2answers
53 views

general questions about algebraic surfaces and Castelnuovo's contraction theorem

I am not really sure where should I ask this question so feel free to move it to other more fit community or add more tags. My master thesis is about Algebraic surfaces and Castelnuovo's contraction ...
0
votes
1answer
25 views

Transverse intersection and conditions on Tor

Consider $X$ and $Y$ varieties inside a smooth variety $M$. I say that $X$ and $Y$ intersect transversally at $m\in M$ if the tangent spaces of $X$ and $Y$ span the whole tangent space of $M$ at $m$. ...
1
vote
1answer
49 views

Intersection points of the 27 lines on smooth cubic surface

Let us work over $\mathbb{C}$. It is a classic result that if $S$ is a smooth cubic surface, then there are 27 lines contained in $S$. My question is: can we compute the number of points which are ...
3
votes
1answer
145 views

An intersection calculation over a finite field

Question: I made a calculation that must be wrong, but am having trouble spotting the error. Which steps below are invalid? Thank you in advance for your attention! Setup: Let $p$ and $\ell \neq p$...
2
votes
1answer
49 views

Intersection of Simple Closed Curves in $\mathbb{R}P^2$

Given a (smooth) simple closed curve $C$ in $\mathbb{R}P^2$, we either have that $\mathbb{R}P^2 \setminus C$ is the disjoint union of a disk and Möbius strip, or that $\mathbb{R}P^2 \setminus C$ is a ...
0
votes
0answers
16 views

The sufficent condition about the direct sum and intersection of subspaces

Let $A,B,C,D$ be subspaces of a vector space over a finite field such that $\dim(A\cap B)=0$, $\dim(C\cap D)=0$, then what is the sufficent conditions that 1) $(A\oplus B)\cap(C\oplus D)=(A\cap C)\...
0
votes
1answer
52 views

Every fiber intersects transversally the section

Let $k$ be an algebraically closed field, let $S$ be a smooth projective surface over $k$, let $C$ be a smooth projective curve over $k$. Let $f:S\to C$ be a genus $1$ fibration, i.e., $f$ is a ...
0
votes
0answers
27 views

Linear systems on singular curves

I have a question about a equality from a proof I found Kollars 'Lecures on resolutions of singularities'. Kollar gave two proofs of Theorem 1.58, my question refers to the second (the 'Noether's' ...
1
vote
1answer
62 views

Explicit Examples of Transversal Intersections of Two Curves

Here, a surface is a nonsingular projective surface over an algebraically closed field, and a curve is any effective divisor on that surface. Let X be surface. Let C and D be two curves on X, and let ...
0
votes
0answers
16 views

Singularity of Null Cone

This is a statement in Lazarsfeld in Positivity in Algebraic Geometry, Example 1.5.24. Let $f: X\rightarrow Y$ be a surjective morphism of irreducible projective varieties or schemes with $\dim X=n$ ...
5
votes
1answer
84 views

The Chow ring of affine space

I want to show that $\operatorname{CH}(\mathbb{A}^n) = \mathbb{Z}$ from first principles. There's a proof of this in 3264 by Eisenbud and Harris that utilizes the definition of rational equivalence ...
-1
votes
2answers
31 views

Intersection / union of sets over R with epsilon

Calculate the following union (proof needed) $\bigcup_{0 < \epsilon \leq 1} [a-\epsilon, b+\epsilon), a<b \in \mathbb{R}$ and the intersection (with proof) $\bigcap_{\epsilon > 0} [a-\...
5
votes
0answers
124 views

Any section of a smooth morphism is regular

Let $f:X\to Y$ be a smooth morphism of relative dimension $n$ of separated schemes which are of finite type over $\text{Spec}(k)$, where $k$ is any field. Suppose that $i: Y\to X$ is a section of $f$, ...
0
votes
3answers
26 views

Find an equation for the line that goes through the point and intersects line1 and line2

Let $L_1$ be a line in $\mathbb R^3$ that is defined by $(x,y,z)=(2,2,0)+t(3,0,2)$ a) Find the plane that includes $L_1$ and the point $A=(9,2,3)$ b) The line $L_2$ is defined by $(x,y,z)=(5,1,0)+t(...
0
votes
0answers
23 views

intersection number on the boundary of a manifold

Let $F: W \to N$ be a smooth map, where $W$ is a compact manifold with boundary, $Z \subset N$ is closed and all manifolds are oriented. Also $\partial F \pitchfork Z$ and $F^{-1}(Z)$ is a compact, ...
0
votes
1answer
28 views

Defining intersection of two surfaces

I am having a bit of trouble beginning this question. The question is as follows: Let $r$ be the curve which is the intersection of the surface $z = \frac{1}{3}x^2 + \frac{2}{3}y^2$ and the surface: ...
2
votes
1answer
69 views

Existence of a transversal map prevents density?

Let $S$ be a $C^{\infty}$-submanifold of $N$ and suppose that $S-N$ is dense in $N$, where $M,N$ are $m$ and $n$ dimensional $C^{\infty}$-manifolds, respectively (without boundaries). In this post ...
0
votes
0answers
25 views

Show that the degree of a function is an intersection number

I'm trying to prove the Brouwer's fixed point theorem and somewhere I need the following result: $M$, $N$ are smooth manifolds of dimensions $m$ resp. $n$, $f: M \to N$, $F: M \to M \times N$ defined ...
2
votes
1answer
40 views

Determine if two annuli intersect by just looking at the generating triangles

Draw a random triangle on the plane and label its vertices $A$, $B$ and $C$: Now draw a circle with $A$ as its center and $\overline{AB}$ as its radius, and one with $\overline{AC}$ as its radius: ...
0
votes
0answers
17 views

Constructing a point with intersection multiplicity $n$.

Working in the projective plane over complex numbers, suppose that we have a line $l : x=0$, is there a systematic way to construct a homogeneous irreducible polynomial $f$ in $\mathbb{C}[x,y,z]$ s. t....
0
votes
0answers
6 views

How to find coordinate intersections between two points.

So I have a 2d plane split that plane into 1m X1m squares. Each square is referenced by the location of the top left corner, e.g. (76,65). Now, on this plane I have two random ponts. I want to know ...
3
votes
0answers
195 views

Intersection Theory and Blow up

The following is from Fulton's Intersection Theory: Theorem 6.7 (Blow-up Formula). Let $V$ be a $k$-dimensional subvariety of $Y$, and let $\widetilde{V} \subset \widetilde{Y}$ be the proper ...
0
votes
0answers
15 views

Common points of algebraic set defined by the common root of a family of polynomials

I am struggeling with a combination of definitions in a paper. https://scholar.google.de/scholar?hl=de&as_sdt=0%2C5&q=Finding+the+real+intersection+of+three+quadrics+using+techniques+from+...
0
votes
0answers
28 views

Questions on Chow Rings of Affine Spaces

I have two questions regarding the Chow Ring $A_*(X)$ of an affine scheme $X$. Find an affine smooth variety $X/\mathbb{C}$ such that $A_*(X) \ncong \mathbb{Z}$: My idea: we know that $A_n(X)=\mathbb{...
2
votes
0answers
34 views

First Chern map not injective

I'm looking for an example that shows that the map $c_1: Pic(X) \rightarrow A_{n-1}(X)$ is in general not injective. Eisenbud/Harris gives an exercise using X a plane cubic nodal in 1.35, but I didn't ...
0
votes
1answer
21 views

Can I merge overlapping intervals this way?

Say if I have the following function: $f_{X,Y}(x,y) \ = \ \left\{ \begin{array}{ll} x^2 + 5x -13, & \mbox{if $2 \leq x \leq 5$ and $0 \leq y \leq x$}, \\ 0, & \mbox{otherwise}. \end{...
1
vote
0answers
51 views

Divisors in complete intersection coming from divisors of the intersected varieties

Consider a variety $C$ which is a complete intersection of $X$ and $Y$ in $\mathbb{P}^n$ such that the degrees of $X$ and $Y$ are coprime, and so the degree of $C$ is the product of the degrees. ...
0
votes
0answers
25 views

Looking for clarification on event intersection $\cap$ and how to explain to non statisticians

I have a few things I am hoping to get clarified. Let's say we want to determine $Pr(A \cap B)$. Can we say $Pr(A \cap B)$ is the joint probability of A and B occurring? Or is that the wrong term ...
0
votes
1answer
40 views

intersection of numbers on wolfram [closed]

After evaluations of the Riemann Hypothesis, with reflective thoughts over a year as to how to solve it, a formula that is very close has been found, if not distinctly the act of the original proof, ...
1
vote
0answers
39 views

A simple proof for sum of intersection multiplicities

The definition for intersection multiplicity used in Fulton's Algebraic Curves is given by: $$I(P,F\cap G):=\dim_{\mathbb{K}}(\mathscr{O}_{P,\mathbb{A}^2}/(F,G)),$$ (see section 3.3) where $\...
-1
votes
1answer
54 views

Self-intersection of a curve after successive blow-ups

Let $P_0,P_1,P_2\in\Bbb{P}^2$ points in general position,consider the lines $\ell_i:=\overline{P_jP_k}$ for $\{i,j,k\}=\{0,1,2\}$ and the blow-up $\pi:S\to\mathbb{P}^2$ at $P_0,P_1,P_2$. I was told ...
0
votes
0answers
51 views

$Tor_1(\mathcal{O}_L,\mathcal{O}_Z)$ where $Z \cap L$ are distinct points and $L$ is a line

Let $Z$ be a scheme of points in $\mathbb{P}^2$ and $L \in \mathbb{P}^2$ a line. I know that $Tor_1(\mathcal{O}_L,\mathcal{O}_Z)$ is a sheaf with support in $Z \cap L$, but can I prove that $Tor_1(\...
4
votes
1answer
115 views

Tangent cone of a non-isolated singular point

Assume $X\subset \mathbb P^n$ is a variety (Edit: let's say $X$ is a hypersurface in $\mathbb P^n$, as pointed out in the comment) and $x\in X$ is a singular point which is not isolated. Intuitively, ...
0
votes
2answers
24 views

Intersection between three events

If I have $S_{1}=(F\cap\bar{G}\cap \bar{H})$, $S_{2}=(\bar{F}\cap G\cap \bar{H})$ and $S_{3}=(\bar{F}\cap\bar{G}\cap H)$, how can I apply the law of total probability for $\mathbb{P}(S_{1}\cup S_{2}\...
3
votes
0answers
53 views

A curious “cancellation” of orientations on the intersection of two hypersurfaces

The situation: Let $Σ^{n-1} ⊂ ℝ^n$, $n ≥ 2$, be an oriented, compact, embedded smooth hypersurface with boundary. For simplicity assume $∂Σ = \{ x ∈ ℝ^n \,|\, (x¹)² + … + (x^{n-1})² = R², x^n = 0 \}$, ...
0
votes
0answers
12 views

Calculate if polygon and circle intersects (expressed in lat,long)

I would like to be able to calculate if a polygon and a circle drawn in Google Maps intersect (represented in latitude,longitude points and the radius of the circle in meters). Let's use this as an ...
1
vote
1answer
52 views

Mori cone: extremal ray intersections

On an algebraic surface, much can be said about the Mori cone, or cone of curves. In this question, I will be particularly interested in intersection properties. Several sweeping statements can be ...
0
votes
0answers
41 views

Intersection product of nef and ample divisor.

Let $X$ be a projective variety of dimension $n$, let $D$ be a nef divisor on $X$ and let $H$ be an ample divisor. Does $$D \cdot H^{n-1} > 0$$ necessarily hold? The context I encountered this is ...
2
votes
1answer
92 views

Intersection of a set of hyperplanes and a curve in $\mathbb{P}^2$

I have struggles with solving the following question: Let $X$ be a curve in $\mathbb{P}^2$ of degree $d$. Show that the set of hyperplanes $H \in (\mathbb{P}^2)^{*}$ such that $X \cap H$ consists of ...
0
votes
0answers
78 views

Grassmannians: Varieties swept out by linear spaces (Eisenbud & Harris: 3264 and All That)

I have a couple of questions on statements from Harris' and Eisenbuds's lecture "3264 and All That" at page 145, Section 4.2.3: Varieties swept out by linear spaces. The content of 4.2.3 answers ...
0
votes
1answer
21 views

Boundaries of a 3 event union

I have read through many of the threads in StackExchange, but I cannot seem to grasp the idea of a 3 event union fully. I have these probabilities to use: $$P(A) = \frac{73}{250}, P(B)={\frac{9}{25}}, ...
5
votes
0answers
66 views

Smoothness of fibers vs smoothness of total space

Let $f:X \to Y= \mathbb P^n$ be a flat morphism. We define condition $S_k$ for such morphisms whose total space over every $k$-dimensional linear space is smooth, namely: $$f\in S_k \text{ if for ...
0
votes
0answers
17 views

How to detect intersections in a vector space represented by logical expressions on a vector?

I have an unusual problem. I have a set of feature vectors that are classified by a set of logical expressions on its elements. Let me give an example: Given a vector $(x_1, x_2, x_3, \cdots, x_n)$. ...
1
vote
0answers
34 views

Projection formula for $\pi: Y \to X$, where the general fiber of $\pi$ is not finite.

Let $X, Y$ be proper (possibly projective, normal) varieties, and let $\pi: Y \to X$ be a morphism, whose general fiber has dimension $s \geq 0$. Let $\dim X = r$, so that $\dim Y = r + s$. Let $(\...
3
votes
0answers
110 views

Smoothness of general divisors in a pencil with base locus

Everything is over $\mathbb C$. Let $f:X \to \mathbb P^2$ be a flat projective morphism and $x_0\in \mathbb P^2$ be a fixed point in $\mathbb P^2$, and all the lines ($\mathbb P^1$) through it are ...
0
votes
0answers
22 views

Doubt regarding homological equivalence of cycles

Suppose $X$ is a complex projective variety, and let $Z_K(X)$ denote the group of $k$-cycles of $X$ (as a free abelian group). The group $CH_k(X) := \dfrac{Z_k(X)}{Z_k(X)_{rat}}$ is the so-called Chow ...
0
votes
1answer
86 views

Intersection number between a curve and a line not a component of the curve

Fulton's Algebraic Curves makes us deal with the following exercise (ex. 3.21): 3.21. Let $F$ be an affine plane curve. Let $L$ be a line that is not a component of $F$. Suppose $L=\{(a+tb,c+td)\,...

1
2 3 4 5
10