# 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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### 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 ...
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### 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 ...
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### 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 $P$. ...
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### 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 ...
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### 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 $m$. ...
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### 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 ...
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### 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 $\ell \neq p$...
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### 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 ...
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### 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 $f$, ...
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### 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 ...
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### 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 ...
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### 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 $∂Σ = \{ x ∈ ℝ^n \,|\, (x¹)² + … + (x^{n-1})² = R², x^n = 0 \}$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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: $$P(A) = \frac{73}{250}, P(B)={\frac{9}{25}}, ... 0answers 66 views ### Smoothness of fibers vs smoothness of total space Let f:X \to Y= \mathbb P^n be a flat morphism. We define condition S_k for such morphisms whose total space over every k-dimensional linear space is smooth, namely:$$f\in S_k \text{ if for ...
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 $(x_1, x_2, x_3, \cdots, x_n)$. ...