# 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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### Concluding the surface is a torus from properties of vector fields on it.

I have a pair of smooth 3-vector fields $B: \mathbb R^3 \to \mathbb R^3$ and $J: \mathbb R^3 \to \mathbb R^3$, with $\nabla \times B = J$, both of which are tangential to a regular surface defined as ...
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### Minimal number of intersections of families of lines

I'm considering two families, $F_1$ and $F_2$, of lines in the plane with $\vert F_1 \vert= N_1$ and $\vert F_2 \vert =N_2$. The families are such that if we pick $g \in F_1$ and $l \in F_2$ we get ...
2answers
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### Intersection degree of noncomplete intersection

Via the theory of hilbert polynomials, I think the following is true: Suppose $X$ is a closed subscheme of $P^n$ of degree $d$, and $Y$ is also such one but is also a complete intersection and also ...
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### Question about Bezout's theorem between varieties and schemes.

Exercise 18.6.K in Vakil's Foundations of Algebraic geometry is: Let $X$ be a projective scheme of dimension $\geq 1$ over a field $k$, with a fixed closed immersion $i : X \rightarrow \mathbb{P}^n_k$....
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### Prove the equation for the point of intersection of two vector lines

This is my first question here and I'm not sure if it's the right place but I'm kinda desperate. I was learning about vector lines and parametric equations. My teacher gave me the assignment below and ...
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### Fulton Ex. 3.2.22: Counting conics intersecting 8 lines in 3-space

I'm trying to understand Fulton's approach to the question how many plane conics in $\mathbb P^3$ meet 8 general lines. Suppose $V$ is a 4-dimensional vector space, and $\mathbb P^3 = \mathbb P(V)$. ...
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### Theorem I.7.5 (Hilbert-Serre) Hartshorne uniqueness of Hilbert polynomial.

I don't understand why the Hilbert polynomial is unique. In the theorem I.7.5 we find a polynomial $P_M(z)$ such that $\varphi_M(l)=P_M(l)$ for all $l\gg0$ ($\varphi_M(l)=dim_K M_l$ as discribed in ...
1answer
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### When existence of transfer map of algebraic $K$-theory implies rational injection.

Given a map of smooth projective varieties $f:X\rightarrow Y$ over fields, there is a projection formula in algebraic $K$-theory given by $f_*(\alpha.f^*(\beta))=\beta.f_*(\alpha)$. I was wondering ...
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