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 ...
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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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Defining Segre classes of a vector bundle using projective bundle of hyperplanes

Let $X$ be an algebraic scheme over a field and $E$ a vector bundle of rank $e+1$ on $X$. Denote by $\mathbf{P}(E) = \mathrm{Proj}(\mathrm{Sym}(E))$ the projective bundle of hyperplanes in $E$ and by $...
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Length of a module, intersection theory

I have a question to Proposition 7.1 "Intersection Theory", Fulton: What is the length of $A/J$? The length $l_R(M)$ is defined for an $R$-module $M$. Over which ring do I view $A/J$?
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Question about intersecting pullback of invertible sheaves by a projective morphism.

This is exercise 20.1.J in Vakil's Foundations of Algebraic Geometry. Let $X$ be a projective scheme over a field $k$, and $F$ be a coherent sheaf with $supp(F)$ proper and $dim(supp(F)) \leq n$. Let $...
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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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Two functions can only partition neighbourhood into at most 4 path connected sets

I have two $C^1$ functions $f$ and $g$ which go from $\mathbb{R}^{N-1}\rightarrow\mathbb{R}$. For some $k_1,k_2\in\mathbb{R}^N$, I know that locally about a point $x_0\in\mathbb{R}^N$ the sets; $$F=\{\...
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How to plot $S:=\{(\Re (x^*Ax),\Im(x^*Ax))\mid\|x\|_2=1,\|Ax\|_2=1\}$?

Given $A\in\mathbb{C}^{n\times n}$, denote set $$S_1:=\{(\Re (x^*Ax),\Im(x^*Ax)) \mid\|x\|_2=1\}.$$ This set is called numerical range and it contains all eigenvalues of $A$. I found a program that ...
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Intersection multiplicity for curves over a DVR

I'm interested in relating the multiplicity of the intersection of curves defined over a local to the corresponding curves in the residue field. Let $A$ be a DVR with maximal ideal $m$ and $f_1,f_2 \...
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Possible dimensions of $V \cap W$ in $\mathbb{R}^n$

If $V$ and $W$ are two dimensional subspaces of $\mathbb{R}^n$, what are the possible dimensions of $V \cap W$ if; $V$ and $W$ have the same dimensions? (for example; $\dim(V)=2, \dim(W)=2, n=3$) $V$ ...
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Intersection index in complex projective plane

Let $Z(f)$ be the zero set of a degree $d$ homogeneous polynomial $f\in\mathbb{C}[x_0,x_1,x_2]$, then it defines a submanifold in $\mathbb{CP}^2$. How can we compute its intersection index with the ...
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Motivation for quantum cohomology rings

I can't seem to find a good source for the motivation for defining the big quantum cohomology ring with its quantum product. Collecting the Gromov-Witten invariants in a generating function seems like ...
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Intersection module 2 exercise

They ask me: Be $X,Z$ compact manifolds and $Z \subset Y$ with dim $X$+dim $Z$ = dim $Y$, $i: Z \to Y$ the inclusion and $f: X \to Y$, prove $I_2(f,i) = I_2(f,Z)$ Well I don't know how can I prove ...
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Generically reduced preimage implies generically transverse intersection

In Eisenbud and Harris' 3264 and all that, part of their statement of Kleiman's theorem is the following: Suppose that an algebraic group $G$ acts transitively on a variety $X$ over an algebraically ...
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Nef divisor $D$ in an elliptic surface, $D\cdot F=0$ for a fibre $F$, then $D\equiv nF$

Let $X$ be a rational elliptic surface over $\Bbb{C}$. It is well known that $-K_X$ is linearly equivalent to a general fiber $F$. I'm trying to prove/disprove the following: let $D$ be an effective, ...
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If two curves have the same tangent at $p$, then $\operatorname{mult}_p(C_1 \cap C_2) > \operatorname{ord}_p(C_1) \operatorname{ord}_p(C_2)$

I have to prove that, if two curves $C_1$ and $C_2$ have the same tangent line at point p then: $\DeclareMathOperator{\mult}{mult}$$\DeclareMathOperator{\ord}{ord}$ $$\mult_p(C_1 \cap C_2) > \ord_p(...
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Find projective transformation

Given four projective lines $L_1,L_2,L_3,L_4$ in projective plane, such that no three lines intersect in the same point and another four lines $M_1,M_2,M_3,M_4$ such that any three also do not ...
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Determine whether two circles intersect?

Given two circles with the same radius, how do I figure out whether or not they intersect? I am only given the integer coordinates (x,y) for the centre of each circle.
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Curves sharing points over finite fields, and their mutual divisibility

Consider in $\mathbb{A}^2(\mathbb{F}_q)$ two $\mathbb{F}_q$-rational curves $\mathcal{X}:f(x,y)=0$ and $\mathcal{Y}:g(x,y)=0$, and let $\mathcal{Y}$ be absolutely irreducible. Suppose also that $\...
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Angles of an ellipsoid intersecting a plane

I have the following two sufaces: A not-rotated and not-translated ellipsoid, i.e. whose axes are parallel to the axes of the coordinate system and whose center is at (0,0,0): $$ S1:\quad\frac{x^...
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Area moments of parts of a rectangle divided by a line

I want to calculate the areas (and possibly their first and second moments) of a rectangle split into two parts by a line. It would be very helpful for me if there're closed form expressions for this ...
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Reference request: cohomology ring of flag varieties

Just when I started understanding the basics of Schubert calculus and how the cohomology ring of Grassmannians $G(k,n)$ works, I figured I needed a generalization in terms of (partial) flags. My goal ...
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Degree of a subvariety of the Grassmannian

This might be a stupid question due to my lack of knowledge in algebraic geometry. So I have a subvariety $V\subset G_k(\mathbb{C}^n)$ of codimension $k$. The only thing I know about $V$ is its ...
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Calculating intersection multiplicities explicitly with Serre's formula

Let's say I want to explicitly calculate the intersection multiplicity of two subvarieties of $\Bbb A^n_k$ using Serre's Tor-formula (involving as little homological algebra as possible). A typical ...
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Coordinate ring of irreducible component

In the proof of affine dimension theorem (you can see the proof I am following here), Hartshorne assume that $A(Y)/\mathfrak{p}$ is the coordinate ring of the irreducible component $W$. Why is this ...
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Intersection of linear varieties in $P^n$

Suppose $Y,Z$ are linear varieties in $P^n$, with $dimY=r$ and $dimZ=s$, then if $Y \cap Z \neq \varnothing$, then $Y\cap Z$ is a linear variety of dimensión $\geq r+s-n$. How can I prove that? I know ...
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Intersection of two varieties need not be a variety

I am introducing my self to algebraic geometry, and by now I am reading Algebraic Geometry by Robin Hartshorne. In the exercise $2.16$, I have to prove that the intersection of two varieties need not ...
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Intersection number of complex curves in a complex surface

Suppose $C_1,C_2$ are embedded complex curves in a complex surface $S$, and $C_1,C_2$ have no common component. Assuming $C_1$ and $C_2$ intersect transversally, the intersection number $C_1\cdot C_2$ ...
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Simplifying intersection of 3 sets when 2 are independent

I am working on a problem involving the probability of the intersection of 3 sets: $P(A \cap B \cap C)$, and am wondering if this can be simplified at all, since we are also given that the sets $A$ ...
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Minimizing cross section area of the intersection between a plane and mesh

I have a 3D triangular mesh and a cutting plane. I want to minimize the area of the cut intersection by rotating the normal of the plane. The image shows the cross section of the cut and the mesh. ...
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Signature of self-intersection submanifold

Let $S\subset M$ be a (complex) codimension $1$ submanifold. $S\cdot S$ denotes the self-intersection manifold of $S$ in $M$. Then it seems that $\text{Sign}(S\cdot S)$ (the signature of $S\cdot S$) ...
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Intersection of two submanifolds with dimension condition

Consider a manifold of dimension $n$, we have two submanifolds $A$ and $B$, with dimension $n_1$ and $n_2$ that satisfies $n_1+n_2=n$. Can we conclude they only intersect at finite points (or isolated ...
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Class of the graph of the Segre embedding in the Chow ring (Eisenbud & Harris, 3264 and all that, Exercise 2.26)

I refer to Exercise 2.26 in the Eisenbud-Harris book "3264 and all that". Work over an algebraically closed field of characteristic zero. Let $\sigma:\mathbb{P}^2\times\mathbb{P}^2\...
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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 ...
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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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For complex subspace $S$, there are vectors $u,w ∈ S$ such that $u\neq zw,\ \forall z\in \mathbb{C}$

The official question reads: Suppose U and W are each 5-dimensional subspaces of $\mathbb{C}^8$. Prove that there are two vectors $u,w ∈ U∩W$ $\hspace{1mm}$ s.t. $\hspace{1mm}$ $u≠zw$ $\hspace{...
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The complement of a neighborhood of a divisor.

Given a projective variety $X$ and a hyperplane section $H$ (intersection of some hyperplane in the ambient projective space with $X$). Is it true that complement of any neighborhood of $H$ is a ...
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How can I use Grothendieck--Riemann--Roch theorem?

For my research, I'm trying to understand how the Grothendieck--Riemann--Roch theorem is used in the paper The Birational Geometry of the Hilbert Scheme of Points on Surfaces by Aaron Bertram & ...
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Intersection form of $\mathbb P^1 \times \mathbb P^1$

This comes from Vakil 20.2 C, self-study. We have to show the intersection form of $X = \mathbb P^1 \times \mathbb P^1$ is given by $l \cdot l = m \cdot m = 0$ and $l \cdot m = 1$, where $l$ is the ...
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Formula of intersection of 2 points with the x axis

I'm trying to figure out how to get the point x = 3 : What's given here are the points S and G . (Assuming the 2 angles are equal) Apparently, we can assume that the ball does not bounce off the x-...
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Why one cannot equate a sphere and a plane but a sphere with a sphere? [closed]

This is a general question about the intersection of a sphere with a plane or sphere which is confusing me. To find the intersection between two spheres K1 and K2, you can equate them, solve the ...
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Scheme theoretic intersection vs intersection of cycles

Let $X$ be a scheme for which we have an appropriate moving lemma to define an intersection product of cycles. For example a smooth projective variety. Then if $V$ and $W$ are properly intersecting ...
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Local geometry near a transverse intersection of two Riemann surfaces

Suppose $C_1$ and $C_2$ are two Riemann surfaces (embedded as submanifolds in some ambient space, say, $M$) which intersect transversally at a point $p_1 \in C_1$. How does one see that near $C_1$, ...
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How to count constraints in algebraic geometry?

I am interested in where to look to find techniques to approach the following problem: Let $M$ be a smooth projective manifold, and let $S \subset M$ be some submanifold. Consider on $M$ a complete ...
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Chow ring of projetive space

The Chow ring of $\mathbb{P}^n$ is $\mathbb{Z}[H]/H^{n+1}$, where $H$ is the linear equivalence lass determined by the zero-set of a linear functional. I have a few (very basic) points of confusion: ...
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Proving that intersection number of divisor is 0

My question arises from proposition 8.3 in the algebraic geometry text by Shigeru Iitaka: Let $f:W\rightarrow S$ be a surjective morphism between nonsingular projective surfaces. Let $E$ be a divisor ...
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Polynomial equivalence

Given a variety $X$, rational equivalence gives an equivalence relation on $Z_i(X)$ leading to the Chow groups and used in the Grothendieck-Riemann-Roch theorem. I was wondering about replacing $\...

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