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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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References and useful results on continuous one-parameter intersection of algebraic surfaces

Consider a one-parameter family of polynomials $\{P_t\in \mathbb{R}[X,Y]\}_{t\in I}$ and a continuous curve $\gamma:J\to \mathbb{R}^2$. Suppose that $$P_t(\gamma(s)) =0, \quad \forall (t,s)\in I\times ...
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how to find the points of intersections between the two complex function

Given two complex functions in one cartesian coordinate system, how to find the points of intersections between the two complex functions?
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A Lemma from Bloch's 1986 paper "Algebraic Cycles and Higher K-Theory"

I am trying to understand the idea and the proof behind Lemma 1.2 of Bloch's paper (link) Here's what it states: Let $G$ be a connected linear algebraic group acting on a quasi-projective variety $X$ ...
user6's user avatar
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Consequences of intersection product equal to $0$

We work over $\mathbb{C}$. Let $X$ be a smooth projective variety, let $D$ be a nef prime divisor and let $C$ be a smooth irreducible curve. I know that, if $D\cap C=\emptyset$, then $D\cdot C = 0$. ...
konoa's user avatar
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Successive quotients of the maximal ideal of the stalk of a variety

If we are given an irreducible variety $X$ of dimension $p$ and a regular point $x\in X$, call $A=\mathcal{O}_{X,x}$ its stalk and $\mathfrak{m}$ the maximal ideal and $k=A/\mathfrak{m}$ the residue ...
Simon Pitte's user avatar
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Self intersection number of diagonal

Suppose I have an elliptic curve $E$. How would I calculate the self intersection in $A_*(E \times E)$ of the diagonal $\Delta$? It seems the formula I need to use is $\Delta . \Delta = c_1(N_{\Delta/...
Slim Shady's user avatar
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For cubic surface, if $\dim(\operatorname{Sing}(X))\geq 1$ then a line is contained a in $\operatorname{Sing}(X)$

Let $X$ to be an irreducible cubic surface in $\mathbb{P}^3_{\mathbb{C}}$. Is it true that if $\dim(\operatorname{Sing}(X))\geq 1$, then a line is contained in $\operatorname{Sing}(X)$? I.e, does a ...
ben huni's user avatar
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Confusion about codimension of a subvariety of a scheme

In Eisenbud's and Harris's "3264 & All That", they define the codimension of a subvariety $Y$ of a variety $X$ as $\operatorname{codim}_X(Y)=\dim(X)-\dim(Y)$. This part is fine and also ...
Anon's user avatar
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Question about Fulton’s “Intersection Theory”, example 8.4.6

In Fulton’s aforementioned book, after stating Bezout’s theorem, he states that a classical application of it is to show that an irreducible projective variety $X\subseteq\mathbb P^n$ of dimension $m$ ...
Simon Pitte's user avatar
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How does an intersection survive through (generic) perturbation?

I am looking for the proof of a folklore statement which I know (or heavily suspect) to be true, but haven't been able to find written down yet. I have a (symplectic) manifold $M$ of dimension $2n$, ...
Azur's user avatar
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How to compute $\operatorname{Length}(k[[x_1,\dots,x_n]]/I)$ for some ideal $I$

Let $k$ be an algebraically closed field and $\mathcal{O}_0=k[[x_1,\dots,x_n]]$ be the ring of formal power series, $I$ be an ideal of $\mathcal{O}_0$ such that $\operatorname{Spec}(\mathcal{O}_0/I)$ ...
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A question on Euler characteristic

Let $X$ be a smooth projective variety and $D$ be a Cartier divisor on $X$. Let $\mathscr{F}$ be a locally free sheaf on $X$. I think the following equality of Euler characteristic is valid $$ \chi(mD,...
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Intersection of superellipse with line

My goal is to do a stereographic-like projection of the plane but on a $L_p$ sphere and with the projection between the pole and the center of the sphere. For that I begin with 2D stereographic-like ...
Mehdi MABED's user avatar
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Finding intersection between straight line and spherical line

I'm trying to find the intersection between two functions. The first function describes the red straight line in the figure: $$\tan\epsilon_1=\frac{d(z)-d_{01}}{z-z_{01}}$$ The second function ...
JoeMama's user avatar
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Understanding why set theoretic intersection is not necceraly a complete intersection

If it is true that for projective varieties one can show that: $Z(f_1, f_2) = Z(f_1)\cap Z(f_2)$ for any homogenous polynomials, than why isn't true than any set theoretic complete intersection of ...
Joe's user avatar
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Inverse direction of Hodge index theorem

The Hodge index theorem states that the intersection matrix associated to curves on a smooth algebraic surface has a specified signature---namely, if the intersection matrix has size $n \times n$ then ...
Harry Richman's user avatar
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For prime divisors $V,W\subseteq X$ in a smooth threefold $X$ and and integral curve $C\subseteq V\cap W$ with $i(V,W;C)>1$, do we have $V.C=W.C$?

Let $X$ be a smooth projective variety of dimension $3$ over an algebraically closed field. Let $V,W\subseteq X$ be prime divisors, i.e. two integral closed subvarieties of dimension $2$. Let $C\...
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Locus of 4 colinear points in $\mathbb{P}^2$

I'm trying to solve exercises 2.34 and 2.35 in 3264 and all that (Eisenbud & Harris, 2016, p.80). Exercice 2.34 Let $\varphi\subset(\mathbb{P}^2)^4$ be the locus of 4-tuples of colinear points. ...
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Cycle class of a smooth complete intersection $X \hookrightarrow \mathbb P^n$

Let $f:X \hookrightarrow \mathbb P^n$ be a $(d_1,\ldots ,d_r)$ smooth complete intersection over an algebraically closed field $k$. Let $\ell$ be a prime number different from the characteristic of $k$...
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Analytical Method for Finding the Closest Point on a 3D Quadrilateral Polygon Face from a Line Segment

I am interested in developing an analytical method to determine the closest point on a convex quadrilateral polygon face, defined by four points (A, B, C, and D), from a given line segment connecting ...
thi's user avatar
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Circumradius of the intersection of two regular simplexes

For $n \geq 2$, let $\Delta^n$ be a regular $n$-dimensional simplex in $\mathbb{R}^n$ centered at the origin $0$ and inscribed in the unit sphere $\mathbb{S}^{n-1}$. Let $v_0,\dots,v_n \in \mathbb{S}^{...
Paul Tristant's user avatar
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Bertini-type theorem for reducedness

Let $f_1, ... f_r$ be homogeneous polynomials in $k[x_0, ... x_n]$ such that $X = Proj(k[x_0, ... x_n] / (f_1, ... f_r))$ is reduced and has pure dimension $d$. Is there an open set $U$ of $\mathbb{P}^...
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How to explicitely reduce the expression of the intersection of plan and sphere from 3D to 2D?

If you have for example a plan ($\mathcal{P}$) and a sphere ($\mathcal{S}$), let say : $$(\mathcal{P}) \enspace \enspace \enspace \enspace z= \frac{1}{2} $$ $$(\mathcal{S}) \enspace \enspace \enspace \...
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Definition of Schubert Variety

Let $V$ be a full flag, $\lambda$ a partition. Consider $$\sigma_\lambda(V) = \{ \Lambda \in G(k,n): \Lambda \cap V_{n-k+i-\lambda_i} \geq i \}.$$ If you have another full flag $V'$, are $\sigma_\...
Hoji's user avatar
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Parameterizing intersection curve of paraboloids

I'm at a loss for ideas on what to do. I'm given two paraboloids and their equations: $z_1 = x^2 + 1$ and $z_2 = 5 - y^2$. I know their intersection is when $z_1 = z_2$. This gives me $x^2 + y^2 = 4$, ...
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Number of lines on singular cubic surfaces

Bruce and Walls in their paper On the classification of cubic surfaces state at the end the "final observation" that a number of distinct lines on cubic surface with $k$ isolated du Val ...
Alexander Golys's user avatar
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$A_{\mathfrak{p}}$-submodules of the residue field $k = A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}}$?

I was trying to calculate the length of the $A_{\mathfrak{p}}$-module $A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}}$. It seems that the only $A_{\mathfrak{p}}$-submodule must be the zero module, but ...
Anthony Lee's user avatar
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How to find the first intersection point of 2 systems of symbolic equations?

Hopefully this is not a fail. But, before I explain the math, I think it will be easier if I give a background of what I'm trying to do... Imagine 2 line segments randomly defined in a $2D$ room. The ...
proj786's user avatar
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2 votes
1 answer
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Normal bundle of transverse intersection of two irreducible components

Let $X$ be an equidimensional reduced scheme of finite type over an algebraically closed field $k$. Assume that $X$ has two irreducible components $X_1$ and $X_2$. Assume also that $X_1$ and $X_2$ are ...
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Background for Gross-Zagier paper

I have been reading the paper "Heegner points and derivatives of $L$-series" by Gross and Zagier. Link to the paper. In section III of the paper, they use intersection theory to express a ...
Joseph Harrison's user avatar
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Three annulus intersection problem

Recently I faced a problem about intersection of three annuli. Imagine having three annuli same dimensions and you put them next to each other into triangular shape like putting together three circles....
Stefan Vujic's user avatar
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Question in Fulton's Intersection Theory, Theorem 6.7 (b)

I have a question about Theorem 6.7 (b) in Fulton's "Intersection Theory". Notation: $X$ is a regularly embedded sub-scheme in $Y$ of co-dimension $d$. Let $\tilde Y\xrightarrow{f} Y$ be the ...
user6's user avatar
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Reference request: The ring $k[x_1, \dotsc, x_n]/I$ as a limit of functions with distinct zeroes

p. 152-153 of "Using Algebraic Geometry" (2nd. ed) by Cox, Little, and O'Shea says: [S]uppose, that a collection of $n$ polynomials $f_1, \dotsc, f_n$ has a single zero in $k^n$, which we ...
Fred Akalin's user avatar
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Why degree equals to the intersection number with general linear space

Let $X\subseteq\mathbb{P}_{K}^{N}$ be an irreducible projective variety where $K$ is an algebraically closed field. Let $\text{dim}(X)=d$. As in Chapter I of Hartshorne's book, we may define $\text{...
LittleBear's user avatar
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127 views

Schubert classes appearing in the class of certain subvarieties of incidence variety

The above picture comes from Fulton's "Introduction to Intersection Theory in Algebraic Geometry". The variety $I$ is the partial flag variety $F(0,d;n)$, also known as the incidence variety ...
Andrea B.'s user avatar
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Definition of Lefschetz Number in Bott&Tu and in Gullemin&Pollack Differs by a sign?

In Bott and Tu's Differential Forms in Algebraic Topology, the Lefschetz number of a map $f:M\to M$ between an oriented compact manifold $M^m$ is defined, as in any algebraic topology text, to be $L(f)...
Tianyi Wang's user avatar
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How to compute the intersection number of a line on a projective surface

I am trying to understand the example mentioned in this question, but I can't follow the argument Shafarevich is using, so I want to compute an explicit example to understand what's going on. Let $X$ ...
14159's user avatar
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Very ample + effective = ample?

It is well known that there exist divisors (on a normal projective variety over say the complex numbers) that are big (= is a sum of an ample and an effective divisor) but not ample. However, if we ...
Calculus101's user avatar
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Can the intersection multiplicity for plane curves be defined computationally?

Wikipedia says There is a unique function ... $I_{p}(P,Q)$ called the intersection multiplicity of $P$ and $Q$ at $p$ that satisfies the following [six] properties: ... Although these properties ...
Fred Akalin's user avatar
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1 answer
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For a decreasing sequence of nonempty closed convex sets in a Banach space, must the intersection be nonempty?

If ${A_n}$ is a sequence of nonempty closed convex sets in a Banach space such that $A_{n+1} \subset A_n$ for all $n \in \mathbb{N}$, must $\bigcap_{n \in \mathbb{N}} A_n$ be nonempty? My Answer: ...
Proloffc6's user avatar
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Intersection multiplicities of varieties expressed with curves

Let $X,Y$ be different irreducible projective varieties in $\mathbb P^n$ (over an algebraically closed field). Let $Z$ be an irreducible component of $X\cap Y$. Then the intersection multiplicity of $...
quantum's user avatar
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Basepoint free line bundle on Blow up

I am reading lecture note 15 of Vakil's intersection theory course (Ref: https://virtualmath1.stanford.edu/~vakil/245/) Let $\mathcal{L}$ be a line bundle on a scheme (let's make everything easier: A ...
Dick. Y's user avatar
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Point of Intersect between Line Centered in Square

I need to find the point of intersection between some square xy + N,N where N is the size of the square and some line x1,y1,x2,y2...
Kayle's user avatar
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2 answers
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Exercise V.1.5. Hartshorne

Let $X$ be a surface of degree $d$ in $\mathbb P^3_K$, we compute $K_X^2$. We have $S\sim dH$ and $K_{\mathbb P^3}\sim -4H$ for a hyper plane $H$. We have by Hartshorne II.8.20 that $$\omega_X\cong \...
raisinsec's user avatar
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4 votes
1 answer
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Euler characteristic and intersection number [closed]

Let $X$ be a complex projective manifold of pure (complex) dimension $n$. Denote its canonical line bundle by $K_X$. Is there a relation between $\int_Xc_1(K_X)^n$ and the topological Euler ...
Doug's user avatar
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How many intersections does n degree two polynomials have?

Given $n$ polynomials $f_i(x) = a_i x^2+ b_ix+c_i$, where $a_i, b_i, c_i \in \mathbb{R}$. And assume that no two polynomials are identical. The minimum number of intersections is 0 since one can let $\...
peng yu's user avatar
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1 vote
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What is the probability of three union if one of the events has probability zero?

my problem is this: We throw $n$ bells randomly into $3$ boxes initially empty. Compute the probability that at least One boxes remaining empty. I have the solution and that is: $$ P(A_1 \cup A_2 \cup ...
Lisa Tassinari's user avatar
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Intersection between torsion cycles and free cycles

Let's consider a closed and oriented 4-manifold $M_4$ and denote $H_2(M_4,\mathbb{Z})$ as the homology group of 2-cycles and $Q_{M_4}(S_{A},S_{B})$ as the symmetric intersection pairing between 2-...
JQ Skywalker's user avatar
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1 answer
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Confusion about line bundles and the intersection product

Let $X = \mathbb{A}_{\mathbb{C}}^{2}$ and let $Y$ be the blowup of $X$ at the origin. Let $E \cong \mathbb{P}^{1}$ the exceptional divisor. I think that we have a canonical inclusion $\mathcal{O}_{Y} \...
Fraktale Fatalität's user avatar
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1 answer
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Proving a system of quadratic forms has no (non-zero) solutions

A system of homogeneous linear equations always has the solution $ x=(0,\dots, 0) $. Suppose we have a system of $ n $ homogeneous linear equations in $ k $ variables. If $ k > n $ then there will ...
Ian Gershon Teixeira's user avatar

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