Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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)

1
vote
0answers
11 views

plane edge intersections embedded in higher dimensional space

Let's say we have some D-dimensional euclidean space, and we have some circles of dimension 0 to D-1 (circle dimensionality meaning the minimum number of vectors needed to fully define it, so a ...
2
votes
1answer
325 views

Codimension 1 points

Today I was reading a proof of the following Lemma from Liu's "Algebraic Geometry and Arithmetic Curves" Recall: A a point $x \in X$ is called a codimension 1 point if $ \overline{ \{x \}}$ has ...
0
votes
0answers
85 views

Transversality in the Zariski tangent space

I am trying to understand how to check whether two algebraic varieties intersect transversally from a purely algebraic standpoint. Is the following argument correct? Say locally a smooth projective ...
0
votes
0answers
45 views

Chow ring isomorphic to homology ring?

I heard an algebraic geometry professor say that the Chow ring is usually isomorphic to the homology ring for cases we care about in application. However, I cannot find many results about this, beyond ...
1
vote
3answers
2k views

Polygon and line intersection

Does anyone help me with the fast algorithm to determine the intersection of a polygon (rotated rectangle) and a line (definite by 2 points)? The only true/false result is needed.
0
votes
1answer
1k views

Finding a 3rd point in a 3D triangle with known plane, two points and lengths of each side

I have a very similar problem to the below question. right triangle in 3D space, vectors, line intersection? Rather than having the unit vector $A$ I have the lengths $i_2$ to $i_3$ and $i_1$ to $...
0
votes
0answers
17 views

How this inequality works to know a point cross a line or not?

First of all, sorry for my bad english.. but i really want to know how this works. as you see in the picture, there are 3 points and if leftside is greater than rightside, It means the point p0 is ...
0
votes
0answers
27 views

Degree of Line bundle and Intersection

Let $(\Sigma,g)$ be a closed Riemann surface with metric $g$. For any holomorphic line bundle $L\to \Sigma$, given a metric we have its curtature in terms of Chern connection $A$. It is well-known ...
0
votes
1answer
146 views

What is a general linear subspace?

When it comes to defining a general plane with respect to a line in $\mathbb R^3$, I can think of this definition as: take any plane not containing the line. Reading Fulton's "Young Tableau" I can't ...
0
votes
2answers
379 views

collision point of circle and line

I'm trying to figure out the collision point of the circle and a line, ultimately it should work in 3D but for now just in 2D to simplify the problem as much as possible. I've created 2 examples here ...
2
votes
1answer
51 views

Finiteness of intersection numbers

I'm trying to understand Shafarevich's definition of intersection numbers: By definition, "$D_1,...,D_n$ general position at $x$" means that $\bigcap_{i=1}^n\text{Supp}(D_i)$ has finitely many ...
6
votes
1answer
85 views

Modified homotopy and relation with intersection theory.

Denote by $\Gamma$ a hypersurface in $\mathbb{C}^2$, i.e. the zero locus of a polynomial of two complex variables. Denote by $X$ the complement of $\Gamma$ in $\mathbb{C}^2$. I am trying to define a ...
2
votes
1answer
58 views

Notation (intersections) in 3264

I am starting to read 3264 by Eisenbud and Harris and I consistently cannot tell whether by $$A \cap B$$ they mean the scheme-theoretic or set-theoretic intersection. For instance, in their ...
2
votes
3answers
64 views

How to find intersections of a curve and tetrahedra?

I have a curve going through tetrahedron elements. How to find the intersections of this curve with the tetrahedrons? The curve is constructed by a few points and represents the axis of the artery ...
2
votes
2answers
2k views

Properly comparing two histograms

I need to implement a function (in Golang) to compare the similarity/distance of two histograms. The histograms were generated from two different images. I have searched on the internet and have found ...
2
votes
0answers
35 views

Understanding the proof of the fact that the Chow group of a scheme $X$ is graded by dimension.

I would like to understand the proof of this fact: If $X$ is a scheme (separated, of finite type over $k=\overline{k}$) then the Chow group of $X$ is graded by dimension; that is, \begin{equation} A(...
1
vote
0answers
32 views

Cycle associated to a closed subscheme

Let $X$ be an algebraic variety (i.e. an integral $k$-scheme, such that $X \to \mathrm{Spec \;}k$ is separated and of finite type). Let $Y$ be a closed subscheme and $Y_1, \dots Y_n$ the irreducible ...
0
votes
0answers
52 views

Rigid curves and reducibility

On a projective complex surface, a curve $C$ may sit in a linear equivalence class $[C]$ that is given as a sum $[C] = [C_1] + [C_2]$, where $C_1$ and $C_2$ are effective. In general the curves in the ...
2
votes
0answers
54 views

Negative intersections between distinct curves: geometric picture?

On a smooth projective complex surface, if $C$ and $C'$ are distinct irreducible curves then their intersection is non-negative, $C \cdot C' \geq 0$. I am interested in cases where negative ...
0
votes
0answers
16 views

Sampling intersection of multiple polynomials

I have a set of 20 multivariate polynomials in 5 dimensions $\big(f_i(x_1,x_2,x_3,x_4,x_5)=0\big)$. They are all 6th degree in each dimension. I am looking for a method to sample the intersection of ...
1
vote
1answer
14 views

Example of a tangent line intersecting with multiplicity three in a smooth point

In these notes it is said that a tangent line to a smooth point $p$ of a curve $C$ can be characterised as the unique line $L$ such that $mult( L\cap C,q_0)\geq 2,$ where mult means intersection ...
3
votes
0answers
60 views

On the proof of the Whitney trick (from Scorpan's book)

I'm trying to study a proof of the h-cobordism theorem from Scorpan's "The wild world of 4-manifolds". Given a handle decomposition for the cobordism, the Whitney trick is used to eliminate every pair ...
1
vote
1answer
53 views

Self intersection and cohomology of boundary of tubular neighbourhood

Let $X$ a compact orientable manifold of dimension $2n$ and $Y$ a compact submanifold of dimension $n$. Further let $U$ a tubular neighbourhood of $Y$. When I did some calculations I got the ...
1
vote
0answers
23 views

Intersection number via tangent spaces

Assume that finite groups $G_1$ and $G_2$ act smoothly on a manifold $M$ in such a way that the fixed point set, $M^{G_1\cap G_2}$, is an oriented closed manifold, $M^{G_1}$ and $M^{G_2}$ are its ...
1
vote
1answer
54 views

Understanding notation in Fulton's Intersection Theory

In Fulton Intersection Theory Second Edition there is the definition of "degree" of a zero cycle. I am referring to Definition 1.4 where he says If $X$ is a complete scheme and $\alpha=...
1
vote
0answers
31 views

singular intersection only comes from tangent?

Let $X$ be a smooth hypersurface in $\mathbb {CP}^n$, $H$ be a hyperplane. If $X\cap H$ is singular, is it true that $H$ is a tangent plane at some point of $X$? Note the converse is always true, if ...
0
votes
0answers
21 views

Intersection Theory in the projective space

Let $X$ be an irreducible projective variety embedded in $\mathbb{P}^N$. Let $V,W\subseteq X$ be two irreducible subprojective varieties intersecting properly in $X$, that is $$\mathsf{codim}_X(V)+\...
4
votes
0answers
51 views

Singular locus of dual hypersurfaces

Everything is over field $\mathbb C$. Let $X$ be a hypersurface of degree $d$ in $\mathbb P^n$. We know that if $X$ is smooth, then its dual $X^\vee$ is still a hypersurface in $(\mathbb P^n)^\vee$, ...
4
votes
1answer
144 views

Locus of tangent lines to a smooth curve of degree $d$ and genus $g$

Suppose $C\subseteq\mathbb{P}^3$ is a smooth curve of degree $d$ and genus $g$ (let's say we are working over $\mathbb{C}$). Let $T(C)$ be the locus of tangent lines to $C$. In other words, $$ T(C) = \...
1
vote
1answer
60 views

“intersection number” of surface with boundary

Consider in an $2n$ dimensional manifold $M$(compact, smooth), for embedded $n$ dimensional surfaces with boundary, denoted by $\Sigma$ and $\Sigma'$, in M, we can consider "intersection number"(...
1
vote
1answer
97 views

Intersection number and cup product

It is known that for a closed oriented smooth manifold $M$, if A and B are oriented submanifolds of $M$, and if A and B intersect transversely, then the Poincare dual of A ∩ B is the cup product of ...
2
votes
1answer
215 views

Intersection of Schubert cycles

I want to compute intersection of the Schubert cell $\sigma_{(3,0)}$ with all the cells $\sigma_{a_1, a_2}$ in the grassmanian $G(2,5)$. I am not sure I am doing correctly but I can't see my mistake. ...
1
vote
0answers
48 views

Automorphisms of generic hyperplane sections

Let $X\subset \mathbb {P}^n=\mathbb {CP}^n$ be a smooth hypersurface of degree $d$, $\{H_\lambda\}_{\lambda\in {\mathbb P^n}}$ be the set of hyperplane sections of $X$. We exclude the case $(d,n-2)=(4,...
2
votes
1answer
253 views

Schubert class in the Grassmannian G(3,6)

How to compute the Schubert class $\sigma$$^2$$_2$$_1$ in the Grassmannian G(3,6)? I remember the result is $\sigma$$_3$$_3$ + 2$\sigma$$_3$$_2$$_1$ + $\sigma$$_2$$_2$$_2$.
3
votes
1answer
308 views

Intersection of the Irreducible Components of Intersections of Schubert Varieties

Let $K$ be an algebraically closed field and $G$ be the Grassmannian of $k$ planes in some $l$ dimensional vector space $V$ over $K$. Let $V_1\subsetneq ... \subsetneq V_l$ be a flag for $V$. A ...
5
votes
0answers
249 views

Classes of Schubert Cycles form a basis

I am reading the not yet published book of Eisenbud and Harris about intersection theory (http://isites.harvard.edu/fs/docs/icb.topic720403.files/book.pdf) and I don't quite understand the following: ...
1
vote
0answers
57 views

Every Schubert cycle a Chern class?

Consider the Grassmann variety $\mathbb{G}(k,n)$ and its Chow ring $A$. It is known that the classes of Schubert cycles form a $\mathbb{Z}$ basis of $A$. Is it known which of these Schubert cycles can ...
9
votes
2answers
233 views

How does 11 split in the ring $\mathbb{Z}[\sqrt[3]{2}]$

I learned that the splitting of primes in a number field $K = \mathbb{Q}(x)/p(x)$ depends on the factorization of $p(x) \pmod p$. While this is not at all obvious to me, let's use it: $$x^3 - 2 \...
3
votes
1answer
91 views

rational points on the quadrifolium $(x^2 + y^2)^3 = (x^2 - y^2)^2$

I have been reading the Wikipedia page on the Quadrifolium there are two of them: \begin{eqnarray*} r &=& \sin 2\theta \\ (x^2 + y^2)^3 &=& 4 x^2 y^2 \end{eqnarray*} and it's $45^\...
0
votes
0answers
26 views

Prove that a function from R to the unit circle is a local diffeomorphism.(2.4.8 G&P)

In order to prove the existence of the function $g$ in the question I want to proof that the following function is a diffeomorphism (I was told a hint that it is a diffeomorphism): $$p(t) = (\cos t, ...
0
votes
0answers
36 views

A difficulty in understanding the proof of boundary theorem in G&P.

The theorem and its proof is given below: But I could not understand the last line in the proof in particular: Why $F^{-1}(Z)$ is a compact one dimensional manifold with boundary? And why this leads ...
0
votes
0answers
21 views

A difficulty in understanding a case in the intersection theory mod 2(p.80 Guillemin and Pollack)

The following is written just before the boundary theorem in Guillemin & Pollack : But I see that if Z is not transversal to X this is not true,why the book did not consider this case? why the ...
3
votes
1answer
58 views

Euler characteristic of matrix manifolds

I'm reading through examples of computing Euler characteristic of manifolds. I know how to compute it for generic manifolds like sphere and torus. But what about matrix manifolds? I'd like to know how ...
1
vote
0answers
49 views

Intersection number for projective plane curves

Context: In our lecture about algebraic geometry we defined the intersection number of two algebraic curves $F,G$ to be $dim_k (\mathcal{O}_P (\mathbb{A}^2)/(F,G))$. Then we proved that it satisfies ...
2
votes
1answer
43 views

Any hyperplanes is covered by non-Lefschetz pencils?

Let $X\subset \mathbb P^n$ be a smooth hypersurface over base field $\mathbb C$. A pencil of hyperplanes is just a projective line $(X_t)$ in $\mathbb P^{n*}$. It is called a Lefschetz pencil if it ...
0
votes
1answer
129 views

How can one show that an elliptic curve has a point whose tangent line also meet another point of the curve

I'm coming from a projective setting of a smooth cubic plane curve over a field $K$ and want to show that I can bring it to the Weierstrass long form. The usual method is to start with a point and ...
7
votes
2answers
268 views

Intersection Theory on a Surface

I have some problems to prove the exercise 20.2.A part (b) in Ravi Vakil's "Fondation of Algebraic Geometry". Here the excerpt: The setting is: We have a surface $X$ (therefore 2-dimensional, proper $...
0
votes
1answer
59 views

General statement about how many lines in Euclidean space will determine a line

It is easy to see that in $3$-dimensional Euclidean space, given $4$ lines in general position, there exists precisely one line who intersects with each of the $4$ lines. We call the $4$ lines ...
1
vote
0answers
86 views

Question about adjunction formula $K_Y = (K_X + Y)|_Y$

I've learnt that if $Y\subset X$ is a smooth subvariety of codimension 1 then the degree of canonical divisors of $Y$ and $X$ are related by \begin{equation} \deg K_Y = \deg((K_X + Y)\cap Y). \end{...
2
votes
1answer
259 views

Use Quadrilateration to locate a point

I'm having lots of trouble with this. At first I thought the problem was a matter of equating four spheres to find their one common point, i.e. point of intersection. I've looked up lots of things on ...