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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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(...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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=...
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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 ...
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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)+\...
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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 $(\mathbb P^n)^\vee$, ...
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“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"(...
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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 ...
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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,...
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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^\...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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{...
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Section of Hirzebruch Surface Semi Ample

We consider the Hirzebruch surface $S = \mathbb{P}(\mathcal{E})$ with locally free sheaf $\mathcal{E} = \mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(2)$. By definition of ...
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Divisor with positive Selfintersection Number Semi Ample

Let $X$ be a surface (therefore $2$-dimensional, proper $k$-scheme) and $D$ a divisor with positive self intersection number $(D \cdot D) >0$. Futhermore it is nef therefore for each irreducible ...
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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 $...
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Why is : $ g \circ f = ( \mathrm{pr}_{XZ}^{XYZ} )_* \big( ( \mathrm{pr}_{XZ}^{XYZ} )^* . \mathrm{pr}_{YZ}^{XYZ} )^* g \big) $?

Let $ \mathcal{E} (k) $ be the enumerative category of smooth projective varieties over a field $k$. This is a $ \mathbb{Q} $ - linear category which has for objects : smooth projective varieties over ...
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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 ...
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Every cycle class a Chern class?

I am currently learning intersection theory of smooth algebraic varieties and I have the following question. Let $X$ be a smooth projective variety and $\mathcal{F}$ a vector bundle on $X$. Then the $...
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Proving non-centered at the origin ellipsoid A is a subset of ellipsoid B

I would like to prove an ellipsoid $E_a$ center at $q_a$ with shape matrix $A$ and radius $r_a$ defined as: $(x-q_a)^T A^{-1} (x-qa) <= r_a$ is included into another ellipsoid $E_b$ -i.e. $E_a$ ...
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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$.
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Can two projective surfaces intersect in points only?

Let $S_1,S_2\subset \mathbb P^n$ be two algebraic surfaces (we may assume that $n=3$). Is it possible that $S_1\cap S_2=\{x_1,\dots,x_N\}$ a finite set of points? I can imagine the surfaces two be ...
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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 ...
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Degree of curve where matrix of polynomials has rank 1

My question is about a step in Exercise 12.8 on page 442 of 3264 & All That by Eisenbud and Harris. Chapter 12 is about Porteous' formula. The exercise reads: Let $A=(P_{i,j})$ be a $2 \times 3$ ...
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Fulton, example 3.2.16

In his "Intersection theory" book Fulton proves in lemma 3.2 the following fact: if a $E$ is a filtered vector bundle of rank $r$ over $X$ with quotients line bundles $L_i$, $s$ is a section of $E$ ...
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Plane intesection with vertical vector given plane normal

There is a plane defined by a normal and an origin. For simplicity's sake, the origin is $(0,0,0)$. And then, there are two coordinates ($x$ and $z$) of a point on this plane. How can I find the ...
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Can a curve intersect 0 to inf without crossing inself?

Can you draw a curve such that it intersects every x at y=0 without ever crossing over itself (that is to say, without hitting the same value twice)? Wouldn't accomplishing this feat mean infinitely ...
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Massey product used to show that Borromean rings are linked

I'm trying to understand an example in "Elements of Homology Theory" from V.V. Prasolov (p. 85-88) where he shows that the Borromean rings represented by three spheres $S_1, S_2, S_3$ in $S^3$ are ...
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Chern classes of tangent bundle over the Grassmannian G(2,4)

What are the Chern classes of the tangent bundle $\tau_G$ of the Grassmannian $G=G(2,4)$ of lines in $\mathbb{P}^3$? This is Exercise 5.37 on page 191 of 3264 & All That by Eisenbud and Harris. ...