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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Intersection of divisor and curve on a subvariety
Let $X$ be a normal $\mathbb Q$-factorial variety (irreducible) over an algebraically closed field $k$ of characteristic $0$. Let $D\subseteq X$ be an irreducible divisor (which must be $\mathbb Q$-...
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Fundamental cycle
Let $X$ be a seperated scheme of finite type over a field. We define the fundamental cycle of an equidimensional subscheme $Z \subset X$ with irreducible components $Z_1, ..., Z_r$ to be the $d$-cycle
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Order of a function on a zero locus
Let $X$ be a scheme (=separated scheme of finite type over a field). For any closed subvariety (=closed integral scheme), we define the order function via
$$
\text{ord}_Z: k(X)^\times \to Z, \quad \...
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Some example in 3264 and all that
I have started to read 3264 and all that from Harris and Eisenbud.
In particular I was eager to understand the concrete examples in it.
I'm at a loss to understand something that must be very basic ...
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Calculating the closest point a line intersects with a circle on a world map?
I am working with the world map and trying to figure out an equation to solve:
Given starting point A which is latitude 51.137933 and longitude -0.267017, I want draw a straight line or head in the ...
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Some intuition behind the properties of intersection number.
In the study of algebraic curves,we define intersection number of $F$ and $G$ at a point $p$ to be $I(F\cap G,p)=\dim_K(\mathcal O_p(\mathbb A^2)/\langle F,G\rangle$.But it is a rather unintuitive ...
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Find a divisor by local intersection multiplicities with another divisor
Let $X$ be a projective surface. Let $D$ be a Cartier divisor which is locally defined by $f$ at a point $P$. Let $Z$ be a closed subscheme of $X$ only support at point $P$, and locally its ideal ...
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Exact sequence of Chow groups for a finite birational morphism
In Fulton's Intersection Theory, Example 1.9.5. Let $f:X'\to X$ be a finite, birational morphism of $n$-dimensional varieties. For each codimension one subvariety $V$ of $X$, let $d(V)$ be the ...
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Complete Intersection of Hypersurfaces are Fano varieties
I am currently studying Algebraic Geometry, and the wikipedia page of Fano Variety says the following "a smooth complete intersection of hypersurfaces in n-dimensional projective space is Fano if ...
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Relative Intersection Pairing (Poincaré-Lefschetz duality)
Let $M$ be a $2$-dimensional orientable compact manifold. (For example, a Riemann surface.) We have an intersection pairing
$$H_1(X,\def\Z{\mathbb{Z}}\Z) \times H_1(X,\Z)\to \Z$$
which is unimodular, ...
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Bezouts Theorem generalization to n dimensions proof reference
For my bachelors thesis in the context of total degree homotopies I am looking for a reference for a Bezout type theorem like this:
Let $V\subseteq \mathbb{C}^n$ be a complex algebraic set of ...
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Reference request: Intersection number using derived tensor product
I just learned there is a version of definition of intersection number using derived tensor product to avoid the moving lemma. I only know intersection theory as presented in Fulton's book. Does ...
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Find polygon after translate/cut process, starting from rectangle
I have a geometrical question, but if need be I mention that I use Qt so I can do some cool geometric stuff if needed, like QPolygon::intersected(QPolygon)
For more ...
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Hyperplane section rationally equivalent to not lying on a hyperplane
Let $X \subseteq \mathbb{P}^n$ be a hypersurface and $L \subseteq \mathbb{P}^n$ be a generic plane of dimension $m-1 < n$. Then $X \cap L$ is a subvariety of dimension $n-m$ lying on the plane $L$.
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Efficiently Determining Surface Intersections Along a Line Segment
Background
In general, I know how to determine the points of intersection between a surface and a line. In my case, I may have a large number of defined surfaces that may (or may not) intersect each ...
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How to determine the 3d shape formed by intersecting planes
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How can I optimally get the list of vertices for the 3d shape (a convex hull) formed by a set of intersecting planes in 3d space that contains all feasible solutions. Assuming that a 3d shape ...
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Historical proof of Bézout's theorem
Bézout's theorem says that given $f, g$ homogeneous polynomials over $\mathbb{CP}^2$ with degrees $n, m$ respectively, the number of intersections of their zero sets is exactly $nm$, counted with ...
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3264 Example: X non-smooth gives dimensional transversality with intersection in reduced point, but not generic transversality
In Eisenbud-Harris' 3264 and All That, on page 33 they state: Subschemes $Y$ and $Z$ of $X$ have generic transversality iff dimensional transversality and each connected component of $Y\cap Z$ has a ...
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Difference between L∩T∩R and (L|T∩R)?
I am trying to understand conditional probability. In this numerical, L is the event of being late to office, T is the event of heavy traffic and R is the event of raining.
I can solve this problem ...
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A small question in the generalization of a enumerative problem (10.4) in Fulton's intersection theory?
This is the Step 1 in section 10.4 of W. Fulton's book Intersection Theory (Page 189).
For $I\subset\mathbb{P}^2\times\check{\mathbb{P}}^2$ as $\{(P,L):P\in L\}$, we have $I=P(E):=\mathbb{P}(E^{\vee})...
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An intersection-theoretic criterion for Moishezon surfaces by Sakai
I have found myself in a situation where I need to show that a particular normal analytic surface is (projective) algebraic. I am trying to use a result by Brenton's 1977 paper Some Algebraicity ...
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Cylinder Triangle Intersection
so i have a personal project, for which i need an algorithm which tells me if a given Cylinder and given Triangle intersect with each other. Along these lines;
Thats an intersection.
This is also a ...
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Intersection of schubert varieties
Let $L_1$ and $L_2$ $\in$ $\mathbb{P}^4$ be two planes that intersect in exactly one point $Q$. Let $P_1 \in L_1$, $P_2 \in L_2$ points, such that $P_1 \neq Q \neq P_2$. Using the duality theorem, ...
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How to calculate intersection point of polyhedron and a line?
this is my first time in this forum. Please guide me if I do something wrong.
I need to make an algorithm to calculate intersection point of polyhedron and a line (in 3D space). The line is made from ...
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Degree of an interesection divisor
Let $\mathcal{C},\Gamma\subseteq \mathbb{P}^2(\mathbb{C})$ be two smooth algebraic curves defined by:
$$\mathcal{C}:F(X_0,X_1,X_2)=0 \ \ \ \ \ \ \Gamma:G(X_0,X_1,X_2)=0$$
such that none of them ...
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Calculating ray intersection with two spheres (with smooth_blending between the two spheres)
I have the following function to define the distance to the interesection between a ray and the surface of a sphere:
...
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Oriented intersection number $I(A, B)$ of $S^2$ and $T^2$ in the oriented product manifold $S^2 \times T^2$
In the product manifold $S^2 \times T^2$ of an oriented 2-sphere and an oriented 2-torus, is the oriented intersection number $I(A, B) = \chi(A) \times \chi(B)$ where $A$ is the submanifold $S^2 \...
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Finding six conics tangent to one conic and four lines in the plane
Indeed the number of conics tangent to five plane curves of degrees $d_1,d_2,d_3,d_4,d_5$ in ${\Bbb P}^2_{\Bbb C}$ is
$$d_{{1}}d_{{2}}d_{{3}}d_{{4}}d_{{5}} \left( d_{{1}}d_{{2}}d_{{3}}d_{{4}
}+d_{{1}}...
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Conjugacy classes of an element that are the same
Recently I have been studying the transfer homomorphism, and it came to mind that whether conjugacy class of an element with respect to some subgroup is the same as the original group. Namely, if $x \...
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Equality for subespaces and relative interiors
Assume the following hypotheses: $U,V$ subspaces of a normed space $X$ and $C\subseteq X$ a convex subset such that $\operatorname{int}_{U}(C) = \operatorname{int}_{V}(C)$ and $C \subseteq U \cap V$, ...
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Intersection form of a connected sum
I am trying to prove the following:
Let $M,M'$ be closed connected oriented 4-manifolds. Then intersection form of a connected sum is the direct sum of intersection forms: $q_{M\#M'}=q_M\oplus g_{M'}$....
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Global sections of algebraically trivial line bundle
Let $\pi \colon X \rightarrow C$ be a smooth minimal elliptic surface over an algebraically closed ground field $k$. Furthermore assume that $\pi$ has a section and that the fundamental line bundle $R^...
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Is there a relatively easy way to find whether two plane curves have a "common component"?
I am interested in determining all intersections between two plane curves f(x,y)=0 and g(x,y)=0. (f is degree 4, and g is degree 3.) I would like to use Bezout's Theorem.
I have found 12 total ...
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Chern classes of tensor product of rank two vector bundles
This answer gives the first three Chern classes of the tensor product of any two locally-free sheaves.
I computed the fourth Chern class as
$$
c_4(E\otimes F) = \frac{1}{2} c_1^4(E) -5c_1^3(E)c_1(F) -\...
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The Self-Intersection Number of the Complex Projective Line $\mathbb{C}\mathbb{P}^1$
Let $\mathbb{C}\mathbb{P}^2$ be the Complex Projective Plane.
Let $C \subset \mathbb{C}\mathbb{P}^2$ be a Rational Curve, i.e., $C$ is isomorphic to the Complex Projective Line $\mathbb{C}\mathbb{P}^1$...
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Confusion about why a specifc Vector Transformation magically fixes the UV mapping of a Rotating ray plane intersection
Alright so, i have been toying around with some basic ray intersection functions as of recently. And in doing so found myself rotating these intersection shapes, such as disks. Which eventually lead ...
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Why doesn't GeoGebra show the points of intersection between two ellipses when they are tangent?
I drew two ellipses in GeoGebra that have a common focus and my purpose is to show all cases of their relative geometries. I used the Intersect(Object,Object) ...
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Blow-up of a Pencil of Cubic Curves (Miranda's basic theory of elliptic surfaces)
In Rick Miranda's "The basic theory of elliptic surfaces" the Example (I.5.1) see page 7 on a pencil of plane curves contains an argument Inot understand yet:
Let $C_1$ be a smooth cubic ...
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Compute the intersection form
Consider a smooth subvariety $\iota:X=V_+(f)\subset\mathbb{P}^2\times\mathbb{P}^1$ with some $f\in H^0(\mathcal{O}(1,2))$, how can I compute the intersection
$$\iota^*\mathcal{O}(2,3).\iota^*\mathcal{...
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Intersection mulitplicity of a curve and a hyperplane in $\mathbb P^n$
Assume we have a curve $X\subseteq \mathbb P^n$ and a hyperplane $H$. Let $x\in X\cap H$. I want to know how to compute the intersection multiplicity $m(X,H,x)$.
As the argument is local, we assume $H ...
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Geometric interpretation of intersection product
Let $E,E^*$ be a pair of dual $n$-dimensional real vector spaces and let $e,e^*$ be a pair of dual basis vectors of $\bigwedge^n E$ and $\bigwedge^n E^*$, respectively. For $u,v\in\bigwedge E$, we can ...
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Graph of a holomorphic map $\Bbb CP^1\to \Bbb CP^1$ of degree $d$
I am reading this lecture note: https://www.mathematik.hu-berlin.de/~wendl/pub/rationalRuled.pdf. In Example 1.1, it is explained that the graph $\Sigma_f \subset S^2\times S^2$ of a degree $d(>0)$ ...
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Intersection numbers on blow-up
Let $p:X\to\mathbb A^3$ be the blow-up of $\mathbb A^3$ along a line $L$. Denote the exceptional divisor as $E$. Let $H$ be a plane containing $L$ and $\tilde{H}$ be the strict transform. I found ...
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probability- minimum and maximum value of a union
Suppose that events A and B exist such that P(A)=0.15, P(B)=0.2 . The maximum and minimum possible value for P(not A or not B)
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Calculating the Intersection Number of 2 Specific Sections on Elliptic Surface
Take the elliptic surface defined by the equation $E_6: y^2= x^3 + t^6 + 1$ over the field $\mathbb{F}_5$ (or the algebraic closure thereof, it does not really matter for this question). I have the 2 ...
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How can overflow be defined for the Klein bottle in $\mathbb{R}^4$?
I was watching Dr. Tadashi Tokeida's lecture series on Youtube:
https://kevinbinz.com/2017/10/25/isotopy/
For submanifolds $L$ and $K$ being placed in ambient manifold $M$, the overflow can be defined ...
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Existence of rigid curves with non-negative self intersection
The question is essentialy the title: is there a smooth algebraic surface $X$, say over $\mathbb{C}$, and an irreducible algebraic rigid curve $C\subset X$ with non-negative self intersection?
Here by ...
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What is the probability that 3 random chords in a circle do not intersect?
I understand that for 2 random chords, the probability of no intersections is 1/3 thanks to this blog post.
What happens when I have 3 random chords? Is there an intuitive explanation for calculating ...
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Chern Class of the Structure Sheaf of a Cartier Divisor
I'm reading 3264 and all that page 486. They calculate the chern class of the structure sheaf of a Cartier divisor.
Let $Y\subseteq X$ be a Cartier divisor. Then we have short exact sequence:
$$0\...
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Are Weil divisors defined over any algebraic scheme?
I notice that:
In fulton's book "Intersection Theory", $r$-cycles are defined in any algebraic scheme (see page 10). In particular, Weil divisors (i.e. cycles of codimension 1) are well ...
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