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$ ...
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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$.
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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 ...
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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/...
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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 ...
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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 ...
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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$ ...
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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$, ...
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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 ...
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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 ...
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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 ...
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