# 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)

189 questions
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### (Weil divisors : Cartier divisors) = (p-Cycles : ? )

Suppose $X$ verifies the suitable conditions in which Weil (resp. Cartier) divisors make sense. The group of Weil divisors $\mathrm{Div}(X)$ on a scheme $X$ is the free abelian group generated by ...
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### Self-Intersection Number $-2$

I am new here, but hopefully you can help me with a concrete problem I have. I try to compute a Self-Intersection Number of a constructed curve in an analytic surface. I know the answer by some ...
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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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### Question about Gysin map (pushforward in cohomology)

The setting is the following: $X \subset \mathbb{CP}^r$ is an algebraic subvariety of dimension $n$, codimension $e=r-n$, and degree $d$. Call $j$ the inclusion. Then, Poincaré duality induces a map \...
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### What is bivariant Chow group?

I have trouble understanding "bivariant Chow groups". Remember that for any morphism of schemes $f:X\rightarrow Y$, we can define a bivariant Chow group $A^*(f:X\rightarrow Y)$. When $Y$ is a point, ...
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### Analytic Grothendieck Riemann Roch

I was wandering if is there an analytic version of the Grothendieck-Riemann-Roch theorem. If so, could you please tell me the references?
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### Why is an intersection product $X.C=0$?

Here is the situation: $S$ is a nonsingular complex projective surface, and $C=X\cup_AY\subset S$ is a uninodal curve of compact type: it is obtained by glueing two nonsingular curves $X,Y\subset S$ ...
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### 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: ...
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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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### Pulling back cycles to closed subschemes

Let $X$ be a (reduced, affine) scheme and consider the closed embedding $i: X \cong X\times 0 \to X\times \mathbb A^1$. Consider the pullback map $i^*: CH^k(X\times \mathbb A^1) \to CH^k(X)$. I know ...
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### Maximum number of singular points on irreducible curve in $\mathbb{CP}^{2}$

Let $C$ be a degree $d$ irreducible curve in $\mathbb{CP}^{2}$. Can we find maximum number of singular points in $C$? For $d=2$, I find that there is no singular point on irreducible conic. (If there ...
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### Geometric intuition under a negative intersection

I am studying Intersection Theory in algebraic geometry and in the following when I say intersection I mean the intersection of a dimension k cycle with a codimension k cycle. I was reading this ...
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### Geometry of cubic 3-fold

I'm having some questions about the geometry of the cubic 3-fold. Every variety is over $\mathbb C$. Take $Y$ a smooth cubic 3-fold in $\mathbb P^4$ and $E$ a curve of degree 6 and genus 1 contained ...
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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 Theory and projection maps

Let $X$ be a manifold and let $Y,Z$ be two compact submanifolds of $K=X\times S^1$. dim $Y+$ dim $Z=$ dim $K$. If $I_2(Y,Z)\neq 0$ prove either $\pi\circ i_Y$ or $\pi\circ i_Z$ is onto where $\pi$ is ...
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### Pull-back of algebraic cycle and moving lemma

Suppose $f:X \rightarrow Y$ is morphism between non-singular projective varieties, and $Z$ is a codimension-$c$ cycle of $Y$. From the chapter on Intersection Theory of Stack Project, for rational ...
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### Proof of a property of curves intersecting transversely

Let $F, G \subset \mathbb{A}^2$ be affine plane curves that share no common tangents at a point $P \in \mathbb{A}^2$. Let $I$ be the ideal of $k[X, Y]$ generated by $X$ and $Y$. By a change of ...
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### Conics meeting 8 general lines

I am trying to show that the number of plane conics in $\mathbb{P}^3$ meeting $8$ general lines is $92$, using what I know about intersection theory. I started considering the tautological bundle $S$ ...
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### Generalizing pullback of cycles and intersection products as suggested in Serre's local algebra

Chapter V in the book Local algebra by Serre, introduces the notion of a "relative intersection product" meaning that if $f: X \to Y$ is a morphism of varieties with $Y$ regular and $x,y$ are cycles ...
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### Explicit form of certain polynomials and intersection of curves

Let $X$ be a smooth degree $d$ hypersurface in $\mathbb{P}^3$ and $C, D$ two effective divisors on $X$ intersecting at finitely many points. Is it true that if $C$ and $D$ intersect in ''low'' number ...
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### Two simple counterexamples in algebraic geometry

Suppose we have a smooth complex algebraic variety $X$. Then in general, $K^a(X)\to K(X^{an})$ is not surjective. Could someone give an example of a topological vector bundle class which contains no ...
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### definition of cycle theoretic fibre

I am studying the definition of Chow variety on Kollar's Rational Curves on Algebraic Varieties, and I am having some trouble in understanding Definition 3.9. Here we have a proper morphism of ...
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### Restriction and intersection

Say $X$ is a smooth threefold, and $A$, $B$, and $C$ are three smooth divisors on $X$. Is it true that the three-way intersection $(A \cdot B \cdot C)_X$ is equal to the intersection \$(A\vert_C \cdot ...