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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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327 views

(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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658 views

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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130 views

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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611 views

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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81 views

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, ...
5
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83 views

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?
5
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127 views

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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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: ...
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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
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64 views

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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536 views

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 ...
4
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123 views

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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144 views

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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289 views

Reference request: generalized Lefschetz-Hopf fixed point theorem

Let $X$ be a smooth projective variety over $\mathbb{F}_p$. A basic (and very important) theorem is that we have equality $$ \# X(\mathbb{F}_{p^n}) = \sum_{i=0}^{2\dim X} (1)^i \operatorname{tr}({\...
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164 views

Negative self intersection and section of the conormal sheaf for a singular complex curve

Let $M$ be a complex $2$-dimensional manifold and $C$ be a compact complex curve in $M$ (possibly singular). Let us suppose that there exists a holomorphic function $f\in\mathcal{O}(M)$ such that $f$ ...
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284 views

When is an intersection of varieties finite

Consider the general Bezout's theorem: If $p_1 \ldots p_n$ are polynomials with degrees $d_1,\ldots, d_n$ in $\mathbb{R}[x_1,\ldots,x_n]$, with $V = \{a=(a_1,\ldots, a_n) | p_i(a) = 0, \forall i\}$ ...
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305 views

Degree of line bundles is additive on curves

In Hartshorne (among other places), it is used that on a nonsingular curve $C$, the degree of line bundles is additive. That is, $$\mbox{deg}_C(L_1 \otimes L_2) = \mbox{deg}_C(L_1)+ \mbox{deg}_C(L_2)....
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61 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 ...
3
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117 views

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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61 views

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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57 views

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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59 views

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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88 views

Why is the dimension of this Grassmann manifold $G_{d, n}$ equal to $(d+1)(n-d)$ formed by the Plucker coordinates of a $d$-plane?

A $d$-plane $L \subset \mathbb{P}^{n}$ is defined as the set of points $P=(p(0), p(1), \ldots, p(n)) \in \mathbb{P}^{n}$ that satisfy equations $\sum_{j=0}^{n} b_{\alpha j}p(j) = 0$, where $\alpha = 1,...
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329 views

How to compute intersection numbers in practice?

When speaking about plane curves, one of the most fundamental and important results is Bézout's Theorem, which states that over an algebraically closed field $k$, two plane projective curves of ...
3
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128 views

When does a homogeneous morphism have only finite fibers?

Suppose that we have a map ${\bf f}:=(f_1,f_2,\cdots ,f_n):\mathbb{C}^n\rightarrow \mathbb{C}^n$ given by $$ \mathbb{C}^n\ni {\bf z}:=(z_1,z_1,\cdots,z_n)\rightarrow \big(f_1({\bf z}),f_2({\bf z}),\...
3
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88 views

Blowing-up of a linear sub-space is Fano?

Consider $P=\mathbb{P}^3\times \mathbb{P}^1$ with coordinates $([x_0:x_1:x_2:x_3],[u:v])$ and let $\varepsilon:X\to P$ be the blow-up of $\mathbb{P}^2\cong A=(x_3=v=0)\subseteq P$, of pure codimension ...
3
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741 views

Grassmannian bundle: any good reference?

I have met the notion of Grassmannian bundle of a vector bundle over a variety in intersection theory, but anywhere I look I just find a brief recall of how the stalks look like (my references so far ...
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41 views

14 excess contribution?

I want to know the number of smooth conics that tangent 4 general lines and pass through 1 general point. Indeed, the number is 2. But from Bezout's theorem we get $1\cdot2^4=16$ on $\mathbb{P}^5$. ...
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58 views

What is $\int_\gamma c_i(E)$ counting?

Let $M$ denote the Kontsevich moduli space of stable maps $\overline M_{0,n}(X,\beta)$, where $\beta\in A_1(X)$ and $X$ is a convex variety. I am trying to understand why $$\dim\, M=\dim\,X+n-3+\int_\...
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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(...
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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 ...
2
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53 views

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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105 views

Smooth quartic surface in $\mathbb{P}^{3}$ that contains a smooth curve of genus 2 and degree 6.

I am reading through an article by Matsumura and Monsky on Automorphisms of Hypersurfaces in which they state that there exist quartic surfaces is $\mathbb{P}^{3}$ which have infinite automorphism ...
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39 views

Coordinates of tangent points from a point outside a spheroid (ellispoid of revolution)

The blue conics in the figure is an ellipse (it would be in a case a circle), that represents the tangency points, from the line drawn from the point $P$ to the ellipsoid surface in all direction, ...
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69 views

self intersection of divisor

My question concernes following former thread in MO: https://mathoverflow.net/questions/117808/the-intersection-multiplicity-of-the-canonical-divisor-of-a-surface-with-a-fibre here we have a (minimal)...
2
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0answers
118 views

Does an immersed curve in general position has finite self-intersections?

This problem comes from Hirsch's differential topology in page 67. "Generically" a $C^1$ immersion $S^1\to \mathbb{R}^2$ has only a finite number of crossing points. Then I want to ask if in ...
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0answers
57 views

Does Splitting principle hold in the category of schemes?

Let $k$ be a field, consider the category of $k$-varieties. If a relation among Chern classes of vector bundles holds when all the vector bundles involved split into direct sums of line bundles, does ...
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110 views

Intersecting $\mathbb{Q}$-Cartier divisors on a singular variety (Cone)

Let $S\subset \mathbf{P}^2$ be a quadric, and let $X \subset \mathbf A^3$ be the cone over $S$ with vertex $P$. As a practice, I want to calculate $$(K_X+L)|_L,$$ where $L$ is a rule of $X$. My ...
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58 views

Chern classes of hyper-Kähler fourfolds in Grassmannians

Both the Fano variety $F$ of lines in a general cubic fourfold and the "Debarre–Voisin" fourfolds $Y_\sigma$ introduce in [DV] are smooth, four-dimensional, hyper-Kähler subvarieties of Grassmannians (...
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48 views

How to show that : $ s_p ( E \otimes L ) = \displaystyle \sum_{ i=0 }^p (-1)^{p - i } C_{e + p }^{e+i} \ s_i ( E ) c_1 ( L )^{p-i} $?

Let $ E $ be a vector bundles of rank $ e+1 $, and let $ L $ be a line bundle. How to establish that the Segre class : $$ s_p ( E \otimes L ) = \displaystyle \sum_{ i=0 }^p (-1)^{p - i } \begin{...
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0answers
69 views

Zero dimensional component of an intersection

Let $X$ be a smooth projective variety and let $A,B$ be closed irreducible subvarieties of complementary dimension in $X$ and with intersection product equal to $n$. The varieties $A$ and $B$ may not ...
2
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0answers
55 views

Self intersection of correspondences of a curve $X$ and degree of isogenies in $\text{End}(J_X)$

I migrated my question from MO because I just got one vote and maybe was too basic. Let $C/\bar{k}$ be a nonsingular irreducible curve of genus $g$ and $\mathfrak{C}(C\times C)\cong \text{CH}^1(C\...
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0answers
193 views

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 ...
2
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0answers
45 views

Tropical top self-intersection numbers of boundary divisors in toroidal embeddings

Let $X_0 \subset X$ a toroidal embedding without self intersections and denote by $\overline{\Sigma}$ its corresponding (weakly embedded) extended conical simplicial complex. Let $D$ be a divisor on $...
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0answers
421 views

How many different ways can a circle intersect a triangle N ways?

Consider a circle intersecting a triangle. The circle and triangle can have between 0-6 total intersection points. Is there a mathematical formula for the number of possible ways they can intersect ...
2
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0answers
291 views

Intersection of hypersurfaces in the projective space

Fix an integer $n>0$. Is it true that for any $k>0$ and a closed point $x \in \mathbb{P}^n$, there exist hypersurface sections of degree $k$ (i.e., global sections of $\mathcal{O}_{\mathbb{P}^n}(...
2
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0answers
33 views

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 ...
2
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0answers
94 views

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 ...
2
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0answers
41 views

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 ...
2
votes
0answers
73 views

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 ...