# Questions tagged [intersection-theory]

The intersection-theory tag has no usage guidance.

348
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### Sufficient condition for pair of real quadrics to have real intersection

In the following, when I talk about the zero of a homogeneous polynomial I always mean a projective zero.
Let $ q $ be a real quadric. Then $ q $ has a real zero if and only if $ q $ has indefinite ...

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### Geometric interpretation of pairing between bordism and cobordism

In page 448 of these notes, a pairing between bordism and cobordism
$$\langle \ ,\ \rangle: U^m(X)\otimes U_n(X)\rightarrow \Omega^U_{n-m}$$
is defined as follows. Assume $x\in U^m$ is represented by $...

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### Are horizontal divisors on abelian fibered hyperkähler manifolds proportional in $NS(X)$ up to vertical divisors?

Oguiso writes[1]
Theorem 1.1 Let $f: X \to \mathbf P^n$ be an abelian fibered HK [hyperkähler] manifold. Let $K = \mathbf C(\mathbf P^n)$ and let $A_k$ be the generic fiber of $f$. Then, $\rho(A_K)= ...

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### Is there some relations between derived category and intersection theory?

After learning the traditional intersection theory (W. Fulton's Intersection Theory and D. Eisenbud & J. Harris's 3264 and all that), I have some biased thinking about what I have learned in this ...

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### Is $K_F\cdot C\leq K_X\cdot C$ for a fibre $F\subseteq X$ containing the curve $C$?

This is a question that I originally posted on Math Stack Exchange. After a couple of days I have not received any comments or answers, and after thinking about it more I realize that this question is ...

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### Intersection of two modules (and sub-modules) under tensors

I have a (hopefully quick) question regarding an intersection of two tensor modules. Let $K$ be a field and $A, B$ finitely-generated modules over the Laurent series $K((X))$. Let $\tilde{A}$ be a (...

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180
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### Finite flat pullback of the diagonal

Let $X, Y$ be smooth projective connected complex varieties of the same pure dimension $d$ and $f : X\to Y$ a finite flat surjective morphism.
Let $\Delta_X$ be the closed subscheme of $X\times X$ ...

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### Arf invariant for characteristic surfaces in closed 4-manifolds depends on homeomorphism type

Let $X$ be a closed smooth oriented 4-manifold with $H_1(X;\Bbb Z)=0$. Then for a smoothly embedded orientable surface $F\subset X$ which is characteristic, there is a well-defined invariant $\text{...

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### How to show that the intersection of two certain affine varieties is reduced?

$\DeclareMathOperator\codim{codim}$Let $X=V(I)$, $Y=V(J)$ be two affine varieties. I'd like to know possibile strategies to understand when their schematic intersection, i.e. $X\cap Y=V(I+J)$, is ...

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### Intersection product of $\mathbb{Q}$-Cartier divisors with irreducible complete curves is well-defined

I am learning the notion of intersection product of a $\mathbb{Q}$-Cartier divisor with an irreducible complete curve on a normal variety. The definition I learned is that if $D$ is a $\mathbb{Q}$-...

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### Prescribed intersection of varieties

Every variety here is complex analytic, or complex algebraic if it solves anything.
Given a germ of a (possibly singular, nor necessarily irreducible) hypersurface $(H,0)\subset(\mathbb{C}^{n+1},0)$ ...

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### Intersection theory on schemes with Gorenstein singularities

Is there a good reference/book on Intersection theory on schemes with Gorenstein singularities? Does the construction of the intersection of cycles discussed in Fulton's book also hold for schemes ...

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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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### Hypersurface of singular plane cubics

In the projective space $\mathbb{P}^9 = \mathbb{P}(\mathbb{C}[x,y,z]_3)$, parametrizing plane cubics, consider the hypersurface $X\subset\mathbb{P}^9$ whose points corresponds to singular cubics. The ...

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### Positivity of the intersection form [closed]

Let $M^4$ be a closed orientable smooth manifold and
$$
I: H_2(M,\mathbb Z) \times H_2(M,\mathbb Z) \to \mathbb Z
$$
its intersection form. If $b^+ \ge 1$, where $b^+$ denotes the number of positive ...

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### Blow-up of a pencil of cubic curves (from Miranda's basic theory of elliptic surfaces)

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 that I do not understand yet.
Let $C_1$ be a smooth ...

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### Hausdorff dimension and non-empty intersections with lines

Let $A\subseteq [0,1]^d$, $d\geq 2$, a set with Hausdorff dimension $\operatorname{dim}_{\mathcal{H}}A=s$. What is the minimum $s$ (if any) which guarantee that $A$ has non-empty intersections with a ...

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### Intersection theory on normal crossing algebraic surfaces

Let $X$ be an algebraic surface with normal crossing singularities. Suppose the singular locus of $X$ is a smooth curve. Let us denote it by $C$. Suppose $D$ is another smooth curve in $X$ which ...

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### If Serre's intersection multiplicity $\chi(R/I, R/J)$ equals $\operatorname{length}_R (R/(I+J))$, then are $R/I, R/J$ Cohen-Macaulay?

Let $(R,\mathfrak m)$ be a regular local ring. Let $I,J$ be proper ideals of $R$ such that $R/(I+J)$ has finite length i.e. $\sqrt{I+J}=\mathfrak m.$ Since $I+J$ annihilates $\text{Tor}_n^R(R/I, R/J)$ ...

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### About Fulton's Intersection theory Appendix Lemma A 4.1

The assumption for Lemma A.4.1 is $A \to B$ is flat. The second assumption is that $A$ and $B$ are Artinian rings. From this Lemma A.4.1 states that $l_B(B) = l_A(A) \cdot l_B(B/mB)$ where $m$ is the ...

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### Intersection of translate of divisors on abelian variety

Setup. Let $K$ be an algebraically closed field of characteristic zero, and let $A/K$ be a simple abelian variety of dimension $n$. Let $\{ x_1,x_2,\dots,x_{m^{2n}}\}$ denote the $m$-torsion points of ...

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### To whom is Bézout's theorem for varieties due?

The following is a modern, fairly general form of Bézout's theorem. (There are forms that are more general and/or more precise; bear with me.)
Define the degree of a reducible variety to be the sum of ...

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### Comparison between residual intersection in Fulton's intersection theory and Aluffi's result on Milnor class

$\textbf{Question}$ I deduced that $m(A \cup B, V) = 0 $ for nonsingular variety $V$ and nonsingular hyper surfaces $A$ and $B$ whose intersection is also nonsingular. But I do not think it is true ...

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### Isomorphism outside of negative curves against the canonical

Let $X$ be a smooth projective complex variety and let us suppose that the closure of the union of curves $C$ on $X$ that are non-positive against the canonical divisor is a closed subset $F\subsetneq ...

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### How to calculate the divisor given by closure of subscheme

Let $X \subset \mathbb{P}^N$ be a nonsingular projective variety over algebraically closed field which is embedded by very ample line bundle $\mathcal{L}$. Let $Y = \mathbb{P}(\mathcal{L}^{\oplus 3})$...

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### Varieties connected by curves in projective spaces of small dimension

Let $X\subset\mathbb{P}^N$ be an irreducible complex variety. Fix an integer $a\geq 2$ and call $P_a$ the following property: given $x_1,\dots,x_a\in X$ general points there exists an irreducible ...

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### Intersecton form of complete smooth Toric surface

Given a complete smooth Toric surface (over $\mathbb C$), is its intersection form well-known? Or is there an algorithm to calculate it? Thanks in advance.

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### Algebraic and homological equivalence relations for $0$-cycles

Let $X$ be a connected smooth projective variety. Let $Z_0(X)_{alg}$ be the group of $0$-cycles algebraically equivalent to $0$ and $Z_0(X)_{\hom}$ be the group of $0$-cycles homologically equivalent ...

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### A question on the Chow group on stacks

Let $X$ be a separated Deligne-Mumford stack finite type over the ground field. Then there is a Chow group $A_*(X)$ of $X$ which is well-behaved under flat pull-back, defined as follows.
Let $\...

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### Mordell conjecture over function fields

So I've read (for instance in the introduction to R.S de Jong's thesis ) that the naive adaptation of the proof of the Mordell conjecture over function fields fails, even using Arakelov intersection ...

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### Intersection of plane with Segre

$\newcommand{\complex}{\mathbb{C}}$ Let $Seg \subseteq \complex^M \otimes \complex^N$ be the set of elements of the form $v \otimes w$. It is well-known that a general linear subspace of dimension $(M-...

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### $0$-dimensional intersection in weighted projective space

Consider homogeneous polynomials $P_0,P_1,P_2,P_3,P_4,P_5$ of degrees $3,3,2,3,2,1$ over $\mathbb{P}^3$, and the map $\phi:\mathbb{P}^3\rightarrow\mathbb{P} = \mathbb{P}(3,3,2,3,2,1)$ given by
$$
\phi(...

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### Moving lemma for countable collection of subvarieties

Fix an integer $n \ge 5$. Let $\mathcal{V}$ be a countable collection of closed subvarieties of $\mathbb{P}^n_{\mathbb{C}}$ of codimension at least $2$. Choose a point $p \in \mathbb{P}^n$. Does there ...

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### Extending effective Cartier divisors

Let $X$ be a non-singular, quasi-projective variety (over $\mathbb{C}$) of dimension at least $3$, $D_1, D_2$ are integral effective divisors in $X$ with $D_1 \cap D_2$ of codimension $2$ in $X$. Let $...

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### On intersections of exceptional divisors

Let $X$ be a smooth, projective variety of dimension $n \ge 2$, $L$ a very ample line bundle on $X$ and $\pi: \widetilde{X} \to X$ be the blow-up along a closed subvariety of codimension at least $2$. ...

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### Is the sum of a radical ideal and the ideal of a generic linear space intersecting that ideal radical?

Let $X \subseteq \mathbb{C}^n$ be an irreducible algebraic set that forms a cone, and let $I=I(X) \subseteq \mathbb{C}[x_1,...,x_n]$. Let $m < n$ and $k\leq m$ be positive integers. Is it true that ...

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### Use of Porteus‘ formula in a paper of Beauville

In “Sur la cohomologie de certains espaces de modules de fibrés vectoriels”, Beauville calculates the Chern class of the diagonal $\Delta$ of the moduli space $M$ of certain stable bundles on a curve $...

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### Detecting non-principal Weil divisors on normal varieties using curves

Let $X$ be a normal projective variety over an algebraically closed field $k$. Given any morphism $f:Y\to X$, there is a pullback homomorphism $f^*:\text{Cl}(X)\to\text{Cl}(Y)$, where $\text{Cl}(X)$ ...

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### Pinch points and dual surfaces

I am currently reading Fulton's expository lectures "Introduction to intersection theory in algebraic geometry".
On pg. 4, Fulton sketches an argument of George Salmon which I don't ...

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### Intersection of curves in non-singular projective algebraic surfaces

Bezout thereom that says that two irreducible algebraic curves $C$ and $D$ in $\mathbb{P}^2_\mathbb{C}$ intersect at $nm$ points (counted with multiplicity), where $n$ and $m$ are the degrees of $C$ ...

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### Are "transverse" hyperplane sections of nondegenerate irreducible projectice varieties always nondegenerate

Let $X \subseteq \mathbb{P}^n$ be a irreducible complex projective variety. It is called nondegenerate if it is not contained in a hyperplane in $\mathbb{P}^n$.
Assuming $X$ is nondegenerate and ...

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### Intersection cycle in a product of Grassmannians

Let $G(k,n)$ denote the Grassmiannian of $k$-planes in $\mathbb C^n$. Let's define
$$ I_j =\{ (\Lambda_1,\Lambda_2 ) \in G(k,n) \times G(l,n) \, | \, \dim(\Lambda_1 \cap \Lambda_2) \geq j \}. $$
These ...

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### Cycle of non-equidimensional scheme

In Fulton's intersection theory, example 1.7.1, he mentioned an example that contradicts to the splitting of cycles with respect to irreducible components. Consider the subscheme $X$ in $\mathbb{A}^3$ ...

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### Algebraic K-theory and intersection theory (Bloch's formula)

It seems to be a well known fact that algebraic K-theory can be used to understand intersection theory, at least for varieties (or stacks!) over a field. A first glimpse of this result seems to be ...

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### A proper morphism restricts to a closure of a point on the generic fiber

Let $\pi:X^{e}\rightarrow Y^{d}(e\geq d)$ be a proper and dominant morphism of projective varieties over field $k$. Moreover, $Y$ is assumed to be smooth. Denote $\eta$ the generic point of $Y$, $X_\...

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### Varieties swept out by Linear Spaces nondegenerated

We working over complex numbers $\mathbb{C}$ keeping our constructions
as geometric as possible.
Let $\Lambda_1, ..., \Lambda_m \cong \mathbb{P}^{n-2} \subset
\mathbb{P}^{n} $ be pairwise distinct, ...

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### Computing Massey products via intersection theory

Let $K$ be an $n$-manifold with boundary and let $x,y,z \in H^*(K)$ be cohomology classes with $x\cup y=y\cup z=0$.
The Massey product $\langle x,y,z \rangle$ is defined as the set of cohomology ...

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### Genus of a curve given by self intersection of a very ample line bundle

Let $X$ be a smooth, integral and projective $d$-dimensional variety over a field $k$ of characteristic 0. Let $H$ be a very ample line bundle over $X$. Assume that there exists a smooth and ...

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402
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### Line segment-triangle intersection algorithm [closed]

currently in my project I'm using signed tetrahedron volume to check whether a line segment intersects a triangle. Initially I've found this approach in the great answer provided by professor O'Rourke:...

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### Curves sharing points over finite fields, and their mutual divisibility

Consider in $\mathbb{A}^2(\mathbb{F}_q)$ two $\mathbb{F}_q$-rational curves $\mathcal{X}:f(x,y)=0$ and $\mathcal{Y}:g(x,y)=0$, and let $\mathcal{Y}$ be absolutely irreducible. Suppose also that $\...