Questions tagged [intersection-theory]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2 votes
0 answers
117 views

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-...
2 votes
0 answers
93 views

$0$-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(...
3 votes
2 answers
189 views

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 ...
3 votes
1 answer
193 views

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 $...
2 votes
0 answers
179 views

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 $2$. ...
1 vote
0 answers
97 views

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 $...
3 votes
0 answers
102 views

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 $\text{Cl}(X)$ ...
1 vote
0 answers
160 views

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 ...
4 votes
1 answer
406 views

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 $C$ ...
2 votes
1 answer
342 views

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 ...
5 votes
1 answer
293 views

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 ...
6 votes
1 answer
435 views

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 ...
2 votes
1 answer
114 views

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 $\mathbb{A}^3$ ...
3 votes
0 answers
90 views

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_\...
1 vote
0 answers
83 views

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, ...
6 votes
1 answer
382 views

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 ...
1 vote
0 answers
210 views

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 ...
2 votes
1 answer
121 views

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 $\...
1 vote
1 answer
491 views

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:...
6 votes
0 answers
241 views

Singling out irreducible components

Let $V\subset \mathbb{A}^n$ be a variety defined by equations of degree $\leq D$, or, what is the same, an intersection of hypersurfaces of degree $\leq D$. Let $V^+_0$ be an irreducible component of $...
3 votes
0 answers
95 views

Singling out lower-dimensional components

Let $V\subset \mathbb{A}^n$ be defined by equations of degree $\leq D$. (That is, $V$ is an intersection of hypersurfaces of degree $\leq D$.) Assume $V$ is not pure-dimensional. Let $V^-$ be the ...
0 votes
0 answers
170 views

Intersection product when one factor is contained in the exceptional divisor

I am trying to calculate some intersection numbers and would appreciate help on the following problem: Consider two divisors $D_1$ and $D_2$. Blowing up their intersection yields $\varphi^{*}(D_i) = \...
5 votes
2 answers
707 views

Reference request: Kleiman's proof of Snapper's Lemma

On page 4 of Nitin Nitsure's paper Construction of Hilbert and Quot Schemes, the author refers to the fact that Hilbert polynomials are indeed polynomials as a special case of Snapper's Lemma, see &...
2 votes
0 answers
87 views

Comparing the Segre classes of a cone with its abelian hull

Let $X$ be a smooth scheme, with a sheaf of graded quasi-coherent algebras $\mathcal{A}^*$, that yields a cone $C$ (in the sense of Fulton's intersection theory). Suppose that $\mathcal{A}^1$ is a ...
3 votes
0 answers
211 views

algebraic vs rational equivalence

Are there classes of algebraic varieties for which algebraic and rational equivalence for algebraic cycles coincide? (references also appreciated)
2 votes
0 answers
323 views

Functoriality of Chern-Fulton's class

Let $X$ be a, possibly singular, algebraic variety embedded as a closed subvariety of a manifold $M$ with map $i : X \rightarrow M$, and $\pi : \tilde{M} \rightarrow M$ be a proper birational map with ...
10 votes
0 answers
313 views

Integrality of primary genus $0$ Gromov-Witten invariants of a Fano manifold

Suppose $(X,\omega)$ is a positively monotone compact symplectic manifold, i.e., after a positive scaling of the symplectic form, we have $c_1(T_X) = [\omega]$ in de Rham cohomology ($T_X$ has well-...
3 votes
0 answers
245 views

Transversal intersection with linear subspaces

Let us work over an algebraically closed field $K$. If $X\subset \mathbb{P}^n$ is a closed subset of dimension $r$, then there should exist a linear subspace $L\subset \mathbb{P}^n$ of dimension $n-r$ ...
3 votes
1 answer
553 views

Intersection theory on singular varieties by embedding to smooth ones

Let $X$ be a normal complex projective variety over $\mathbb C$. In order to define the intersection product of the Chow ring, one usually requires $X$ to be smooth. How to weak the smoooth assumption ...
0 votes
0 answers
77 views

EXACT number of intersection points of two algebraic curves

As the picture shows2(the paper's link is in 1),it seems that I can use tools including Bezout's theorem to solve the EXACT number of intersection between two algebraic curves(F(x,y) is of degree two ...
3 votes
1 answer
177 views

Equivalence relations among algebraic cycles

In the book 3264 and All That by Eisenbud & Harris, the authors claim that for smooth projective varieties admitting an affine stratification, the algebraic equivalence relation and the rational ...
1 vote
0 answers
187 views

How to compute the genus of the (singular) intersection of three quadratics in $\mathbb{C}P^4$?

Consider three quadratics in $\mathbb{C}P^4$: $$ x_0^2+4x_1^2+\frac{x^2_2}{4}=0,~ x_1x_4+x_2x_3=0, ~ x_0^2+x_3^2+x_4^2=0. $$ If there intersection was non-singular, then the intersection should be a ...
5 votes
0 answers
161 views

How to compute the class defined by intersection with a square?

$\DeclareMathOperator\Gr{Gr}$Let $\Gr(k,n+k)$ (of course, one can do also for $\Gr(k,\infty)$) be the complex Grassmannian of $k$-planes in $n+k$-dimensional linear space. It is well-known that ...
2 votes
0 answers
188 views

Rank of the top Chow group

Let $X$ be a regular integal scheme of finite type over $\mathbb Z$ and assume that $X$ has dimension $d$. In general it is not known if the Chow groups $CH^q(X)$ ($q$ is the codimension) are finitely ...
1 vote
1 answer
592 views

Non-transverse intersection of submanifolds

What can we tell about non-transverse intersection points of (smooth) submanifolds? Especially, in the case of complementary dimensions, how can we define and calculate the ''multiplicity'' of an ...
1 vote
0 answers
58 views

Locus of linear spaces with prescribed contact order

Let $X\subset\mathbb{P}^{n}$ be a smooth projective variety of pure dimension $d$. Let $Z\subset \mathbb{G}(n-d,n)\times\mathbb{P}^{n}$ be the space of pairs $(P,x)$ of a linear space $P\cong\mathbb{P}...
4 votes
3 answers
513 views

Irreducible components: associativity for intersections?

Let $A$, $B$, $C$ be closed irreducible subvarieties of $\mathbb{A}^n$. Let $V_1$ be an irreducible component of $B\cap C$, and $V$ an irreducible component of $A\cap V_1$. Must there necessarily be ...
5 votes
1 answer
401 views

Statements related to Thurston's work on the surface

If we have simple closed curves $\alpha$ and $\beta$ on a surface $\Sigma_g$, the intersection number $i(\alpha ,\beta)$ is defined to be the minimal cardinality of $\alpha_1\cap\beta_1$ as $\alpha_1$ ...
1 vote
1 answer
611 views

Push-forward of divisors and intersections

Let $f:X\rightarrow Y$ be a surjective finite morphism of varieties, with $X$ normal and $Y$ smooth. Let $D\subset X$ be a divisor and $C\subset Y$ a curve. Does the equality $$C\cdot f_{*}D = f^{*}C\...
1 vote
0 answers
101 views

Minimising kurtosis (non-convex). Can I use algebraic geometry or alternate methods to show uniqueness of a particular solution?

I consider a weighted sum of $n$ identically-distributed correlated random variables. The weights in the sum, $w_i$ for $i=1, 2,...,n$, satisfy $w_i>=0$ and $\sum_{i=1}^{n}w_i=1$. I am ...
6 votes
1 answer
552 views

Intersection theory in analytic geometry

This might be a weird/stupid question, but it came to me a couple of times, and I would like to get an answer for that. In some papers I read, constantly the authors define some analytic subspaces, ...
6 votes
0 answers
250 views

If cohomology theory corresponds to intersection theory, valuation theory corresponds to -?

This is a meta question I asked myself. Cohomology theory is dual to an intersection theory. Is there anything valuation theory corresponds to in general? For instance, McMullen's polytope algebra is ...
7 votes
0 answers
175 views

Intersection numbers via residue formula

$\newcommand{\sslash}{\mathbin{/\mkern-7mu/}}$With a friend we are trying to understand residue formulas in the article "Cohomology pairings on singular quotients in geometric invariant theory&...
10 votes
1 answer
756 views

Interpretation of "27" lines for cubic surface with rational double points

It is well known that a smooth cubic surface has $27$ distinct lines. Explicitly, if we choose a planar representation, i.e., blowup $\mathbb P^2$ at $6$ general points $p_1,...,p_6$, the $27$ lines ...
4 votes
1 answer
374 views

Relative canonical class of blowing-up a flag ideal

Let $X$ be a smooth complex projective variety of dimension $n$. Consider a flag ideal $I$ on $X\times \mathbb{P}^1$, namely, $$ I=I_0+I_1t+I_2t^2+\cdots+I_{N-1}t^{N-1}+(t^N)\,, $$ where $t$ is the ...
2 votes
0 answers
213 views

Localization of Chow groups and flat base change

For any flat morphism $f:X\rightarrow Y$, we have a flat pullback of Chow groups $$Ch^i(Y)\rightarrow Ch^i(X).$$ A particular example of this is of course an open immersion $U\rightarrow X$. In that ...
2 votes
0 answers
274 views

Analytic and algebraic definitions of intersection multiplicity of two complex algebraic curves coincide

There are two definitions of intersection multiplicity of two complex algebraic plane curves. One of them is given in An introduction to algebraic curves by Griffiths. Let $t \mapsto (t^k, y(t) )$ be ...
2 votes
1 answer
263 views

Upper semi-continuity of intersection numbers

Consider a smooth projective morphism of schemes $X \rightarrow S$ with relative dimension $n$ (the application I have in mind is with $S$ = an open subset of $\text{Spec } \mathbb{Z}$) and assume ...
1 vote
0 answers
154 views

The morphisms induced by two Cartier divisors

Let X be a projective variety. We consider two Cartier divisors $D,E$ such that $E\geq D$ and the relative morphisms $\phi_D: X - - -> \mathbb{P}(H^0(X, O_X(D))^*)$ and $\phi_E: X- - -> \mathbb{...
3 votes
0 answers
347 views

Linear system on singular plane curve

Let $C \subset \mathbb{P}^2_k$ an irreducible plane curve of degree $d >1$ over algebraically closed field $k$. That is $C=V(f(x,y,z))$ where $f \in k[x,y,z]$ homogeneous of degree $d$. Let $\{...

1
2
3 4 5
8