Questions tagged [intersection-theory]

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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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  • 301
3 votes
2 answers
172 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 ...
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3 votes
1 answer
147 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 $...
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  • 1,939
2 votes
0 answers
92 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 $2$. ...
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  • 1,940
4 votes
2 answers
211 views

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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  • 874
1 vote
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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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3 votes
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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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1 vote
0 answers
129 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 ...
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4 votes
1 answer
242 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 $C$ ...
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  • 49
2 votes
1 answer
155 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 ...
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5 votes
1 answer
256 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 ...
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  • 484
2 votes
1 answer
86 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 $\mathbb{A}^3$ ...
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  • 253
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1 answer
266 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 ...
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  • 151
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0 answers
84 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_\...
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  • 113
1 vote
0 answers
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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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6 votes
1 answer
288 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 ...
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1 vote
0 answers
147 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 ...
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1 vote
1 answer
98 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:...
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2 votes
1 answer
110 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 $\...
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6 votes
0 answers
212 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 $...
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  • 16.7k
3 votes
0 answers
93 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 ...
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  • 16.7k
0 votes
0 answers
103 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) = \...
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  • 1
5 votes
2 answers
543 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 &...
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2 votes
0 answers
63 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 ...
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3 votes
0 answers
165 views

algebraic vs rational equivalence

Are there classes of algebraic varieties for which algebraic and rational equivalence for algebraic cycles coincide? (references also appreciated)
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  • 3,665
3 votes
0 answers
139 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 $n-r$ ...
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9 votes
0 answers
246 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-...
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3 votes
1 answer
351 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 ...
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0 answers
47 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 ...
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  • 1
1 vote
0 answers
142 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 ...
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  • 8,079
5 votes
0 answers
158 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 ...
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2 votes
0 answers
138 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 ...
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  • 1,217
1 vote
1 answer
256 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 ...
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1 vote
0 answers
52 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}...
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4 votes
3 answers
481 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 ...
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  • 16.7k
5 votes
1 answer
361 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 $\alpha_1$ ...
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1 vote
1 answer
330 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\...
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1 vote
0 answers
47 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 ...
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  • 173
7 votes
1 answer
435 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, ...
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  • 317
6 votes
0 answers
220 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 ...
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  • 1,156
7 votes
0 answers
146 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&...
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9 votes
1 answer
511 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 ...
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  • 1,013
4 votes
1 answer
327 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 ...
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2 votes
0 answers
158 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 ...
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2 votes
1 answer
188 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 ...
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2 votes
0 answers
168 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 ...
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  • 804
1 vote
0 answers
142 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{...
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2 votes
1 answer
220 views

Polarization of an abelian variety made by the sum of two divisors

Let $X$ be an abelian variety of dimension $n$, and let $L$ be a polarization, that is, an ample line bundle on $X$, with $\chi(L)=3$. In my specific case, I have that $L=\mathcal{O}_X(\Theta + D)$, ...
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3 votes
0 answers
266 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 $\{...
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  • 1,015
2 votes
1 answer
408 views

Looking for examples of not injective maps and not surjective maps of the form $ A_{k} (X) \to H_{2k} ( X , \mathbb{Z} ) $

Here: https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/alggeom-2002-c9.pdf, on pages: $ 1 $ and $ 2 $, we find the following paragraph: For any scheme of finite type over a ground field ...
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