Questions tagged [intersection-theory]
The intersection-theory tag has no usage guidance.
376 questions
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Self-intersection and generic point
The Wikipedia entry on intersection theory contains the following statement:
[for C a curve, on a surface] "the self-intersection points of C is the generic point of C, taken with multiplicity C · C."...
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A question about an intersection number
Let $\pi:Y\rightarrow \mathbb{P}^3$ be the blow-up of two points $p,q\in\mathbb{P}^3$, and then of the strict transform of the line $L$ spanned by them. Now, Let $E_p,E_q, E_{p,q}$ be respectively the ...
user56259
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Intersection Matrix of a resolution
Probably this is a very easy question. Let $f:X\rightarrow S$ be a resolution of a projective surface such that
$$K_X = f^{*}K_S+\sum_ia_iE_i$$
with $a_i>0$. By Grauert-Mumford theorem the ...
user58018
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When is the intersection number well-defined?
According to 1.34 of Birational Geometry of Algebraic Varieties(J.Kollár, S.Mori),
there are at least 4 ways(classical approach, cohomological approach, general intersection theory and topological ...
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Is complex surface in CP(3) a two handlebody?
Consider a complex surface given by homogeneous equation in $\mathbb{C}P^3$. Without loss of generality, take
\begin{equation}
S = \{[x:y:z:w] \in \mathbb{C}P^3~ |~ x^d + y^d + z^d + w^d = 0\}
\end{...
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Sufficient conditions to get complete intersection curves
Let $H_1,H_2\cdots,H_{d-1}$ be hypersurfaces in $\mathbb{P}^d$, if the intersection $B:=H_1\cap H_2\cap \cdots \cap H_{d-1}$ is $1$-dimensional then it is called a complete intersection curve.
What ...
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birational equivalence of linear sections of algebraic varieties
Let $X$ be an irreducible algebraic variety and suppose that $L$ is a linear space defined by the linear forms $l_1,l_2,\ldots,l_k$. I want to study $L\cap X$. I would like to know whether the ...
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Strategy to prove formula for top chern class from knowlege of chern character
I am trying to prove a conjecture that involves an enumerative problem. In the course of doing so, the following situation came up.
I have a sequence of (smooth, complex, rationally connected) ...
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curve through a point avoiding an hypersurface
Let $H$ be a closed hypersurface in $\mathbb{A}^{n}$, $n$ big enough over $\mathbb{C}$. Let $U$ be the complementary open subset.
Let $x\in H$, Is it possible to find an curve $C\subset\mathbb{A}^{...
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excess intersection theory
Can the excess intersection theory be applied to the following problem:
I have a non-singular irreducible variety $X$ of dimension $k$ and degree $d$ and $k+1$ hyperplane sections of $X$, $H_1,H_2,\...
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Explicit formula for the Poincare dual of a CM endomorphism of an elliptic curve
Let $E/\mathbf{C}$ be an elliptic curve with CM by the maximal order $\mathcal{O}_K$ of $K=\mathbf{Q}(\sqrt{-D})$ where $D$ is positive and square-free integer. To make it even more precise, let us ...
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Zero Dimension Intersection
Let $M$ be smooth and purely $r$-dimensional, $E$ be a vector bundle of rank $r$ over $M$, $s$ be a regular section of $E$ and $Z$ the zero scheme of $s$. Then $[Z]$ is dual to the top Chern class $...
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Analogy between variable substitution and the moving lemma
While reading the HoTT book, I came up with the following (very) vague analogy:
Consider $f(x) :\equiv \lambda y.x+y$ of type, say, $f: \mathbb{N} \to \mathbb{N} \to \mathbb{N}$. Also assume you ...
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Rational normal curves as set-theoretic complete intersections
Let $C\subset\mathbb{P}^n$ be a rational normal curve of degree $n$. It is know that $C$ is a set-theoretic complete intersection and that, if $n\geq 3$, is a not a scheme-theoretic complete ...
user49214
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Chern and Segre classes
I've recently started to learn about Chern and Segre classes, and it seems to me that they are very similar, sharing the same important properties and having closely related definitions.
Fulton's ...
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Intersection theory on M_{g,n}
Is there a paper\book that lists the top intersections of Hodge classes and tautological classes on $\overline{\mathcal{M}}_{g,n}$ for small $g$ and $k$, e.g. $g=2,3$ and $k=0,1,2$ ?
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Blow-up of $\mathbb{P}^4$ along a quadric surface
Let $Q\subset\mathbb{P}^3\subset\mathbb{P}^4$ be a smooth quadric surface, and let $X = Bl_Q\mathbb{P}^4$ the blow-up of $\mathbb{P}^4$ along $Q$. Let $H$ be the pull-back of the hyperplane section of ...
user49214
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Relation between intersection multiplicity and Hilbert-Samuel multiplicities
Suppose $X$, $Y$, $Z$ are projective varieties in $\mathbb{P}^n_K$ of dimension $n-1$, where $K$ is a field. $X$, $Y$, $Z$ intersect properly, and $P$ is one of their intersection irreducible ...
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Deformation space form the point of view of intersection theory
I'm interested in deformations of subvarieties of a toric variety $X$.
Suppose we know a subvariety $V$ in the Chow group of $X$, for example, $V$ is a linear combination of powers of hypersurfaces. ...
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Positivity question on K3 surfaces
Let $X$ be a smooth projective complex K3 surface and $L, D$ two effective divisors, $L^2\geq0$ and $D^2\geq0$.
(Q1). do we have $L\cdot D\geq0$ ?
If either one has positive self-intersection, the ...
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Does $h^1(D)=0$ imply numerical connectedness on K3 surfaces?
Let $X$ be a complex K3 surface and $D$ an effective divisor on $X$.
We shall say: $D$ is connected if its support is connected. $D$ is numerically connected if for any non-trivial effective ...
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A question on infinitesimal deformation (related to intersection theory)
Let $X$ be a connected projective scheme in $\mathbb{P}^n$. Assume, $2 \le \dim X \le n-2$. Let $H$ be a general hyperplane in $\mathbb{P}^n$. Denote by $Z:=X.H$ and $Z'=X.H^m$ for $m \gg 0$. Then ...
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bijection of moduli space of equivariant holomorphic embeddings
Consider the moduli space $\mathcal{M}$ of equivariant holomorphic embeddings of closed oriented Riemann surfaces into a generic quintic three-fold $X$ in $\mathbb{P}^4,$ of given degree $d \in H_2(X,...
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Depth of Schubert cycles
For $a:a_1\geq \cdots\geq a_c$, let $\sigma_a$ be the corresponding Schubert cycle over $Gr(c,\infty)$. We say $a$ is of depth $k$ if $a_1-a_c=k$ ($c>1$). Let $a$ and $b$ be of depth $k_1$ and $k_2$...
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Deformation of transversal intersection
Fix a positive integer $n \ge 2$. Let $\pi:\mathcal{X} \to B$ be a family (flat, projective and surjective morphism) of projective subschemes of $\mathbb{P}^n$.
Assume $B$ is reduced, irreducible.
...
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Upper bound for the product of Schubert cycles
Let $Gr(c,\infty)$ be the complex grassmannian of $c$-dimensional subspaces of the infinite dimensional complex space. Every finite dimensional grassmannian, $Gr(c,N)$, can be thought as a subspace of ...
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Intersections of complex submanifolds in $\mathbb{C}^N$
This is an exercise from Gromov's Partial differential relations. (page 5)
Let $V$ and $V'$ be two closed complex submanifolds in $\mathbb{C}^N$ of complimentory dimension. Prove that $V$ and $V'$ ...
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What is the relation between Lefschetz fixed point theorem and Poincare-Hopf theorem on vector fields?
In Dubrovin/Fomenko/Novikov Modern geometry--Methods and applications, Part II, the (Poincare-)Hopf theorem is treated in section 15.2 (see theorem 15.2.7 on page 129), while the Lefschetz theorem on ...
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Which actions preserve non-complete intersections?
Let $X$ be a smooth projective variety and $Z$ is a closed subscheme in $X$ which is not a complete intersection in $X$. Assume the dimension of $X$ (resp. $Z$) is greater than $3$ (resp. $1$). Then,
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Computing intersection of cycles on the product of Grassmannians/Deligne-Lusztig varieties
My collaborators and I are preparing an interesting manuscript where the computation leads to something related to what we believe to be in the area of Schubert calculus; but none of us knows much ...
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Cycle classes that are killed by pushing forward from normalization
Let $X$ be a non-normal algebraic variety and $f \colon X' \to X$ its normalization. Is there a general description $\mathrm{ker}\left(\mathrm{CH}_k(X') \to \mathrm{CH}_k(X)\right)$? Are there ...
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Does every ample divisor "span" a hyperplane?
Let $X\subset\mathbb{P}^n$ be a smooth projective variety of dimension $\geq 2$ and assume that it is not contained in any hyperplane. Now, take some hyperplane $H\subset\mathbb{P}^n$ and consider the ...
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F-curves and divisors on $\overline{\mathcal{M}}_{g,n}$
It is conjectured that a divisor on $\overline{\mathcal{M}}_{g,n}$ would be ample if $D\cdot C >0$ for all F-curves $C\subset \overline{\mathcal{M}}_{g,n}$. Does the intersection degree with all F-...
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Bounds for intersection multiplicity
Let's for simplicity work in $\mathbb{C}^n$. Suppose that $f_1,\dots, f_n$ are polynomials and $0$ is an isolated solution of the system $f_1(z)=\dots=f_n(z)=0$. I want to bound from below the ...
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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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Relationship between virtual cohomological dimension and tautological rings for moduli spaces of curves
Here's the short version of the question. For $M_{g,n}$, $M_{g,n}^{rt}$, $M_{g,n}^{ct}$ and $\overline M_{g,n}$ it seems that the virtual cohomological dimension is given by the complex dimension plus ...
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Smooth, complete varieties on which "zero is effective"
I will say zero is effective on a complete, smooth variety $X$ if some positive linear combination of irreducible varieties is rationally equivalent to zero. In other words, zero is effective if there ...
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Chow ring of extended tropicalizations
In Allermann-Rau '09, the authors define the Chow groups of an arbitrary abstract tropical cycle. In particular, one may take the tropical cycle to be the tropicalization of a subvariety of a torus. ...
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Nef classes on abelian varieties in positive characteristic
Thomas Bauer shows in http://arxiv.org/pdf/alg-geom/9712019v1.pdf that for a complex abelian variety a nef line bundle is numerically equivalent to an effective divisor (this is shown in Lemma 1.1). ...
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Chow ring of a $\mu_2$-gerbe
Suppose that $X$ is a stack, and $Y \to X$ is a $\mu_2$-gerbe. Is there any relationship between the integral Chow rings (in the sense of Edidin and Graham) of $X$ and $Y$?
(I assume they become ...
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Action of an isomorphism in cohomology as the intersection with the class of the graph
Let $X$ and $Y$ be two complex manifolds of dimension 2 and let $\varphi:X\rightarrow Y$ be an isomorphism.
I have read that the action of $\varphi^*:H^2(Y,\mathbb{Z})\rightarrow H^2(X,\mathbb{Z})$ ...
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Axiomatic intersection theory
Is there an axiomatic intersection theory?
What I expect is something like:
An intersection theory is a functor from the category of schemes(or other spaces) to the category of algebras, with well-...
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Bezout's theorem for non-proper intersections?
Is there a version of Bézout's theorem for non-proper intersections?
For my specific problem, the setup is as follows: Let $P_1,P_2,P_3,P_4\in\mathbb{C}[z_1,z_2,z_3,z_4]$, and suppose that (as a ...
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Reference request: Samuel's multiplicity and degree
I am looking for references for the following simple facts.
Let $Y\subset \mathbb{P}^n$ be a variety (or pure-dimensional algebraic set). For $P\in Y$ denote by $e_p(Y)$ the (Samuel's) multiplicity ...
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Divisibility of all entries in an intersection form
What are situations where one can conclude that all entries of an intersection form are divisible by a fixed integer?
More precisely: $F \subset S$ is a proper connected (usually reducible) half-...
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Relation between the index of (the sum of) lattices in euclidean space, and their orthogonal complement.
I am trying to figure out something concerning the index of lattices.
The question came about after reading the paper of W.Fulton and B.Sturmfels, ("Intersection theorey on toric varieties"). To ...
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Non-proper intersection of projective schemes
Let $X, Y$ be projective varieties in $\mathbb{P}^n$ for $n>10$. Assume that dimensions of $X,Y$ are greater than $n/2$. My first question is as follows:
Is there any criterion (...
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euler class of the normal bundle and self intersection number [duplicate]
Let $S$ be a compact submanifold of $X$ smooth manifold. I know that $T_X|_S=T_S\oplus N_{S/X}$ where $N_{S/X}$ is the normal bundle. I have read that the euler class $e(N_{S/X})$ corresponds (via ...
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Local model of virtual fundamental cycle
The following baby version of virtual fundamental cycle is well known:
Let $M\subset V$ be the zero locus of a section $s$ of a vector bandle $E \to V$, in general $s$ is not transversal to the zero ...
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Flat morphisms whose fibers are affine spaces
Let $f:X \to Y$ be a flat morphism, such that each fiber is isomorphic to the affine space $\mathbb{A}^n$. Then is is true that $f$ is a Zariski affine bundle? If not, is it at least an ètale affine ...
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