Questions tagged [intersection-theory]
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376 questions
13
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1
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Intersection of subvarieties versus ranks of Chow groups modulo numerical equivalences
A nice property of $\mathbb P^n$ is:
Property 1: Two subvarieties $U,V$ such that $\operatorname{dim} U +\operatorname{dim} V \geq n$ always intersect.
(for example, any 2 curves in $\mathbb P^2$ ...
- 30.6k
4
votes
2
answers
234
views
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 ...
1
vote
1
answer
375
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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). ...
- 663
15
votes
1
answer
996
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Link: Serre's intersection formula <-> Bloch-Quillen Thm / When only intersecting divisors, is there 'shorter' approach of proof known?
In very short:
When proving the equivalence of intersection theory constructed through (Milnor) K-sheaves and their product vs. defining the product via Serre's local multiplicity formula + moving, I ...
- 934
1
vote
1
answer
465
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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 ...
- 403
2
votes
0
answers
94
views
Tropical self intersection number of boundary divisor on toroidal embedding
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 $...
- 357
1
vote
1
answer
312
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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 ...
- 11
3
votes
1
answer
952
views
Genus of non-complete intersections
Suppose $X\subset \mathbb{P}_k^N$ a nonsingular curve is a complete intersection of hypersurfaces $F_1, \cdots, F_{N-1}$ (of degrees $d_1, \cdots, d_{N-1}$ resp). Then, we know that the canonical ...
- 3,555
0
votes
1
answer
177
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Continuity of Intersection Multiplicities
I’m looking for a correct technical version (and in the best case a reference) for a statement of the following type:
Consider a complex algebraic variety $X\subset\mathbb{P}^n$ and a sequence of ...
- 174
3
votes
1
answer
245
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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 $k=0,1,2$ ?
4
votes
0
answers
225
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Intersection numbers on blow ups of toric varieties
Suppose we have a smooth, complete toric varietiy $X_{\Sigma}$ of dimension $n$. Let $\sigma_k \in \Sigma(k)$ a smooth $k$-dimensional cone in $\Sigma$ and suppose we make the toric blow up at the ...
- 357
0
votes
1
answer
184
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Intersection multiplicty and global sections
Let $X$ be a smooth projective variety, $V, W$ closed subschemes in $X$ such that $V \cap W$ is finitely many points. Let $\mathcal{L}$ be a line bundle on $X$. Is there any relation between $h^0(\...
- 503
5
votes
0
answers
293
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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) ...
- 1,509
1
vote
0
answers
285
views
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 ...
2
votes
1
answer
307
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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-...
- 3,779
7
votes
1
answer
714
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Calculating chern numbers yields a contradiction, why?
I am really stuck on this one. Let $Y=\mathbb{P}^n$ be the complex projective space and let $\tilde Y$ be the blow-up of $Y$ along a linear subvariety $X$ of codimension $d$. We get the following blow-...
- 4,902
1
vote
0
answers
34
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Hodge classes generated in degree $1$
Let $X$ be a smooth projective variety over the complex numbers, and $\text{Hdg}^p(X)_{\mathbf{Q}}$ the abelian group of Hodge classes in $H^p(X,\mathbf{Q}(p))$.
Denote by $\text{Hdg}^*(X)$ the ...
user95222
3
votes
0
answers
193
views
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{...
- 367
0
votes
0
answers
237
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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,\...
- 325
2
votes
1
answer
352
views
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.
...
- 2,323
3
votes
1
answer
530
views
Lefschetz Fixpoint theorem for non-orientable manifolds
The classical lefschetz fixpoint theorem is stated for oriented compact manifolds $M$ and a smooth map $f:M\to M$ as follows:
the intersection number $I(\Delta, \mathrm{graph}(f))$ is equal to the ...
6
votes
0
answers
320
views
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 ...
- 833
4
votes
1
answer
502
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examples of Chow rings of surfaces
Can somone provide me (articles/literature) with examples of Chow rings of surfaces?
(e.g. here: http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf Chapter 9)
What I want is a list of (smooth ...
user19475
6
votes
0
answers
426
views
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 ...
- 40.3k
2
votes
1
answer
141
views
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 $k_2$...
3
votes
1
answer
378
views
Intersection form on quotient manifold
Let $E_{1},E_{2}$ be elliptic curves over $\mathbb{C}$. We denote by $\iota_{i}$ the translation by a 2-torsion point on $E_{i}$. Then $G=\mathbb{Z}/2\mathbb{Z}$ acts freely on the the product $E_{1}\...
- 65
1
vote
1
answer
711
views
Degree of a real algebraic variety and regular morphisms
I'm reading Fulton's "Intersection theory", which i need for some applied needs.
And i have two questions on general definition of degree used in Fulton.
1)Let us we have a real algebraic variety ...
- 413
2
votes
1
answer
543
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Minimal resolution of Log del Pezzo surfaces
Suppose $X$ is a log del pezzo projective surface of index $l$. As far as I understand it will have a finite number of singular points all of which can be resolved by sucessive blow-ups.
Let $E_i$ be ...
- 2,215
1
vote
1
answer
988
views
A Theorem in Intersection theory.
Fulton's Book on intersection theory (Pg.223, theorem 12.3) asserts the following result:
For r pure dimensional schemes in P^n, whose co-dimensions add to at most n, the product of their degrees is ...
- 55
1
vote
1
answer
2k
views
How I calculate degree of the algebraic curve?
Let F be algebraically closed field. Let C be a curve in F^n defined as zeroes of polynomials $p_1(x_1,\ldots,x_n),..,p_{n-1}(x_1,\ldots x_n)$.
Let us define degree of the curve as $\max_S \{ S\cap C ...
- 2,179
0
votes
1
answer
131
views
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}^{...
- 3,472
3
votes
2
answers
279
views
is intersection of a curve and a family of curves generically constant as a scheme?
(everything below is defined over an algebraically closed field)
Let $D$ be a (smooth) surface, and let $X \subset T \times D$ be a flat family of curves on $D$, where $T$ is irreducible. Let $E$ be ...
- 4,070
0
votes
0
answers
161
views
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 ...
- 325
1
vote
1
answer
390
views
Intersection of curves
Let $f(x,y)=0$ and $g(x,y)=0$ be curves in $\mathbb R^2$. Assume that the origin $(0,0)\in \mathbb R^2$ is a $d$-fold point of $f$ and an $e$-fold point of $g$, respectively. Let $f_d(x,y)$ be the sum ...
- 11
0
votes
2
answers
400
views
definition of group operation in elliptic curves
Hi,
Using the isomorphism between an elliptic curve $E$ and its $Pic_1(E)$ group, one can
easily give $E$ the structure of a group variety after choosing a point $O\in E$. The
operation that one gets ...
- 647
4
votes
1
answer
844
views
intersection number
I vaguely recall the following fact that I'd like to use in my research. It should be easy to see that this holds (if it does) but I can't seem to prove it.
Let $p:X\longrightarrow S$ be a (regular) ...
- 73
3
votes
0
answers
434
views
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 ...
- 193
2
votes
1
answer
104
views
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,
...
- 2,032
10
votes
0
answers
331
views
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 ...
- 7,338
2
votes
0
answers
333
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
$f$ ...
- 1,205
2
votes
0
answers
187
views
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,...
- 533
2
votes
1
answer
457
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Intersection powers of the exceptional divisor (and the transform of a hyperplane)
In light of my previous question, I am interested in the following scenario: Let $\tilde Y$ be the blow-up of $Y=\mathbb{P}^n$ along a linear subvariety $X\subseteq Y$ of codimension $d$, i.e. $X\cong\...
- 4,902
0
votes
0
answers
145
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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 $...
- 105
5
votes
0
answers
217
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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 ...
- 40.3k
3
votes
0
answers
298
views
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 ...
- 1,832
0
votes
0
answers
440
views
non proper intersection
Let X and Y two smooth closed subschemes of a smooth projective scheme Z over a field.
Let $W:=X\cap Y$.
I suppose that W is non empty and that the intersection of X and Y is non proper, i.e
codim(...
- 3,472
1
vote
1
answer
212
views
Putting two complete varieties in a family over the projective line
Let $X$ and $Y$ be two proper varieties of dimension $n$ over a field $k$. I'm looking for "reasonable" conditions, under which, there exists a proper and dominant morphism $f:V\to \mathbb{P}^1_k$, ...
- 137
3
votes
1
answer
310
views
Are there n polynomials for which all intersection multiplicities are at least m?
I don't know whether this is known or not, but I was thinking of the following problem.
Let $n$ and $m$ be natural numbers. Are there $n$ polynomial $f_1,...,f_n\in \mathbb{C}[x]$, such that all of ...
- 6,348
3
votes
0
answers
194
views
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 ...
- 61
7
votes
0
answers
188
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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-...
- 6,162