Questions tagged [intersection-theory]
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376 questions
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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 ...
- 1,803
4
votes
1
answer
379
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 ...
- 329
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0
answers
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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 ...
- 4,418
2
votes
1
answer
330
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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 ...
- 100
2
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0
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327
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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 ...
- 1,168
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vote
0
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157
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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
1
answer
276
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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 $L=\mathcal{O}_X(\Theta + D)$, ...
3
votes
0
answers
375
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 $\{...
- 5,966
2
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1
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585
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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 ...
- 325
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0
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174
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Automorphism group of a hypersurface and its sections
This question is moved from my StackExchange.
Assume the base field is algebraically closed. Let $X\subset \mathbb P^n$ be a fixed smooth hypersurface of degree $d$. For any hyperplane $H\subset \...
- 131
3
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0
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278
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Noether intersection multiplicity for complete intersections
If I take two curves $C,D$ on a surface $M$ with isolated intersection point $p$, then Noether gives a formula equating the intersection multiplicity $i_p(C,D)$ of $C$ and $D$ at $p$ in terms of their ...
- 573
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0
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149
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Formula for fibre square (from Fulton's Intersection Theory)
I have a question about argumentation used in the proof of Proposition 1.7 in Fulton's book on Intersection Theory on page 18:
Proposition 1.7 Let
$\require{AMScd}$
\begin{CD}
X' @>{g'}>> ...
- 5,966
1
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1
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301
views
Schubert cycles that intersect generically transversely
Let $\mathcal{V}= 0 \subset V_1 \subset \cdots \subset V_{n-1}\subset V_n=V$, $\mathcal{W}=0 \subset W_1 \subset \cdots \subset W_{n-1} \subset W_n=W$ be two flags. We say that $\mathcal{V}$ and $\...
- 115
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K-theoretic derivation of Bézout theorem
In the paper "$K$-Theory and Intersection Theory" by Henri Gillet in Handbook of K-theory, 2005 (link behind paywall at Springerlink), the author says:
"When the ground field $k = \mathbb C$, Bézout’...
11
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0
answers
607
views
The virtual fundamental class as derived intersection
Say $X$ is a smooth projective variety and $\beta\in H_2(X)$ is a class. Then there is a finite-type proper scheme (or in general, stack) $SM : = \overline{\mathcal{M}}_{g,n}(X,\beta)$ of stable maps ...
- 7,972
1
vote
0
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73
views
Intersection inside normal cone
For a regular embedding $X\subset Y$, one can think the intersection class $[X]\cdot [X]$ as the intersection of the perturbation of the zero section inside $N_X Y$, intersect with itself. For non-...
- 287
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1
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282
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Existence of divisor in the Jacobian of smooth curve of genus two whose intersection with theta divisor is 1
Let $C$ be a smooth projective curve of genus $2$ and $J$ denotes the Jacobian of $C$. Let $\theta$ be the image of $C$ under the abel Jacobi map.
Is there exist a divisor $D$ in $J$ such that $D.\...
- 137
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1
answer
484
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Join of two intersecting varieties
Suppose I have two smooth projective varieties $X$ and $Y$ in $\mathbb{P}^n$, that intersect along a smooth subvariety $Z$. Is there a formula to compute the degree of the join variety $J(X,Y)$ of $X$ ...
- 3,779
1
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0
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91
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Factorizations of closed embeddings of smooth schemes
All schemes will be of finite type over a field $k$. Say I have a closed embedding $\iota: X \hookrightarrow Y$ of smooth $S$-schemes for some scheme $S$ (in particular it is a regular embedding). ...
- 595
1
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0
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116
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Numerical and rational equivalences on intersection of divisors
Let $X$ be a smooth projective variety over a finite field. Since $Pic^0(X)$ is finite and $Pic^0(X)$ can be identified with numerically equivalent to zero divisors this implies that for divisors on $...
- 5,901
2
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1
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324
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Exact sequence of normal cones
Suppose that we have a sequence $i: X \hookrightarrow Y$, $j: Y \hookrightarrow Z$ of closed embeddings of varieties such that $i$ is regular. In this case, do we have an exact sequence of cones of ...
- 2,197
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0
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293
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Log-canonical bundle of a smooth curve with marked points
I am not sure if this question is appropriate for this site, but here it goes. I am not a geometer, so I am not familiar with notation in the area.
I am interested in the moduli space of $r$-spin ...
user35360
6
votes
1
answer
328
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Severi Formula for Intersection Multiplicities
I say in advance that I am a novice in Intersection Theory, so forgive me if my question is trivial.
Let $X\subseteq\mathbb{P}^N$ be a smooth irreducible projective variety of dimension $n$ and $V, W\...
- 1,252
0
votes
0
answers
181
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Topological vs algebraic intersection forms
Let $X$ be a simply connected complex projective surface (hence a real $4$-manifold). Let $(H^2(X,\mathbb Z)/\mathrm{tors}, q_X)$, $(A^1(X)/\mathrm{tors},q_X')$ be the corresponding lattices in ...
- 23.4k
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0
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321
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Surjectivity of pushforward on Chow rings for stacks
Let $f:X\rightarrow Y$ be a proper morphism of smooth Deligne-Mumford stacks of finite type over $\mathbb{C}$ that is birational, but not flat. The coarse spaces of $X$ and $Y$ are both not smooth. Is ...
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334
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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 ...
- 151
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0
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74
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Is it possible to represent a closed substack as a fundamental cycle?
Let $X$ be an Artin stack and $Z \subset X$ be a closed substack. Can we represent $Z$ as a fundamental cycle? i.e. $[Z] = \sum_i a_i [Z_i]$ where $Z_i$ are integral substacks of $X$. In other word, ...
- 421
3
votes
0
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80
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Quartic link in a 5-sphere
In this post I would like to propose a quartic link in a 5-sphere.
Let us start with the following gluing into a 5-sphere:
$$S^5=(D^2_{} \times T^3_{}) \cup_{T^4} ({S^5 \smallsetminus D^2 \times T^3})...
- 10.5k
2
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0
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170
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Triple link in a 5-sphere -- Proposal
In this post I would like to propose a triple link in a 5-sphere.
Let us start with the following gluing into a 5-sphere:
$$S^5=(D^2_{} \times T^3_{}) \cup_{T^4} ({S^5 \smallsetminus D^2 \times T^3})$...
- 10.5k
2
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1
answer
249
views
Vector bundles on henselian schemes
Let $X$ be a smooth and projective scheme over $\mathbf{Z}_p$.
We call $\mathfrak{X}$ the ringed space whose topological space is the topological space of the special fiber of $X$, and whose ...
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0
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Splittings in the difference bundle construction of Atiyah-Hirzebruch
I'm reading the construction of difference bundle(generalized) in the paper of Atiyah-Hirzebruch(Analytic cycles on complex manifolds)
There is one thing I cannot understand. The followings are in ...
- 421
3
votes
1
answer
185
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 ...
- 1,252
3
votes
1
answer
494
views
On connectedness of intersection of subgroups
I am quite interested in any partical answer to the following general (maybe a little bit vague) question: Is there some criterion about the connectedness of the intersection of two connected ...
- 31
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0
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177
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Continuity of Intersection Pairing on Chow monoids
Let $X$ be a smooth irreducible complex projective variety. As we know, if $\alpha,\beta$ are two cycles intersecting properly in $X$, we can define, via Serre's Intersection Formula, their ...
- 1,252
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0
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39
views
Counting Zeros Under Unitary Action
Assume we have two polynomials $f_1$ and $f_2$ with Newton polytopes $A_1 , A_2 \in \mathbb{Z}^2$. Also suppose that coefficients of $f_1$ and $f_2$ are generic. Then we pick a unitary matrix $Q$ and ...
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0
answers
121
views
What happens to a variety after a change of variables?
Suppose I have an irreducible affine variety $X \subseteq \mathbb{A}^n_k$.
Let us denote $X = \{ x \in k^n : f_j(x) = 0 \ (1 \le j \le M) \}$. $k$ is an algebraically closed field. Let $a_i \in k$, $...
- 3,625
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1
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663
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A "boundary map" for the algebraic equivalence relation of cycles
In what follows by projective variety I will mean the zero locus of homegeneous polynomials in some projective space. Anyway, feel free to deal with a sufficiently good scheme if you want.
Let $X$ be ...
- 1,252
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0
answers
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Applications of E8 manifold
The $E_8$ Cartan matrix is given by,
$$
K_{E_8}=\begin{pmatrix}
2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\
-1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\
...
- 10.5k
3
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1
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774
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Homotopy of paths at the boundary
Denote by $\Gamma$ a hypersurface in $\mathbb{C}^2$, i.e. the zero locus of a polynomial of two complex variables. Denote by $X$ the complement of $\Gamma$ in $\mathbb{C}^2$. I am trying to define a ...
- 293
4
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1
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322
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Pairing on arithmetic surfaces
Let $f: X \to S$ be an arithmetic surface, where $S=\operatorname{Spec } O_K$ for a number field $K$. It is well known that if we want to introduce a reasonable intersection theory on $X$ we have to ...
- 321
1
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0
answers
45
views
Hypersurface whose "square" level sets intersect all linear subspace of "high" dimension
Let $k$ be an infinite perfect field (e.g. I'm happy to assume that $k$ has characteristic $0$. On the other hand, the algebraically closed case is not interesting for this question). The question is ...
- 1,017
4
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1
answer
221
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Proper locally trivial bundle is injective on Chow groups
If $X\to Y$ is a map of varieties that is Zariski-locally isomorphic to a projection $U\times P\to U$ with $P$ (smooth) proper, I think the pullback $A_{\bullet}(Y)\to A_{\bullet}(X)$ is supposed to ...
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6
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1
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Fulton's deformation to the normal cone vs Verdier's
Let $X$ be a smooth variety over a field $k$, and let $Y$ be a smooth subvariety. In the literature, I've seen two versions of the deformation to the normal cone:
Verdier's version: $\tilde{X}_Y^\...
- 3,079
6
votes
0
answers
241
views
Bezout theorem for germs of holomorphic functions
UPDATE.
It was pointed out by @Dmitri that two smooth curves given by $f=y$ and $g=y+x^k$ in $\mathbb C^2$ provide a simple counterexample.
Let $f_1, \ldots, f_p, g_1, \ldots, g_q$ be germs of ...
- 1,238
6
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0
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355
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Schubert cycles on Grassmannian bundles
Let $X$ be smooth variety and let $\mathcal{E}$ be a vector bundle on $X$ of rank $n$. On the total space of the Grassmannian bundle $\pi:G(k,\mathcal{E})\to X$ we have the tautological exact sequence ...
- 735
3
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0
answers
440
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The Chow ring of a blow-up along a badly embedded subscheme
Let $X$ be a variety and $i:Y\hookrightarrow X$ be a regularly embedded closed subvariety, and write $\tilde X$ for $Bl_Y X$ the blow-up of $X$ along $Y$. Let $A^*(X)$ refer to the Fulton-Macpherson ...
- 528
3
votes
1
answer
434
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Intersections with a Power of an Ample Divisor on an Abelian Variety
Let $A$ be a $g$-dimensional, complex abelian variety, let $H$ be an ample divisor, let $D\in Pic^0(A)$, and let $0\leq k\leq g$.
Question 1: Does $D^k\neq 0\in CH^k(A,\mathbb{Q})$ imply that $H^{g-k}...
- 776
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0
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94
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$ch(L f^*\epsilon)$
I know the famous principle (or theorem depending on how you define) of Chern classes for locally free sheaves and a morphism $f: X'\rightarrow X$,
$ch(f^* \epsilon)=f^* ch(\epsilon)$.
But if $f$ ...
4
votes
2
answers
347
views
Cycle class of zeroes of a global section
Let $\mathcal{F}$ be a locally free sheaf of rank $n$ on an $n$ dimensional complex manifold $X$. If the zero locus of a generic global section of $\mathcal{F}$ is $0$ dimensional, then its cycle ...
- 2,788
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0
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405
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Twisted sheaves on tower of $\mathbb{P}^n$
Take the projective space $\mathbb{P}^n$ over a ring $W$.
We call $\mathcal{O}(q)$ the usual twisted line bundle.
Now take the map $f: \mathbb{P}^n\to\mathbb{P}^n$ defined by
$$[x_0,\ldots, x_n]\...
user124171