All Questions
Tagged with intersection-theory ag.algebraic-geometry
148 questions with no upvoted or accepted answers
3
votes
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119
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Finding two hypersurfaces of the same degree that intersect $X/\mathbb{F}_q$ smoothly
Let $X$ be a smooth projective variety over a finite field.
In [Poonen - Bertini theorems over finite fields] it is shown that one can find a smooth geometrically integral hypersurface $S$ of degree $...
- 479
3
votes
0
answers
158
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Self-intersection of sum of Eff cone generators on Picard rank 2 surfaces
Let $S$ be a smooth, projective, complex surface with Picard rank 2, whose effective cone is generated by two curves of negative self-intersection, $C_1$ and $C_2$ (i.e. $C_1^2<0$ and $C_2^2<0$)....
- 31
3
votes
0
answers
759
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The self intersection class of exceptional divisor of 3-fold blown up along a curve
Suppose $X$ is a smooth complete variety of dimension $3$, let $\sigma\colon\widetilde{X}\to X$ the blow-up along smooth curve $C\subset X$, let $\sigma^{-1}(C)=E$ be the exceptional divisor, let $f$ ...
user39380
3
votes
0
answers
371
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Intersection Multiplicity
Let $X$ be an hyper-surface in an affine space defined by an equation $F$. We can assume that the ground field is $\mathbb{C}$ and $X$ is normal. Take functions $f_1,\dots, f_n$ on $X$ and let $B$ ...
- 2,384
3
votes
0
answers
212
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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 ...
- 2,263
3
votes
0
answers
298
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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 ...
- 1,832
3
votes
0
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434
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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 ...
- 193
3
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0
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221
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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 ...
- 2,263
3
votes
0
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387
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intersection theory
Assume that $X$ is a smooth 3-fold (over $\mathbb C$). Let $V$ be a smooth divisor on $X$ and let $S_1,S_2$ be prime divisors on $X$. Assume that given a curve $C$ on $X$ not contained in $V$, then
$...
- 31
3
votes
1
answer
952
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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
2
votes
0
answers
149
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Non-proper intersection between divisors on $\mathbb{P}^1$-bundle of Hirzebruch surfaces
We are working on algebraic closed field $k$. Let $\mathbb{F}_1$ be the Hirzebruch surface $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1))$, $C_0$ and $C_{\infty}$ are its zero and infinity sections ...
- 41
2
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145
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Is there a name for a normal, projective variety where every effective divisor is ample?
Is there a name for a normal, projective variety such that every effective divisor is ample? Examples of such varieties are projective space, weighted projective spaces, and simple Abelian varieties ...
- 912
2
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0
answers
109
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Reference for numerically non-negative polynomials for nef vector bundles
Let $K$ be a field. A polynomial $F \in \mathbb{Q}[X_1, \dots, X_r]$ which is weighted homogeneous of degree $n$ with respect to the grading $\deg(X_k) = k$ is called numerically non-negative for nef ...
user513306
2
votes
0
answers
172
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Intersection theory on normal crossing algebraic surfaces
Let $X$ be an algebraic surface with normal crossing singularities. Suppose the singular locus of $X$ is a smooth curve. Let us denote it by $C$. Suppose $D$ is another smooth curve in $X$ which ...
- 494
2
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130
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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-...
- 980
2
votes
0
answers
245
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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 $2$. ...
- 2,032
2
votes
0
answers
95
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 ...
- 121
2
votes
0
answers
207
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 ...
- 1,237
2
votes
0
answers
239
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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 ...
- 4,428
2
votes
0
answers
327
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 ...
- 1,168
2
votes
0
answers
323
views
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 ...
- 776
2
votes
0
answers
335
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 ...
- 151
2
votes
0
answers
186
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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
2
votes
1
answer
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
2
votes
0
answers
219
views
Liftability of varieties, after fpqc base change
Let $X$ be a smooth projective variety over a finite field, with a closed immersion to some other smooth projective variety $S$, with $S$ liftable.
Suppose there exists an fpqc cover $S'\to S$, such ...
user124171
2
votes
0
answers
136
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Explanation of proposition 6.7 (a) of Fulton's Intersetion Theory
Suppose $X$ is a smooth variety over a field $k$ of characteristic zero, and $Z$ is a smooth subvariety of codimension d. Now let $\tilde{X}$ be the blow-up of $X$ at $Z$, and let the exceptional ...
- 2,971
2
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0
answers
256
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Neron Severi under specialization
Let $X$ be a smooth projective variety over $\mathbf{Q}$, and $\mathcal{X}$ a smooth projective model over $\mathbf{Z}[1/N]$ for $N$ large enough.
Call $\eta$ the generic point $\text{Spec}(\mathbf{Q}...
user95222
2
votes
0
answers
656
views
Specialization maps for Chow groups
Let $S$ be a finite type regular integral affine scheme of finite type over $\text{Spec}(\mathbf{Z})$, and $\mathcal{X}\to S$ a smooth projective morphism. Let $\eta$ be the generic point of $S$, $s\...
user119470
2
votes
0
answers
239
views
Group completion of Chow varieties
Let $X$ be a quasi-projective variety over a perfect field $k$.
Given a projective embedding $j : X\to \mathbf{P}(\mathscr{E})$, the Chow variety $\text{Chow}_r(X, j)$ is a quasi-projective variety ...
user113393
2
votes
0
answers
261
views
Codimension restrictions on intersections
This is a question I stumbled across earlier this week. I see a similar one has been asked here.
Suppose I have a smooth quasi projective variety $X$ over a field $K$, and I call $\text{Chow}^r(X\...
user113452
2
votes
0
answers
228
views
On a class of loci in Chow varieties
Let $k$ be a field, $i:X\hookrightarrow \mathbf{P}(\mathscr{E})$ be a fixed projective embedding of a smooth projective $k$-variety $X$, whose dimension is pure and equals $d\ge 0$.
For $0\le p\le d$,...
user95222
2
votes
0
answers
200
views
Top intersections on the Hilbert scheme of points on a surface
The Picard group of $S^{[n]}$ is generated by the Picard group of $S$ (via a map $L \mapsto L_n$) and $E$, where $E = -\frac{B}{2}$, where $B$ is the exceptional divisor of the Hilbert Chow morphism.
...
- 1,509
2
votes
0
answers
250
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Computing higher dimensional intersection numbers for complete intersections of $\mathbb P^n$
Let $X_1,X_2$ be two smooth hypersurfaces of degree $d$ in $\mathbb P^{n}$. Let $B=X_1\cap X_2$. Assume $B$ is smooth. Let $\mathcal N_{B/\mathbb P^n}$ be the normal bundle to $B$. Let $H$ be the ...
- 2,215
2
votes
0
answers
99
views
Class of the locus where two sections are proportional
Let $X$ be a smooth (complex) projective $n$-dimensional variety ($n\geq 3$) and $\mathcal E$ a vector bundle of rank $r<n$ generated by its global sections on $X$. Let $\sigma\in H^0(\mathcal E)$ ...
- 1,463
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
2
votes
0
answers
288
views
Is the Gysin pullback of an effective cycle effective?
Suppose $E$ is a rank $r$ vector bundle over a projective variety $X$, denote the zero section by $i\colon X\to E$. Given an effective cycle $a\in A_{k+r}(E)$, the Gysin pullback gives us a class $i^
- 73
2
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0
answers
187
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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,...
- 533
2
votes
0
answers
333
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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
$f$ ...
- 1,205
2
votes
0
answers
245
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Segre class of cones and Base change of projective cones
I'm trying to work out a result in Fulton's intersection theory and I think I need the following basic result about base change of projective cones (whose support may not be the entire base scheme).
...
- 3,555
1
vote
0
answers
80
views
Computing Chow groups of affine, simplicial toric varieties
Let $k$ be an algebraically closed field. Let $X$ be an $n$-dimensional affine, simplicial toric variety over $k$. There exists an $n$-dimensional simplicial cone $\sigma$ in $\mathbb{R}^n$ such that $...
- 639
1
vote
0
answers
104
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Calculation of intersection multiplicity after the restricting to a fiber
Let $X\to\operatorname{Spec} \mathbb Z$ be an arithmetic surface which is projective, regular and integral. Let $D$ and $E$ two divisors intersecting at a point $x\in X$ that lies over the prime $p$. ...
- 321
1
vote
0
answers
207
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Todd class of blow-up
Let $i:X\hookrightarrow Y$ be an embedding of two non-singular projective varieties over $\mathbb{C}$. Consider the blow-up $f:Y' = Bl_XY \to Y$, and the corresponding embedding $j:E\hookrightarrow Y'$...
- 91
1
vote
0
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145
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Multiplicity and the perfect projective line
Let $\mathbf{F}_p$ be the field with $p$ elements, and $X = (\mathbf{P}^1_{\overline{\mathbf{F}}_p})^\text{perf}$ the inverse perfection of the projective line over $\mathbf{F}_p$.
Let $\Gamma$ be the ...
- 85
1
vote
0
answers
244
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Pull and push formula for degree for non-flat morphism
Let $\varphi\colon X_1\to X_2$ be dominant proper morphism of finite degree (in particular $\dim X_1=\dim X_2$) between varieties.
Let $D \subset X_2$ be a Cartier divisor.
Is it true that $$\varphi_*...
- 219
1
vote
0
answers
291
views
Is there some relations between derived category and intersection theory?
After learning the traditional intersection theory (W. Fulton's Intersection Theory and D. Eisenbud & J. Harris's 3264 and all that), I have some biased thinking about what I have learned in this ...
- 249
1
vote
0
answers
103
views
Is $K_F\cdot C\leq K_X\cdot C$ for a fibre $F\subseteq X$ containing the curve $C$?
This is a question that I originally posted on Math Stack Exchange. After a couple of days I have not received any comments or answers, and after thinking about it more I realize that this question is ...
- 131
1
vote
0
answers
70
views
Prescribed intersection of varieties
Every variety here is complex analytic, or complex algebraic if it solves anything.
Given a germ of a (possibly singular, nor necessarily irreducible) hypersurface $(H,0)\subset(\mathbb{C}^{n+1},0)$ ...
- 333
1
vote
0
answers
71
views
Comparison between residual intersection in Fulton's intersection theory and Aluffi's result on Milnor class
$\textbf{Question}$ I deduced that $m(A \cup B, V) = 0 $ for nonsingular variety $V$ and nonsingular hyper surfaces $A$ and $B$ whose intersection is also nonsingular. But I do not think it is true ...
- 121
1
vote
0
answers
81
views
How to calculate the divisor given by closure of subscheme
Let $X \subset \mathbb{P}^N$ be a nonsingular projective variety over algebraically closed field which is embedded by very ample line bundle $\mathcal{L}$. Let $Y = \mathbb{P}(\mathcal{L}^{\oplus 3})$...
- 121