Questions tagged [intersection-theory]
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376 questions
6
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Blow-up and the Chow group of zero cycles
Let $\tilde{X}\to X$ be a blow-up of a variety $X$ (over an algebraically closed field).
Is it true that the Chow group of zero cycles of $\tilde{X}$ is isomorphic to that of $X$? What if $X$ is a ...
- 69
1
vote
0
answers
250
views
Do A-infinity algebra(in Floer theory)have some kind of intersection theory and Poincare duality?
In Lagrangian Floer theory, we can define an A-infinity algebra. It is by first choosing a subset $X_L$ of chains in the Lagrangian submanifold $L$, and then defining boundary maps on(Actually, sum of ...
2
votes
0
answers
130
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intersections between closed curves on surfaces
I would like to find a result telling me that two simple closed curves $\alpha$ and $\beta$ (on a non-orientable surface $S$) are in minimal position if and only if there is not a disk in $S$ whose ...
- 21
8
votes
1
answer
792
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What is the main failure in using Naive Chow group in Artin Stack
I'm reading Andrew Kresch's paper, Cycle groups in Artin Stacks.
The author defined Chow groups of Artin stacks by very technical way, instead of ordinary ways which he called 'naive chow group', ...
- 421
0
votes
0
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213
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Linear section of an algebraic variety
Let $\pi$ be a linear subspace of $\mathbb{P}^n$ and $X$ a reduced, irreducible variety of $\mathbb{P}^n$. Suppose that $\pi \cap X$ is reducible, hence $\pi \cap X=Y_1\cup Y_2 \cup \cdots Y_k$. When ...
- 325
0
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0
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112
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Reducible sections of algebraic varieties
Let $X$ be an irreducible variety. Is there some necessary condition on a hyperplane $H$ such $X\cap H$ is reducible? Also, suppose that $H\cap X$ is reducible, i.e., $H\cap X=Y_1\cup Y_2 \cup \cdots \...
- 325
1
vote
1
answer
215
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Zero dimensional components of an intersection
Let $X$ be a smooth projective algebraic variety over an algebraically closed field and let $A,B$ be closed irreducible subvarieties of complementary codimension in $X$. Let $n$ denote their ...
- 13
2
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0
answers
200
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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
1
vote
0
answers
103
views
Degree of an isogeny in the endomorphism ring of the jacobian of a curve and self intersection index in its ring of correspondences
I hope this question is not too basic.
Let $C/\bar{k}$ be a nonsingular irreducible curve of genus $g$ and $\mathfrak{C}(C\times C)\cong \text{CH}^1(C\times C)$ be its ring of correspondences.
I am ...
0
votes
1
answer
342
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Intersections of divisors in blow-ups of $\mathbb{P}^n$
Let $p_1,p_2,p_3\in\mathbb{P}^n$ be three general points, $X$ the blow-up of $\mathbb{P}^n$ at $p_1,p_2,p_3$, then along the lines $\left\langle p_i,p_j\right\rangle$, and finally along the plane $\...
user91659
4
votes
1
answer
215
views
Segre Classes of reducible variety
Suppose I have a singular projective variety $X\subset \mathbb{P}^n$ that is reducible with $X=\bigcup_i X_i$ smooth irreducible components. That is, the irreducible components are smooth but $X$ is ...
- 3,779
1
vote
0
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279
views
How to think about the quotient field of an integral stack?
This is the definition given in Vistoli's paper.
Let $F$ be an integral stack. A rational function of $F$ is a morphism $G \rightarrow A^1_S$ defined on a nonempty open substack $G$ of $F$.
...
- 231
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
6
votes
2
answers
345
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Nonempty intersection in Grassmannian
Where can I find a proof of the following fact:
If $X_1$ and $X_2$ are subvarieties of $\mathbb{G}(k,n)$ of codimension $c_1$ and $c_2$ satisfying $c_1+c_2<n+1-2k$, then the intersection $X_1\cap ...
- 1,537
2
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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
3
votes
0
answers
138
views
Non-generic intersections of Schubert varieties?
Let $G$ be a linear algebraic group, $B$ a Borel subgroup, $P$ a parabolic subgroup containing $B$, and $W$ the Weyl group. For $w \in W$, the Schubert variety $X_w^P$ is the closure of the Schubert ...
- 191
2
votes
1
answer
139
views
Intersection multiplicity of limit linear spaces
Let $X\subset\mathbb{P}^N$ be a smooth projective variety. Let us fix a general point $q \in X$, and let $C\subseteq X$ be a smooth curve passing through $q$.
Now let $\Lambda_{\xi, q}$, with $\xi \...
user91659
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
3
votes
0
answers
119
views
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 $C_2^2<0$)....
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7
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551
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Semi-continuity of intersection numbers
I always trusted the following quite vague statement:
If you have a family of effective divisors $D_1(t),\dots , D_k(t)$ on a $k$-dimensional projective variety $X_t$, where $t$ is a paramater say ...
- 2,384
1
vote
0
answers
139
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A strong form of Bezout theorem
Let $X$ be a smooth projective variety of dimension $n$, $U \subset X$, non-empty open set. For any integer $k>0$, does there exist $n$-hypersurface sections $Z_1,...,Z_n \in |\mathcal{O}_X(k)|$ ...
- 2,126
4
votes
1
answer
165
views
The volume around a singular isolated root when equalities are loosened
Suppose I have a system of polynomial equations in $n$ real variables $f_i(x_1,\ldots,x_n)=0$, $i=1,\ldots,m$, such that $0$ is an isolated solution. Now I replace each of the equations with a double-...
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21
votes
1
answer
981
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$8$-ary operation $(\mathbb{P}^2)^8 \text{ }-\to \mathbb{P}^2$, can we say anything about what this formula would look like?
My friend, who is currently taking an algebraic geometry course from an unnamed prolific poster on MO, told me about the following bonus question on one of his problem sets a few weeks ago.
...
user61522
2
votes
0
answers
132
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Common Point of Intersection of n-dimensional ellipsoids [closed]
Suppose we have two ellipses in 2-dimensions centered at the origin. It is easy to visualize that (unless one is contained in the other) they will have 4 points of intersection. Can we say that in ...
- 21
5
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0
answers
486
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Computing intersection number of two arithmetic line bundles
I have some questions in Arithmetic Arakelov geometry
Let $\mathcal X\to Spec(\mathcal O_K)=C$ be an arithmetric projective variety over $C$ , where $\mathcal O_K$, ring of number filed $K$ and $\...
user21574
2
votes
2
answers
219
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Intersection of Subspaces with $O(3)$
Sorry for the confusion from earlier. I tried to fix the thread. The old version can be found below.
For $6$-dimensional subspaces $V$ of the space $\mathbb{R}^{3\times 3}$ of real three-times-three ...
- 9,863
0
votes
0
answers
141
views
Chern classes of a family and Chern classes of a member
Let $X$ be a smooth projective variety over an algebraically closed field $k$ and $\mathcal E$ a family on torsion free coherent sheaves on $X$ parametrized by a smooth curve (over $k$) i.e. a ...
- 1,463
1
vote
1
answer
406
views
Intersection product of pull back under projection
Let $X$ be a surface and $Y$ be a curve over $\mathbb{C}$. Let $L$ and $L'$ be ample line bundles on $X$ and $Y$ respectively. Consider the product $X\times Y$. Let $p$ and $q$ be the projection from $...
- 643
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0
answers
156
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Showing that closure of all lines through a projective variety $Y$ has degree strictly less than $Y$
Let $Y$ be a variety of dimension $r$ and degree $d>1$ in $\mathbf{P}^n$. Let $P\in Y$ be a nonsingular point. Define $X$ to be the closure of the union of all lines $PQ$, where $Q\in Y$, $Q\ne P$. ...
- 1,217
2
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0
answers
288
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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^
- 101
5
votes
1
answer
516
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Smooth quadric hypersurface, Hilbert scheme is blowup of Grassmannian?
Let $Q \subset \mathbb{P}^n$ be a smooth quadric hypersurface. Where can I find a proof of/can anyone supply a proof of$$\text{Hilb}_{2m + 1}(Q) \cong \text{Bl}_{OG(3, n+1)}G(3, n+1)?$$Can we conclude ...
- 51
1
vote
1
answer
387
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Tor-amplitude [0, 1] in the setting of intersection theory on a regular surface?
The question is coming from Definition 1.5 in Deligne's Expose X in SGA 7 on intersection theory.
Let $X$ be a connected regular scheme of dimension $2$ and $Y \subset X$ a reduced divisor that ...
- 825
4
votes
1
answer
237
views
Bézout's theorem for arcs in the plane
Consider two polynomials $p,q \in {\mathbb R}[x,y]$, both of degree $d$. Let $\gamma_p$ and $\gamma_q$ be the two curves in ${\mathbb R}^2$ that are defined by these polynomials, and assume that these ...
- 1,072
0
votes
0
answers
639
views
Transversal intersection in the moving lemma
Let $X$ be a smooth projective variety over an algebraically closed field and let $A,B$ be closed irreducible subvarieties of $X$. Chow's moving lemma which is proved in the book by Eisenbud and ...
- 176
3
votes
1
answer
324
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Calculating the distinguished varieties of intersection product
In Fulton's Intersection theory Example 6.1.2，one considers two divisors on $\mathbf{P}^2$ given by $D_1=A+2B,D_2=2A+B$, where $A,B$ are lines meeting at a point.
Let $X=D_1\times D_2,Y=\mathbf{P}^2\...
user39380
6
votes
1
answer
1k
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What does the Chern-Schwartz-MacPherson class of a singular variety look like?
Let $A_\ast$ and $F_\ast$ be the functors $\textrm{Var}_\mathbb C\to \textrm{Ab}$ of Chow groups and constructible functions, respectively, with respect to proper maps. Then the Chern-Schwartz-...
- 1,534
6
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0
answers
298
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On the local Euler obstruction for singular varieties
Let $X$ be a complex algebraic variety (not necessarily irreducible, nor reduced). Then the local Euler obstruction is a group isomorphism $$\textrm{Eu}: Z_\ast X\to F_\ast X,$$ where $Z_\ast X$ is ...
- 1,534
6
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0
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363
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Why write GRR with the relative tangent sheaf?
The first published version of the Grothendieck-Riemann-Roch theorem, GRR for short, was written in the form
$$
\operatorname{ch}(f_!\alpha).\operatorname{Td}(Y)
=
f_*\left(\operatorname{ch}(\alpha).\...
- 2,011
4
votes
0
answers
225
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Intersection numbers on blow ups of toric varieties
Suppose we have a smooth, complete toric varietiy $X_{\Sigma}$ of dimension $n$. Let $\sigma_k \in \Sigma(k)$ a smooth $k$-dimensional cone in $\Sigma$ and suppose we make the toric blow up at the ...
- 357
0
votes
1
answer
177
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Continuity of Intersection Multiplicities
I’m looking for a correct technical version (and in the best case a reference) for a statement of the following type:
Consider a complex algebraic variety $X\subset\mathbb{P}^n$ and a sequence of ...
- 174
0
votes
1
answer
381
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Samuel multiplicity
Let $X$ be the hyper-surface defined by
$$f:=\sum_{i=1}^k x_i^n=0$$
in $\mathbb{C}^k$. Let $Y$ be the non-reduced sub-scheme of $X$ defined by the ideal
$$I=(x_1^{n-1},\dots , x_k^{n-1}) $$
What is ...
- 2,384
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
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
1
vote
1
answer
457
views
Non-proper intersection of surfaces
I'm interested in the first basic case of excess intersection in intersection theory:
Let $X$ be a smooth projective 4-fold and let $S,T$ be two surfaces in $X$. Assume that the intersection $S\cap T$...
- 587