All Questions
Tagged with intersection-theory ag.algebraic-geometry
329 questions
7
votes
0
answers
551
views
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 ...
CommunityBot
- 1
1
vote
0
answers
139
views
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-...
CommunityBot
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21
votes
1
answer
981
views
$8$-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.
...
CommunityBot
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2
votes
0
answers
132
views
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
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 $...
- 38k
3
votes
1
answer
324
views
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\...
- 203
15
votes
3
answers
2k
views
Can a curve intersect a given curve only at given points?
Clearly the question in the title has a positive answer for analytic (or smooth, or continuous ...) curves, but what about the algebraic category? More specifically, given an irreducible polynomial ...
- 5,359
0
votes
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
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^
- 4,111
1
vote
1
answer
387
views
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 ...
- 4,111
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 ...
- 149k
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
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-...
- 3,290
6
votes
0
answers
298
views
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
votes
0
answers
363
views
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
$$
\operatorname{ch}(f_!\alpha).\operatorname{Td}(Y)
=
f_*\left(\operatorname{ch}(\alpha).\...
- 2,011
4
votes
0
answers
225
views
Intersection numbers on blow ups of toric varieties
Suppose we have a smooth, complete toric varietiy $X_{\Sigma}$ of dimension $n$. Let $\sigma_k \in \Sigma(k)$ a smooth $k$-dimensional cone in $\Sigma$ and suppose we make the toric blow up at the ...
- 357
0
votes
1
answer
177
views
Continuity of Intersection Multiplicities
I’m looking for a correct technical version (and in the best case a reference) for a statement of the following type:
Consider a complex algebraic variety $X\subset\mathbb{P}^n$ and a sequence of ...
- 27.9k
0
votes
1
answer
381
views
Samuel multiplicity
Let $X$ be the hyper-surface defined by
$$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}) $$
What is ...
3
votes
0
answers
371
views
Intersection Multiplicity
Let $X$ be an hyper-surface in an affine space defined by an equation $F$. We can assume that the ground field is $\mathbb{C}$ and $X$ is normal. Take functions $f_1,\dots, f_n$ on $X$ and let $B$ ...
- 2,384
1
vote
1
answer
457
views
Non-proper intersection of surfaces
I'm interested in the first basic case of excess intersection in intersection theory:
Let $X$ be a smooth projective 4-fold and let $S,T$ be two surfaces in $X$. Assume that the intersection $S\cap T$...
- 597
0
votes
1
answer
184
views
Intersection multiplicty and global sections
Let $X$ be a smooth projective variety, $V, W$ closed subschemes in $X$ such that $V \cap W$ is finitely many points. Let $\mathcal{L}$ be a line bundle on $X$. Is there any relation between $h^0(\...
- 39.3k
2
votes
1
answer
358
views
Self-intersection and generic point
The Wikipedia entry on intersection theory contains the following statement:
[for C a curve, on a surface] "the self-intersection points of C is the generic point of C, taken with multiplicity C · C."...
CommunityBot
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1
vote
1
answer
499
views
A question about an intersection number
Let $\pi:Y\rightarrow \mathbb{P}^3$ be the blow-up of two points $p,q\in\mathbb{P}^3$, and then of the strict transform of the line $L$ spanned by them. Now, Let $E_p,E_q, E_{p,q}$ be respectively the ...
CommunityBot
- 1
0
votes
1
answer
411
views
Intersection Matrix of a resolution
Probably this is a very easy question. Let $f:X\rightarrow S$ be a resolution of a projective surface such that
$$K_X = f^{*}K_S+\sum_ia_iE_i$$
with $a_i>0$. By Grauert-Mumford theorem the ...
- 8,998
15
votes
1
answer
995
views
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 ...
- 12.9k
2
votes
0
answers
325
views
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 ...
- 73
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 ...
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
5
votes
0
answers
293
views
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
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 ...
CommunityBot
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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}^{...
CommunityBot
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0
votes
0
answers
237
views
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
5
votes
1
answer
269
views
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 ...
- 149k
0
votes
0
answers
145
views
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
2
votes
1
answer
615
views
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 ...
2
votes
1
answer
606
views
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 ...
- 8,998
8
votes
2
answers
5k
views
Self-intersection of exceptional divisor
Suppose that $X$ is a smooth threefold, and $C \subset X$ a smooth curve. Let $Y$ be the blowup of $X$ along $C$, with exceptional divisor $E$. What is the intersection number $E^3$ on $Y$? (in ...
- 8,998
3
votes
1
answer
245
views
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 $k=0,1,2$ ?
- 8,998
4
votes
1
answer
680
views
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 ...
- 8,998
4
votes
1
answer
4k
views
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 ...
- 34.7k
1
vote
1
answer
465
views
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
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
5
votes
1
answer
332
views
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 ...
- 761
1
vote
1
answer
377
views
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. ...
- 1,595
5
votes
1
answer
304
views
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 ...
- 1,054