All Questions
Tagged with intersection-theory ag.algebraic-geometry
329 questions
2
votes
0
answers
130
views
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
3
votes
2
answers
197
views
Moving lemma for countable collection of subvarieties
Fix an integer $n \ge 5$. Let $\mathcal{V}$ be a countable collection of closed subvarieties of $\mathbb{P}^n_{\mathbb{C}}$ of codimension at least $2$. Choose a point $p \in \mathbb{P}^n$. Does there ...
- 2,323
3
votes
1
answer
225
views
Extending effective Cartier divisors
Let $X$ be a non-singular, quasi-projective variety (over $\mathbb{C}$) of dimension at least $3$, $D_1, D_2$ are integral effective divisors in $X$ with $D_1 \cap D_2$ of codimension $2$ in $X$. Let $...
- 2,323
2
votes
0
answers
245
views
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
4
votes
2
answers
328
views
Is the sum of a radical ideal and the ideal of a generic linear space intersecting that ideal radical?
Let $X \subseteq \mathbb{C}^n$ be an irreducible algebraic set that forms a cone, and let $I=I(X) \subseteq \mathbb{C}[x_1,...,x_n]$. Let $m < n$ and $k\leq m$ be positive integers. Is it true that ...
- 980
1
vote
0
answers
170
views
Pinch points and dual surfaces
I am currently reading Fulton's expository lectures "Introduction to intersection theory in algebraic geometry".
On pg. 4, Fulton sketches an argument of George Salmon which I don't ...
4
votes
1
answer
464
views
Intersection of curves in non-singular projective algebraic surfaces
Bezout thereom that says that two irreducible algebraic curves $C$ and $D$ in $\mathbb{P}^2_\mathbb{C}$ intersect at $nm$ points (counted with multiplicity), where $n$ and $m$ are the degrees of $C$ ...
- 49
2
votes
1
answer
389
views
Are "transverse" hyperplane sections of nondegenerate irreducible projectice varieties always nondegenerate
Let $X \subseteq \mathbb{P}^n$ be a irreducible complex projective variety. It is called nondegenerate if it is not contained in a hyperplane in $\mathbb{P}^n$.
Assuming $X$ is nondegenerate and ...
- 1,141
5
votes
1
answer
309
views
Intersection cycle in a product of Grassmannians
Let $G(k,n)$ denote the Grassmiannian of $k$-planes in $\mathbb C^n$. Let's define
$$ I_j =\{ (\Lambda_1,\Lambda_2 ) \in G(k,n) \times G(l,n) \, | \, \dim(\Lambda_1 \cap \Lambda_2) \geq j \}. $$
These ...
- 344
3
votes
0
answers
91
views
A proper morphism restricts to a closure of a point on the generic fiber
Let $\pi:X^{e}\rightarrow Y^{d}(e\geq d)$ be a proper and dominant morphism of projective varieties over field $k$. Moreover, $Y$ is assumed to be smooth. Denote $\eta$ the generic point of $Y$, $X_\...
- 113
1
vote
0
answers
220
views
Genus of a curve given by self intersection of a very ample line bundle
Let $X$ be a smooth, integral and projective $d$-dimensional variety over a field $k$ of characteristic 0. Let $H$ be a very ample line bundle over $X$. Assume that there exists a smooth and ...
- 51
6
votes
0
answers
250
views
Singling out irreducible components
Let $V\subset \mathbb{A}^n$ be a variety defined by equations of degree $\leq D$, or, what is the same, an intersection of hypersurfaces of degree $\leq D$. Let $V^+_0$ be an irreducible component of $...
- 20.2k
3
votes
0
answers
95
views
Singling out lower-dimensional components
Let $V\subset \mathbb{A}^n$ be defined by equations of degree $\leq D$. (That is, $V$ is an intersection of hypersurfaces of degree $\leq D$.) Assume $V$ is not pure-dimensional. Let $V^-$ be the ...
- 20.2k
0
votes
0
answers
201
views
Intersection product when one factor is contained in the exceptional divisor
I am trying to calculate some intersection numbers and would appreciate help on the following problem:
Consider two divisors $D_1$ and $D_2$. Blowing up their intersection yields $\varphi^{*}(D_i) = \...
- 1
6
votes
2
answers
789
views
Reference request: Kleiman's proof of Snapper's Lemma
On page 4 of Nitin Nitsure's paper Construction of Hilbert and Quot Schemes, the author refers to the fact that Hilbert polynomials are indeed polynomials as
a special case of Snapper's Lemma, see &...
- 1,805
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
3
votes
0
answers
224
views
algebraic vs rational equivalence
Are there classes of algebraic varieties for which algebraic and rational equivalence for algebraic cycles coincide? (references also appreciated)
- 3,779
3
votes
0
answers
291
views
Transversal intersection with linear subspaces
Let us work over an algebraically closed field $K$. If $X\subset \mathbb{P}^n$ is a closed subset of dimension $r$, then there should exist a linear subspace $L\subset \mathbb{P}^n$ of dimension $n-r$ ...
- 7,722
10
votes
0
answers
327
views
Integrality of primary genus $0$ Gromov-Witten invariants of a Fano manifold
Suppose $(X,\omega)$ is a positively monotone compact symplectic manifold, i.e., after a positive scaling of the symplectic form, we have $c_1(T_X) = [\omega]$ in de Rham cohomology ($T_X$ has well-...
- 1,761
3
votes
1
answer
699
views
Intersection theory on singular varieties by embedding to smooth ones
Let $X$ be a normal complex projective variety over $\mathbb C$. In order to define the intersection product of the Chow ring, one usually requires $X$ to be smooth. How to weak the smoooth assumption ...
- 125
1
vote
0
answers
201
views
How to compute the genus of the (singular) intersection of three quadratics in $\mathbb{C}P^4$?
Consider three quadratics in $\mathbb{C}P^4$:
$$
x_0^2+4x_1^2+\frac{x^2_2}{4}=0,~ x_1x_4+x_2x_3=0, ~ x_0^2+x_3^2+x_4^2=0.
$$
If there intersection was non-singular, then the intersection should be a ...
- 9,019
5
votes
0
answers
163
views
How to compute the class defined by intersection with a square?
$\DeclareMathOperator\Gr{Gr}$Let $\Gr(k,n+k)$ (of course, one can do also for $\Gr(k,\infty)$) be the complex Grassmannian of $k$-planes in $n+k$-dimensional linear space.
It is well-known that ...
- 719
2
votes
0
answers
206
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
1
vote
0
answers
58
views
Locus of linear spaces with prescribed contact order
Let $X\subset\mathbb{P}^{n}$ be a smooth projective variety of pure dimension $d$. Let $Z\subset \mathbb{G}(n-d,n)\times\mathbb{P}^{n}$ be the space of pairs $(P,x)$ of a linear space $P\cong\mathbb{P}...
- 735
4
votes
3
answers
525
views
Irreducible components: associativity for intersections?
Let $A$, $B$, $C$ be closed irreducible subvarieties of $\mathbb{A}^n$. Let $V_1$ be an irreducible component of $B\cap C$, and $V$ an irreducible component of $A\cap V_1$. Must there
necessarily be ...
- 20.2k
1
vote
1
answer
729
views
Push-forward of divisors and intersections
Let $f:X\rightarrow Y$ be a surjective finite morphism of varieties, with $X$ normal and $Y$ smooth. Let $D\subset X$ be a divisor and $C\subset Y$ a curve. Does the equality
$$C\cdot f_{*}D = f^{*}C\...
user168627
6
votes
1
answer
622
views
Intersection theory in analytic geometry
This might be a weird/stupid question, but it came to me a couple of times, and I would like to get an answer for that.
In some papers I read, constantly the authors define some analytic subspaces, ...
- 287
7
votes
0
answers
183
views
Intersection numbers via residue formula
$\newcommand{\sslash}{\mathbin{/\mkern-7mu/}}$With a friend we are trying to understand residue formulas in the article "Cohomology pairings on singular quotients in geometric invariant theory&...
10
votes
1
answer
903
views
Interpretation of "27" lines for cubic surface with rational double points
It is well known that a smooth cubic surface has $27$ distinct lines. Explicitly, if we choose a planar representation, i.e., blowup $\mathbb P^2$ at $6$ general points $p_1,...,p_6$, the $27$ lines ...
- 1,803
4
votes
1
answer
379
views
Relative canonical class of blowing-up a flag ideal
Let $X$ be a smooth complex projective variety of dimension $n$. Consider a flag ideal $I$ on $X\times \mathbb{P}^1$, namely,
$$
I=I_0+I_1t+I_2t^2+\cdots+I_{N-1}t^{N-1}+(t^N)\,,
$$
where $t$ is the ...
- 329
2
votes
0
answers
239
views
Localization of Chow groups and flat base change
For any flat morphism $f:X\rightarrow Y$, we have a flat pullback of Chow groups
$$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
1
answer
330
views
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
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
1
vote
0
answers
157
views
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
views
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 $\{...
- 6,006
2
votes
1
answer
585
views
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
3
votes
0
answers
174
views
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
votes
0
answers
279
views
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
1
vote
0
answers
149
views
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'}>> ...
- 6,006
1
vote
1
answer
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 $\...
- 115
4
votes
0
answers
226
views
K-theoretic derivation of Bézout theorem
In the paper "$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
votes
0
answers
608
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
-2
votes
1
answer
282
views
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
2
votes
1
answer
484
views
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
vote
0
answers
91
views
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
vote
0
answers
116
views
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
votes
1
answer
324
views
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
1
vote
0
answers
293
views
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
views
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