The intersection-theory tag has no wiki summary.
7
votes
1answer
32 views
Elliptic Surfaces: a naive question
Ground field $\Bbb{C}$. Algebraic category. Smooth surfaces.
Let $S$ be a minimal elliptic surface $p:S\rightarrow C$ the elliptic fibration (general fiber = elliptic curve). Suppose the ...
4
votes
0answers
48 views
Dimension of the image of the morphism associated to a Divisor
Ground field $\Bbb{C}$. Let $D$ be a divisor on an algebraic smooth surface $S$.
Suppose the complete linear system $|D|$ is base-point free and $h^0(D)=N+1$ with $N>0$. To $D$ is then associated ...
3
votes
1answer
61 views
Linear equivalence of Divisors on a Surface
Studying algebraic geometry, while the abstract theory is pretty clear to me I often feel puzzled in practice. Here I am trying to understand linear equivalence of divisors in some practical ...
2
votes
0answers
63 views
When is an intersection of varieties finite
Consider the general Bezout's theorem: If $p_1 \ldots p_n$ are polynomials with degrees $d_1,\ldots, d_n$ in $\mathbb{R}[x_1,\ldots,x_n]$, with $V = \{a=(a_1,\ldots, a_n) | p_i(a) = 0, \forall i\}$ ...
3
votes
3answers
68 views
Transverse intersection of multiple submanifolds
Let $M$ be a smooth manifold and suppose that we have three (or more) submanifolds $N_1,N_2,N_3\subset M$.
What is the right notion of "transverse intersection" of $N_1,N_2,N_3$, i.e. what is the ...
5
votes
0answers
60 views
Why is an intersection product $X.C=0$?
Here is the situation: $S$ is a nonsingular complex projective surface, and $C=X\cup_AY\subset S$ is a uninodal curve of compact type: it is obtained by glueing two nonsingular curves $X,Y\subset S$ ...
6
votes
1answer
37 views
Injectivity of the push-forward of cycles modulo rational equivalence
Let $Y$ be a closed subscheme of a scheme $X$, and let $i: Y \rightarrow X$ be the inclusion morphism. We have the push-forward $i_{\ast}: A_{k}(Y) \rightarrow A_{k}(X)$, where $A_{k}(X)$ denotes the ...
3
votes
0answers
50 views
Degree of line bundles is additive on curves
In Hartshorne (among other places), it is used that on a nonsingular curve $C$, the degree of line bundles is additive. That is,
$$\mbox{deg}_C(L_1 \otimes L_2) = \mbox{deg}_C(L_1)+ ...
2
votes
1answer
63 views
Showing that intersection multiplicity at a point is finite for prime divisors
My question has two parts two it: one vaguely more elementary, one perhaps less so.
In Beauville (Complex Algebraic Surfaces), we define the multiplicity of intersection of two (irreducible, no ...
3
votes
2answers
80 views
Effective does not imply nef?
The question is really simple, its just terminology.
For simplicity we work on smooth algebraic surfaces and we consider the intersection form on curves on the surface.
So let $S$ be a surface and ...
-1
votes
1answer
72 views
How to remove intersection of ideal $I$ and $J$ from union of ideal $I$ and $J$
after get the intersection of ideal $I$ and ideal $J$
how to remove this intersection from union of ideal $I$ and ideal $J$
in order to do prime decomposition
how can it do in maple?
actually i ...
0
votes
1answer
84 views
Less restrictive set intersection
I have a question that may be trivial but I just can't find an appropriate answer on the Internet. The inclusion-exclusion principle can be used to discern the cardinality of the union among sets ...
6
votes
0answers
92 views
Poincaré duality and intersection
Let's take $X$ and $Y$ K3 surfaces and $Z\subset X\times Y$ an algebraic cycle of dimension 2. I know that the Poincarè dual of $Z$, namely $[Z]$, is in $H^4(X\times Y,\mathbb{Z})$ and by Kunneth ...
8
votes
0answers
125 views
Self-Intersection Number $-2$
I am new here, but hopefully you can help me with a concrete problem I have. I try to compute a Self-Intersection Number of a constructed curve in an analytic surface. I know the answer by some ...
5
votes
1answer
98 views
Intersection paring on homology and cup product cohomology?
One advantage of working with cohomology rather than homology is that cohomology is naturally endowed with a ring structure via cup product. However the cup product is often understood via ...
4
votes
1answer
49 views
Equivalence of definition for polarized K3
In the literature there are two different definitions of polarized K3 surfaces.
1) A polarized K3 is the data $(X,\omega)$. Where $X$ is a K3 surface and $\omega$ is an ample class in ...
4
votes
0answers
28 views
What is bivariant Chow group?
I have trouble understanding "bivariant Chow groups". Remember that for any morphism of schemes $f:X\rightarrow Y$, we can define a bivariant Chow group $A^*(f:X\rightarrow Y)$. When $Y$ is a point, ...
7
votes
0answers
82 views
(Weil divisors : Cartier divisors) = (p-Cycles : ? )
Suppose $X$ verifies the suitable conditions in which Weil (resp. Cartier) divisors make sense.
The group of Weil divisors $\mathrm{Div}(X)$ on a scheme $X$ is the free abelian group generated by ...
1
vote
1answer
75 views
Degree of varieties in $\mathbb{P}^n$
The degree of a variety $X$ of dimension $r$ is defined by $r!$ times the leading coefficient of its Hilbert polynomial. This is the defination given in Hartshorne, but I find it is very hard to ...
2
votes
0answers
95 views
Why does Fulton's Intersection Theory define $x \cdot_f y$ in this way?
His definition 8.1.1: Let $f:X\rightarrow Y$ be a morphism, with $Y$ non-singular of dimension $n.$ Let $p_X: X' \rightarrow X$, $p_Y: Y' \rightarrow Y$ be morphisms of schemes $X',Y'$ to $X$ and $Y$ ...
2
votes
0answers
64 views
Example of excess intersection theory?
Let $M$ be a smooth manifold of dimension $m$ and $\pi:E\rightarrow M$ a vector bundle of rank $e$. Given a section $s$ of the bundle $\pi:E\rightarrow M$, we expect that the zero locus $Z(s)$ of $s$ ...
2
votes
1answer
112 views
Computation of normal cones
I am reading Fulton's intersection theory but I have a poor intuition about the normal cone. I know the cone $C_{Y/X}$ is the normal bundle if $Y\subset X$ is smooth. I would appreciate it if someone ...
1
vote
2answers
170 views
When to read of the degree of a variety from its defining polynomials
The question concerns algebraic varieties.
I just read the question
The degree of an algebraic curve in higher dimensions
and great answer by user M P. One of the thing he says is that if a curve in ...
5
votes
1answer
698 views
Intersection form on quotient manifold
I have a simple algebraic topology question. Let $M$ and $N$ be 2-dimensional oriented manifolds (say $H^{2}(M,\mathbb{Z})\cong \mathbb{Z}\alpha_{M}$ and $H^{2}(M,\mathbb{Z})\cong ...
0
votes
0answers
36 views
Reducing a flat morphism $\psi:X\rightarrow\mathbb{A}_{\mathbb{C}}^1\;$ to $\;\psi|_{Y}: X\cap Y\rightarrow \mathbb{A}_{\mathbb{C}}^1$
Suppose $\psi: X\rightarrow \mathbb{A}_{\mathbb{C}}^1$ is a flat morphism, where $X\subseteq \mathbb{A}_{\mathbb{C}}^n$ with $X$ not needing to be smooth, with
$\psi^{-1}(0)$ being a complete ...
6
votes
2answers
137 views
Actually calculating an Intersection of Variety and Divisor
This has been bugging me for a while now.
Say I have a projective variety given by some polynomial $P$ and the canonical divisor of the projective space. How can I concretly calculate the ...
0
votes
0answers
132 views
Determine angle of arc from intersecting chord theorem
I'm trying to figure out how to draw an arc in CoreGraphics. Ok, I understand which method call to make and how to compute the angles when in the following scenario,
...
2
votes
0answers
67 views
Classes of Schubert Cycles form a basis
I am reading the not yet published book of Eisenbud and Harris about intersection theory (http://isites.harvard.edu/fs/docs/icb.topic720403.files/book.pdf) and I don't quite understand the following:
...
1
vote
1answer
30 views
Rational embedded irreducible curves in a complex surface.
Given a complex surface $X$ and an embedded irreducible compact curve $C$ with its arithmetical genus $g(C) = 0$, how can one show that $C$ is non-singular ?
Thanks for your answers!
0
votes
1answer
56 views
Topological description of map coming from correspondence
An algebraic correspondence between two varieties, $V,W$ is a kind of multi-valued map from $V$ to $W$, or, in other words, a map from a covering $U$ of $V$ to $W$. Apparently, such a map gives a ...
3
votes
1answer
113 views
Very special rational points on curves over number fields
For some reason, I'm convinced the answer to the following question should be (obviously) negative, but I can't come up with a good reason.
Does there exist a number field $K$, a smooth projective ...
1
vote
1answer
88 views
self-intersections in a product of two curves
Let $X$ be a smooth projective curve of genus $g$ over an algebraically closed field and consider the intersection pairing on the surface $X \times X$. I remember hearing that $\Delta^2 = 2-2g$: how ...
2
votes
1answer
124 views
Intersection of the Irreducible Components of Intersections of Schubert Varieties
Let $K$ be an algebraically closed field and $G$ be the Grassmannian of $k$ planes in some $l$ dimensional vector space $V$ over $K$. Let $V_1\subsetneq ... \subsetneq V_l$ be a flag for $V$. A ...
5
votes
1answer
144 views
Chern numbers of projective space blown up in a linear subvariety
This is a follow-up to my previous question about chern numbers. Write $\mathbb{P}^n:=\mathbb{P}^n_\Bbbk$ for projective space over some field $\Bbbk$ and assume that $X\subseteq\mathbb{P}^n$ is a ...
5
votes
2answers
261 views
Chern numbers of Projective Space
Consider the $k$-th chern class $c_k:=c_k(\mathcal{T}_{\mathbb{P}^n})$ of the tangent sheaf of projective space $\mathbb{P}^n=\mathbb{P}^n_\Bbbk$ over some (algebraically closed, if you want) field ...
