All Questions
Tagged with intersection-theory complex-geometry
21 questions
1
vote
1
answer
370
views
Self-intersection of the diagonal on a surface
Let $X$ be a smooth projective curve over the complex numbers, and take $\Delta$ the diagonal divisor on $X\times X$. Using the adjunction formula, one computes $\Delta\cdot\Delta =2-2g$ for $g$ the ...
- 85
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
2
votes
0
answers
172
views
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
votes
1
answer
243
views
Intersecton form of complete smooth Toric surface
Given a complete smooth Toric surface (over $\mathbb C$), is its intersection form well-known? Or is there an algorithm to calculate it? Thanks in advance.
- 343
2
votes
1
answer
262
views
Algebraic and homological equivalence relations for $0$-cycles
Let $X$ be a connected smooth projective variety. Let $Z_0(X)_{alg}$ be the group of $0$-cycles algebraically equivalent to $0$ and $Z_0(X)_{\hom}$ be the group of $0$-cycles homologically equivalent ...
- 519
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
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
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
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
3
votes
1
answer
774
views
Homotopy of paths at the boundary
Denote by $\Gamma$ a hypersurface in $\mathbb{C}^2$, i.e. the zero locus of a polynomial of two complex variables. Denote by $X$ the complement of $\Gamma$ in $\mathbb{C}^2$. I am trying to define a ...
- 293
6
votes
1
answer
717
views
Fulton's deformation to the normal cone vs Verdier's
Let $X$ be a smooth variety over a field $k$, and let $Y$ be a smooth subvariety. In the literature, I've seen two versions of the deformation to the normal cone:
Verdier's version: $\tilde{X}_Y^\...
- 3,079
6
votes
0
answers
241
views
Bezout theorem for germs of holomorphic functions
UPDATE.
It was pointed out by @Dmitri that two smooth curves given by $f=y$ and $g=y+x^k$ in $\mathbb C^2$ provide a simple counterexample.
Let $f_1, \ldots, f_p, g_1, \ldots, g_q$ be germs of ...
- 1,238
4
votes
2
answers
347
views
Cycle class of zeroes of a global section
Let $\mathcal{F}$ be a locally free sheaf of rank $n$ on an $n$ dimensional complex manifold $X$. If the zero locus of a generic global section of $\mathcal{F}$ is $0$ dimensional, then its cycle ...
- 2,788
2
votes
0
answers
136
views
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
8
votes
0
answers
569
views
Bloch Ogus spectral sequence
Let $X$ be a smooth projective variety over $\mathbf{C}$, and $p : X_{\rm an}\to X_{\rm Zar}$ the obvious map of sites.
The Leray spectral sequence
$$H^r(X_{\rm Zar}, R^sp_*\mathbf{C})\Rightarrow H^{...
user92332
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
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}^{...
- 3,472
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 ...
- 7,609
2
votes
0
answers
333
views
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
4
votes
1
answer
4k
views
How many points determine a line?
Consider the affine space $\mathbb C^n$ and then, because of reasons, compactify it to obtain the projective space $\mathbb P^n$. One of the most basic axioms or propositions of geometry is that ...
- 4,343
11
votes
4
answers
4k
views
Question on Kähler/ample cone, cone of curves....
Assume $X$ is smooth "simply connected" complex projective variety and $Y\subset X$ a smooth hyperplane section. ( $Y= X\cap H$, $H\subset \mathbb{P}^n$).
Let's $NE(X)$ be the cone of effective 1-...
