All Questions
Tagged with complex-geometry schemes
23 questions
13
votes
1
answer
542
views
Is there an analogue of projective spaces for proper schemes?
Does there exist a countable set of connected proper smooth $\mathbb{C}$-schemes such that any connected proper smooth $\mathbb{C}$-scheme admits a $\mathbb{C}$-immersion into one of them?
user137671
13
votes
1
answer
553
views
On non-representability of certain hom schemes
Let $k$ be an algebraically closed field. It is well-known that the isom-sheaf Isom$(\mathbb{A}^1_k,\mathbb{A}^1_k)$ is not representable by an algebraic space. (To be clear, the functor Isom$(\mathbb{...
- 131
12
votes
4
answers
4k
views
Nonalgebraic complex manifolds
I'm turning here (a variation of) a question asked by a friend of mine. For the purposes of this question I will say that a compact complex manifold is projective if it is isomorphic to a subvariety ...
- 14.7k
8
votes
0
answers
230
views
A Hartogs-type criterion for flatness
Let $U$ be a smooth affine connected variety over $\mathbb C$ and let $V\subset U$ be an open whose complement is of codimension at least two.
Now, let $Y$ be a smooth quasi-affine connected variety ...
- 81
7
votes
1
answer
587
views
Complex analytic space with no (positive dim.) subscheme ?
Is there an example of a complex analytic space $X$ that doesn't have any (not necessarily open or closed) positive dimensional subspace $Y$ which is analytically isomorphic to (the complex analytic ...
- 23.3k
6
votes
1
answer
269
views
Criteria for when Gauss-Manin sheaves are vector bundles
Let $(X,D_X)$ and $(S, D_S)$ be smooth normal crossings pairs over $\mathbb C$; i.e. smooth schemes of finite type over $\mathbb C$ with a normal crossings divisor. If $f:X \to S$ is a proper, flat ...
5
votes
2
answers
719
views
Relationship between Line Bundles with isomorphic ring of sections
Suppose two positive holomorphic line bundles $L_1 \to X_1, L_2\to X_2$ over two projective complex manifold $X_1, X_2$ have isomorphic ring of sections $R=R_1=R_2$ where $R_i=\oplus_{m=0}^\infty\...
user2529
5
votes
1
answer
323
views
Locally affine varieties and du Val singularities
Let me start with an apologetic disclaimer: I am very far from an algebraic geometer, so this question might be crudely formulated.
I have a specific question about du Val singularities, but while ...
- 390
3
votes
2
answers
500
views
Varieties that do not extend to flat families
Is it easy to give an example of a function field $K$ and a smooth proper variety $X$ over $K$ that does not extend to a flat scheme over $B$, where $B$ is a smooth proper variety with function field $...
- 31
2
votes
1
answer
220
views
Does cohomology with compact support contain "ample" elements
Let $X$ be a quasi-projective integral variety over $\mathbb{C}$. If $X$ is projective, then $\mathrm{H}^2(X,\mathbb{Z})$ contains "ample" classes. These "ample" classes are defined as being the image ...
- 181
2
votes
1
answer
178
views
Blowup formula for a morphism
Let $f: X\to S$ be a smooth projective morphism between smooth schemes over $\mathbb C$, $i: Z \to X$ a closed subscheme of codimension $c$, also smooth over $S$, and let $g: Y\to S$ be the blowup ...
2
votes
0
answers
171
views
Principle of degeneration as precursor of Zariski's connectedness theorem (geometric intuition)
I have following question about so-called "principle of degeneration"
in algebraic geometry (which in modern terms is an immediate consequence
of Zariski's main theorem and goes in it's ...
- 6,028
2
votes
0
answers
124
views
About finite dimensionality of Chow groups of zero cycles
Let $S$ be a connected smooth complex projective surface.
Let $Sym^{d}(S)$, $d\in \mathbb{Z}^+_0$, be the $d$-th symmetric product of $S$ parametrizing $0$-cycles of degree $d$.
Let $Sym^{d,d}(S)=...
- 519
1
vote
1
answer
217
views
Meaning of torsion points in a Roitman's theorem
I am having some problems to understand the meaning of the following theorem due to Roitmann. I found this theorem in Voisin's book: Hodge Theory and Complex Algebraic Geometry, Volume II, page ...
- 519
1
vote
0
answers
19
views
Zariski Connectedness Theorem: From Analytic & Topological Viewpoint
Let $p:Y \to X$ be a proper, surjective map between smooth connected complex varieties $Y,X$, esp. $X$ unibranch. Assume that there exist a Zariski open (esp. dense) $U \subset X$ over which every ...
- 6,028
1
vote
0
answers
177
views
Is there an analytic criterion for quasi-compactness of a scheme?
Let $X$ be a locally finite type scheme over $\mathbb C$.
I'm looking for the analogue of the notion "finite type" for $X^{an}$ and an SGA 1 Exp. XII type of criterion which says that
The scheme $X$...
- 11
0
votes
1
answer
335
views
Self-intersection of zero section of line bundle over elliptic base curve
Let $C$ be an elliptic curve over $k=\mathbb{C}$ and $\mathcal{L}$ a line bundle of degree $d$. It induces naturally a $\mathbb{A}^1$-fibration $L \to C$ where $L=\underline{\operatorname{Spec}} (\...
- 6,028
0
votes
1
answer
528
views
Artin approximation of a diagram
Let consider $f:(X,x)\to (Z,z)$ and $g:(Y,y)\to (Z,z)$ morphisms of pointed $k$-schemes of finite type ($k$ is a field). Suppose that there exists a map on the level of formal neighborhoods $\phi:X_{x}...
- 3,472
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
0
votes
0
answers
110
views
Lefschetz Theorem in Dolgachev's On automorphisms of Enriques Surfaces
Let $F$ be a Enriques surface over $\Bbb C$. I have a question about a detail in the proof of Proposition 2.1. from Dolgachev's On automorphisms of Enriques surfaces.
This 2.1. Proposition. states ...
- 6,028
0
votes
0
answers
224
views
The genus of hyperplane sections
Let $S$ be a connected smooth projective surface over $\mathbb{C}$.
Let $\Sigma$ be the complete linear system of a very ample divisor $D$ on $S$, and let $d=\dim(\Sigma)$ be the dimension of $\...
- 519
0
votes
0
answers
75
views
General fiber and the symmetric product of an ample hypersurface
Let $Sym^m(X)$ be the $m$th symmetric product of a smooth projective variety $X$, $n=\dim(X)$, $Y_1$ an ample hypersurface of $X$, and $CH_0(X)_{hom}$ the Chow groups of $0$-cycles of degree $0$....
- 519
0
votes
0
answers
666
views
Quicker way to show that the restriction to a open subvariety is again proper?
Dear all,
Let $f: X \rightarrow Y$ be a morphism of projective varieties over $\mathbb{C}$. Also let $V \subset Y$ be a nontrivial open subvariety and set $U:= f^{-1}(V)$.
I would like to show that $...
- 469