All Questions
Tagged with complex-geometry algebraic-surfaces
52
questions
1
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0
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105
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Enriques-Kodaira classification of minimal resolution of surface with quotient singularities
Let $X$ be a normal projective complex surface with at worst quotient singularities. Let $\bar{X}\to X$ be the minimal resolution. Further assume that $b_2(X)=1$ and $b_1(X)=b_3(X)=0$. Since quotient ...
- 189
1
vote
0
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42
views
Positivity of self-intersection of dicisor associated to meromorphic function
In the book "Holomorphic Vector Bundles over Compact Complex Surfaces" by Vasile Brînzănescu, in the proof of theorem 2.13 there is the following claim
Let $X$ be a compact non-algebraic ...
- 111
0
votes
1
answer
169
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BMY inequality for surfaces of general type in characteristic 0
Let $X$ be a smooth, complex, projective, minimal surface of general type, i.e. the canonical (line) bundle $K_X$ is big and nef.
It is known that $3c_2\geq c_1^2$ (the Bogomolov-Miyaoka-Yau ...
- 828
8
votes
1
answer
266
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Fundamental group of a smoothing of a complex surface
Let $X_0$ be a compact complex algebraic surface with an isolated singularity and let $X_t$ be a smoothing of $X_0$ over the disc. How can we compute the fundamental group of $X_t$ say in terms of the ...
- 2,970
7
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0
answers
201
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Global generation of $S^n \Omega_X$ for a fake projective plane
Let $X$ be a fake projective plane, namely, a compact complex surface with
$$p_g(X)=q(X)=0, \quad K_X^2=9$$ and $K_X$ ample.
Since $K_X^2=9 \chi(\mathcal{O}_X)$, Yau's celebrated proof of the Calabi ...
- 65.7k
4
votes
0
answers
155
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Surface with $\Omega_X$ globally generated and singular Albanese image
This question is inspired by abx's comment to my previous question MO430933.
Let $X$ be a complex surface of general type, and denote by $$a \colon X \to \operatorname{Alb}(X)$$ the Albanese map of $X$...
- 65.7k
6
votes
0
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172
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Lower bound for $h^0(X, \operatorname{Sym}^n \Omega_X)$
This is a weaker version of my previous (unanswered) question MO429574.
Let us start with a smooth, ample divisor $X$ in an abelian threefold $A$. It is a surface of general type such that $\Omega_X$ ...
- 65.7k
7
votes
1
answer
252
views
Different algebraic structures on complements to divisors
Complements to square-zero curves in projective surfaces sometimes have several non-isomorphic algebraic structures. Serre’s example is possibly the most famous illustration of this phenomenon (see f....
- 2,305
5
votes
0
answers
229
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Computation of $H^i(X, \, \operatorname{Sym}^n \Omega_X)$ for a surface of general type $(i=0, \, 1)$
Let $X$ be a smooth, complex surface of general type such that $\Omega_X$ is globally generated, and let $n \geq 2$ be a natural number.
Question. Is there a way to compute $h^i(X, \, \operatorname{...
- 65.7k
1
vote
0
answers
115
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Abelian subvarieties corresponding to vector subspaces
Let $S$ be a connected smooth projective surface.
Let $C$ a smooth curve on $S$
In page 9 of the paper "https://arxiv.org/abs/1704.04187v1" a read the following:
Let
\begin{equation*}
r: ...
- 519
4
votes
2
answers
193
views
Constructions of complex surfaces covered by the ball of $\mathbb{C}^2$
Let $S$ be a compact complex surface. It is well-known that the following two facts are equivalent
$c_1^2(S) = 3 c_2(S)$ and $S \neq \mathbb{CP}^2$
The universal cover of $S$ is biholomorphic to the ...
- 2,636
7
votes
1
answer
2k
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Relating the holomorphic Euler characteristic of a family of algebraic varieties to properties of the base and fibers
Let $f : X\rightarrow Y$ be a proper flat morphism (of schemes) with connected fibers over a smooth projective curve $Y$ over $\mathbb{C}$. Let $X_{y_0}$ denote a smooth fiber over $y_0\in Y$.
If $f$ ...
- 10k
8
votes
3
answers
1k
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Seeking concrete examples of "generic" elliptic fibrations of K3 surfaces
For me a K3 surface will be a smooth complex projective variety of dimension 2 that is simply-connected and has trivial canonical bundle. Given a K3 surface $X$, an elliptic fibration $\pi \colon X \...
- 21.8k
3
votes
1
answer
648
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Cohomology of singular projective cubic surface
Let $X\subset \mathbb{P}_{\mathbb{C}}^3$ be a projective singular cubic surface with two singular points. Is the rationalcohomology of such objects known? As an example of the type of surfaces I'd be ...
2
votes
1
answer
457
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Behavior of canonical divisor under a finite group quotient
Given a smooth algebraic surface $X$, and a group $G$ acting on it and letting $Y := X / G$, how can we compute $K_Y^2$ from from $K_X^2$?
Current progress: In Borisov and Fatighenti - New explicit ...
- 29