All Questions
27 questions
24
votes
3
answers
1k
views
Hyperbolic Coxeter polytopes and Del-Pezzo surfaces
Added. In the following link there is a proof of the observation made in this question: http://dl.dropbox.com/u/5546138/DelpezzoCoxeter.pdf
I would like to find a reference for a beautiful ...
- 28.9k
13
votes
4
answers
2k
views
Algebraic surfaces and their (intrinsic) geometry
Recently I began to consider algebraic surfaces, that is, the zero set of a polynomial in 3 (or more variables). My algebraic geometry background is poor, and I'm more used to differential and ...
- 579
10
votes
1
answer
1k
views
A proper smooth surface is projective
My question is a reference request for the following fact: if $k$ is a field and $X$ a proper smooth surface over $k$, then $X \rightarrow \mathrm{Spec}\, k$ is projective. Where is this well-known ...
- 2,663
7
votes
1
answer
799
views
Liftability of Enriques Surfaces (from char. p to zero)
Let $k$ be an algebraically closed field of characteristic $p > 0$, $X$ a variety over $k$.
We say $X$ lifts to characteristic zero, if there exists a local ring $R$ containing $\mathbb Z$ with ...
- 839
6
votes
1
answer
342
views
What is Mumford's example of a normal complex algebraic surface $X$ with non-torsion elements in $H^2_{et}(X,\mathbb{G}_m)$?
I have heard that Mumford has constructed an example of a normal complex algebraic surface $X$ such that $H^2_{et}(X,\mathbb{G}_m)$ contains a non-torsion element. But I cannot find the reference.
...
- 9,019
5
votes
2
answers
469
views
Reference for Automorphisms of K3 surfaces
I am looking for some introductory reference concerning Automorphisms (of finite order) on K3 surfaces. Any suggestion?
- 761
5
votes
1
answer
420
views
Certain double covers of cubic surfaces
Let $S$ be a smooth cubic surface in $\mathbb{P}^3$. I would like to understand that variety $V$ that parametrizes lines $\ell$ such that $\ell \cdot S=3P$ with $P \in S$. At any point $P \in S$, let $...
- 3,142
5
votes
0
answers
326
views
Max Noether's theorem for algebraic surfaces
The well-known Max Noether's theorem for curves [see Arbarello-Cornalba-Griffiths-Harris Geometry of Algebraic Curves vol.1, p. 117] states that, if a smooth curve $C$ is non-hyperelliptic, then the ...
- 66.3k
4
votes
1
answer
175
views
Every elliptic surface contains only finitely many negative self-intersection rational curves?
By a properly elliptic surface, I mean an algebraic surface $X$ with Kodaira dimension $\kappa(X)=1$. It has a natural elliptic fibration $\pi\colon X\rightarrow S$.
According to section 5.2 of this ...
- 141
4
votes
0
answers
100
views
Fundamental groups of Hirzebruch's line arrangement varities
Let $\Lambda$ be a line arrangement in $\mathbb{P}^2$ and $n > 0$ an integer. Then Hirzebruch defined a smooth projective surface $H(\Lambda, n)$ as the minimal desingularization of a covering $Y \...
- 3,625
3
votes
3
answers
515
views
Does there exist a holomorphic fibration of genus two over $\mathbb{P}^{1}$ with $7$ nodal singularities?
This is a problem about the holomorphic fibration on a complex manifold.
Does there exist a holomorphic fibration of genus two over $\mathbb{CP}^{1}$ with 7 nodal singularities?
If you are aware of ...
- 31
3
votes
1
answer
414
views
Modern reference for the theory of correspondences for curves
The classic theory of correspondences between smooth algebraic curves can be found in André Weil's Foundations of algebraic geometry. However, this reference works in a pre-modern algebraic geometry ...
- 2,871
3
votes
1
answer
203
views
"Direct" calculation of $K_0$ for surfaces, 3-folds
I apologize in advance for the vagueness of my question, but I am looking for sources (if they exist) where $K_0$ (the Grothendieck group of coherent sheaves) is computed "by hand" for some low-...
- 528
3
votes
1
answer
215
views
Enriques classification of algebraic surfaces in characteristic zero
I am searching for a reference about the classification of algebraic surfaces over an arbitrary algebraically closed field of characteristic zero. In the 1949 book "le superficie algebriche" by ...
- 31
3
votes
1
answer
702
views
Is each rationally chain connected surface rational?
Consider a 2-dimensional smooth projective algebraic surface S over complex numbers. Could you recommend any exact references to the proofs of the following assertions (of course, if they are true):
...
2
votes
1
answer
732
views
Where to find "Families of curves on a surface of general type" (MR0457450)?
I am currently doing some research on surfaces of general type and I need some results from Bogomolov's paper:
Bogomolov, F. A.
Families of curves on a surface of general type.
Dokl. Akad. Nauk SSSR ...
- 363
2
votes
2
answers
363
views
Why does a complex linear normalization of a real algebraic surface inherit a real structure?
Could you recommend any references to (some of) the following very basic assertions in algebraic geometry? (It seems unreasonable to reprove them in a research paper.)
(1) Let a surface $X$ in $\...
2
votes
1
answer
299
views
Explicit families of elliptic curves
I am interested in finding families $X\to C$ of elliptic curves over a projective curve $C$. The fibers are not all isomorphic, not all smooth, and the singular fibers should be nodal stable curves. ...
- 29
2
votes
1
answer
714
views
Castelnuovo and Artin contractibility criteria for families
In the course of a proof, I need some version of Castelnuovo and Artin's contractibility criteria for a family of surfaces. Say I have a family (flat is probably needed, in order to compare ...
- 625
2
votes
1
answer
467
views
Absorbing ramification and factoring finite flat maps
In his Algebraic surfaces book, Beauville gives a result allowing one to "absorb ramification" for certain maps (see below). There are also something similar one can do with number fields. I would ...
- 3,555
2
votes
1
answer
470
views
Resolution of "nice" and zero-dimensional singularities on a surface
Assume I have a singular algebraic surface $X$ over an algebraically closed field (characteristic zero if you want) which is singular in a finite set of points. I am looking for a condition as to the ...
- 4,902
2
votes
0
answers
95
views
Reference request The support of $f$-nef divisor
I'm seaching for a proof of the theorem below.
Do you know any reference?
Consider $f:X\rightarrow Y$ projective birational map between normal varieties and $\mathbb{R}$ cartier divisor $D$ whose ...
- 328
2
votes
0
answers
671
views
description of very ample bundle of Hirzebruch surface
I learned some basic properties of Hirzebruch surface mainly from Vakil's notes "the rising sea", section 20.2.9. the Hirzebruch surface is defined as $\mathbb{F}_n:=\operatorname{Proj} (\...
- 343
1
vote
1
answer
247
views
Infinitesimal deformation of strict transform
Let $X$ be an affine, complex surface with isolated singularities and $\pi:\widetilde{X} \to X$ be a resolution of singularities (not necessarily minimal) i.e., $\widetilde{X}$ is non-singular and $\...
- 2,126
1
vote
1
answer
493
views
Fundamental group of Log del Pezzo surfaces
A Log del Pezzo surface is a normal complex surface with ample anti-canonical bundle and at worst quotient singularities.
It is known that such surfaces are rational. This is proven, for example, ...
- 14.3k
1
vote
2
answers
196
views
What are the finite étale coverings of a quasi-hyperelliptic surface?
Let $X$ be a quasi-hyperelliptic surface in characteristic 3 where the canonical bundle $K_X$ is trivial.
Question: Is there a finite étale covering $Y \rightarrow X$ such that
$Y$ is an abelian ...
- 9,501
1
vote
0
answers
73
views
Explict equations for unirational Enriques surface with a nonzero 1-form
I am hoping to write down very explicitly the equations for the following data:
an Enriques surface $X$ of type $\mathrm{Pic}^{\tau} = \mathbb{Z} / 2 \mathbb{Z}$ such that its canonical $\mu_2$-cover ...
- 3,625