All Questions
50 questions
16
votes
1
answer
1k
views
Is the set of surfaces over Spec Z with ample canonical sheaf empty
Main question. Does there exist a smooth projective morphism $X\to$ Spec $\mathbf Z$ of relative dimension two such that the canonical sheaf $\omega_{X_{\mathbf Q}}$ of the generic fibre $X_{\mathbf Q}...
- 9,497
15
votes
2
answers
2k
views
Surfaces containing curves of arbitrarily negative self-intersection
Olivier Wittenberg and I are curious about the following :
Let $S$ be a smooth projective complex surface. Are the self-intersection numbers of integral curves on $S$ always bounded below ? Or can $S$...
- 6,543
14
votes
1
answer
364
views
Are any of these complex surfaces ever projective?
Let $C$ and $T$ be compact connected Riemann surfaces (or: smooth projective connected curves over $\mathbb{C}$) of genus at least two and let $X:=C\times T$. Let $(c,t)$ be a point of $X$, and let $...
- 9,497
11
votes
1
answer
1k
views
Restriction of the Picard group of a surface to a curve
In a paper by Griffiths and Harris on the Noether-Lefschetz theorem, they use the following fact which they don't comment as if it is obvious:
For a general (smooth) surface $S$ in $\mathbb{P}^3$ ...
- 111
10
votes
2
answers
839
views
Uniformization of Kodaira fibered surfaces
Consider a Kodaira fibration. i.e. a smooth non-isotrivial fibration $X\rightarrow C$ with $X$ a smooth complex surface and $C$ a smooth complex curve, such that both the genus of $C$ and genus of the ...
- 2,221
10
votes
1
answer
418
views
A diffeomorphism of complex surfaces mapping subvarieties to subvarieties
Let $X$ and $Y$ be smooth projective complex surfaces. If a diffeomorphism from $X$ to $Y$ maps subvarieties to subvarieties does it have to be holomorphic or antiholomorphic? Can we at least verify ...
user145520
9
votes
1
answer
284
views
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 ...
- 3,290
8
votes
3
answers
1k
views
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 \...
- 22.3k
7
votes
3
answers
969
views
Basepoints in the canonical system of algebraic surfaces
Let $X$ be a smooth projective variety defined over $\mathbb{C}$. In the context of the minimal model program it is often important to understand the geometry of the maps defined by the complete ...
- 2,097
7
votes
2
answers
478
views
Show Fiber Product of Rational Elliptic Surfaces is Calabi-Yau
In a handful of contexts people study Calabi-Yau threefolds formed by taking the fiber product of two rational elliptic surfaces. I can't find any detailed explanation of why such geometries are ...
- 1,701
7
votes
1
answer
2k
views
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 $f$ ...
- 10.7k
7
votes
1
answer
263
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
7
votes
0
answers
204
views
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 ...
- 66.3k
6
votes
2
answers
2k
views
Generalisations of Riemann-Roch for surfaces
Let $X$ be a smooth projective algebraic surface (over $\mathbb{C}$ ). For all $L\in \mathrm{Pic}(X)$, we have
$$\chi(L)=\chi(\mathcal{O}_X)+\frac{1}{2}(L^2-L\cdot \omega_X).$$
This is the famous ...
- 2,215
6
votes
1
answer
646
views
Additivity of Kodaira dimension for a nice fibration
Consider a surjective holomorphic map between two complex projective manifolds $\pi :X \rightarrow Y$. Iitaka conjectured the subadditivity of Kodaira dimensions: $\kappa(X)\geqslant\kappa(Y)+\kappa(...
user60772
6
votes
0
answers
175
views
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 $\Omega_X$ ...
- 66.3k
6
votes
0
answers
218
views
Can the base of an elliptically fibered Calabi-Yau threefold be an Enriques surface?
For this question, a Calabi-Yau manifold or variety of dimension $n$ is defined as a non-singular projective variety with trivial canonical bundle and $h^{i,0} = 0$ unless $i = 0$ or $i = n$.
If ...
- 515
5
votes
1
answer
224
views
Condition for two surfaces to not live inside a common threefold
Let $Y_1$, $Y_2$ be two complex smooth projective surfaces, are there some restrictions for $Y_1$ and $Y_2$ to be embedded in a common smooth projective threefold?
The first thought is to use ...
- 6,622
5
votes
1
answer
350
views
Quotient of a smooth projective surface by an involution
Is the quotient of a smooth complex projective surface by an involution projective? Suppose the quotient happens to be smooth; does that change the situation?
- 1,493
5
votes
0
answers
229
views
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{...
- 66.3k
5
votes
0
answers
218
views
Reducible surface as a degeneration
I am interested in the following situation. If $S_1\cup_D S_2$ is a union of two irreducible smooth projective surfaces over $k=\overline{k}$(over $k=\mathbb{C}$ is enough, if it's relevant) glued ...
- 71
4
votes
1
answer
982
views
Do there exist double points on an algebraic surface in $\mathbb{P}_{\mathbb{C}}^3$ that are not rational?
The title explains it all.
I'm familiar with the du val singularities on surfaces, also known as rational double points. In http://homepages.warwick.ac.uk/~masda/surf/more/DuVal.pdf, 2.1, they are ...
- 469
4
votes
3
answers
1k
views
Divisor class group on blowup of nodal surface
The following got no answer on mathstackexchange. I believe it not to be hard, but maybe it is a little specialized?
All varieties will be over $\mathbb{C}$ and projective unless stated otherwise.
...
- 469
4
votes
2
answers
197
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 ...
- 2,696
4
votes
1
answer
712
views
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 ...
- 1,061
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
160
views
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 $X$...
- 66.3k
3
votes
1
answer
288
views
Kähler classes for surfaces of general type with $c_1^2=3c_2$
Given a smooth, compact complex surface with ample canonical bundle satisfying $c_1^2=3c_2$, is it true that every Kahler class is a multiple of $c_1$? This seems to be the case for fake projective ...
- 3,383
3
votes
1
answer
1k
views
Elliptic fibrations with few singular fibers
It is known that non-isotrivial fibrations of genus $g>0$ curves over the projective line have a bunch of singular fibers. There are at least three of them.
It is not difficult to prove that an ...
- 502
3
votes
1
answer
177
views
What can a quartic surface in $\mathbb{P}^3$ with an ordinary quadruple point look like?
All varieties will be projective and over $\mathbb{C}$.
If $S$ is any surface in $\mathbb{P}^3$ of degree 2 that posseses an ordinary double point, it follows easily that $S$ is projectively ...
- 469
3
votes
0
answers
221
views
Historical proof of Leschetz Hyperplane Theorem
I browse in Phillip Griffiths' Slides
on historical development of
Hodge-theory and these include a sketch of the original approach
with Lefschetz used to study complex surfaces in his famous
...
- 6,028
3
votes
1
answer
242
views
Linear homogenous polynomials that generates several quadratic polynomials
This is a generalization of this question.
Let $P_1, \ldots, P_m$, $Q_1, \ldots, Q_k \in \mathbb{C}[x_0,\ldots,x_n]$ be linear homogenous polynomials. Let $f_1, \ldots, f_s$ be a homogenous ...
- 2,576
3
votes
0
answers
176
views
Elliptic fibration arising from a higher genus linear system
Let $H$ be a very ample linear system on a smooth compact complex surface $X$ whose Kodaira dimension is $\geq 0$. A general element of $H$ is smooth and has genus $\geq 2$.
Let $L\subset H$ be a ...
- 502
3
votes
0
answers
260
views
Contracting rational curves on surfaces and getting something non-algebraic
Recently I posted an "announcement" on arxiv where I said something to the effect of "this is the first example we know where contracting (a tree of) rational curves from a non-singular algebraic ...
- 5,339
2
votes
1
answer
435
views
Smoothing transverse intersections
Let $S$ be a complex surface with ample canonical class. Let $C_1$ and $C_2$ be smooth complex curves in $S$ that intersect transversally at $n $ points. Furthermore, assume that the self-intersection ...
- 23
2
votes
1
answer
623
views
Dual of a Complex 2-Torus
Is a complex torus $A$ of dimension 2 always isomorphic to its dual torus (i.e. the torus obtained by taking the dual lattice), or are there counterexamples to this?
- 761
2
votes
1
answer
502
views
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
2
votes
3
answers
341
views
Linear homogenous polynomials that generates one quadratic polynomial
Let $P_1, \ldots, P_m$, $Q_1, \ldots, Q_k \in \mathbb{C}[x_0,\ldots,x_n]$ be linear homogenous polynomials. Let $f$ be a homogenous quadratic polynomial of degree $2$.
Assume that for every $i$ and ...
- 2,576
2
votes
1
answer
499
views
Normal bundle to fibers of a rational morphism
Let $f:X\dashrightarrow C$ be a rational fibration from a 3-dimensional variety $X$ to a curve $C$ such that generic fiber is smooth and different fibers intrsect in smooth curves. Take $S$ to be a ...
- 165
2
votes
1
answer
696
views
On complex surfaces with Kodaira dimension 1
Let $S$ be a complex surface of Kodaira dimension $1$ and $\pi_{1}(S) \neq 1 $.
What is known on possible diffeomorphism types of such $complex$ surfaces with a given fundamental group? Is it true ...
- 73
2
votes
1
answer
308
views
Pencils in very ample linear systems without curve in its base locus
If $L$ is a very ample line bundle over a smooth complex projective surface $X$ and $s_0, \dots, s_n$ is a basis of the global sections of $L$, is there some choice of $i,j$ such that the pencil ...
- 502
2
votes
1
answer
216
views
Principal elliptic bundles over curve with Kähler total space
I wonder what could be a Kähler surface, which is a total space of principal elliptic bundle over a curve. I believe that there is a classification and that it must be pretty simple, but I cannot find ...
- 1,170
2
votes
0
answers
643
views
canonical bundle of Abelian surface fibrations
For minimal surfaces admitting an elliptic fibration over a smooth curve,
there is a famous analysis of possible singular fibers and a canonical bundle formula due to Kodaira.
There are two papers of ...
2
votes
0
answers
330
views
surfaces with effective first Chern class
Let $S$ be a smooth complex surface, If $c_1(S) \in N_1(S)$ is nef and non-torsion, then we know that this would imply some restrictions on the cone of effective curves (and surface itself)--see the ...
1
vote
1
answer
226
views
Non-isotrival fiber bundle over compact Riemann surface
In this paper, Kodaira constructed a fiber bundle $\Phi:M_{m,n}\to S$ from a compact complex surface $M_{m,n}$ to a compact Rieman surface $S$ of genus $>0$. In particular, (on p.212) for any point ...
- 343
1
vote
0
answers
116
views
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
1
vote
0
answers
168
views
Rational classes of $(-2)$-curves in a minimal surface of general type
Let $X$ be a minimal surface of general type over $\mathbb{C}$. One can show that if for any set of $(-2)$-curves $C_1,\cdots,C_l$ on $X$, there exists $k$, $1\le k\le l$ such that $$\sum_{i=1}^k\...
- 41
0
votes
1
answer
176
views
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
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
265
views
Explicit adjunction formula and local top form
I am reading the section 4.2 in Kollar-Mori, where they provide the explicit equations for Du Val Singularities. In the course of the proof, they reduce to studying the equation $x^2+f(x,y)=0$ in a ...
- 625