3
votes
1answer
244 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 ...
2
votes
1answer
110 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?
2
votes
1answer
281 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 ...
8
votes
2answers
319 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 ...
3
votes
1answer
113 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 ...
2
votes
3answers
330 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. ...
12
votes
1answer
570 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 ...
3
votes
0answers
197 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 ...
2
votes
0answers
265 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
0answers
201 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 ...
2
votes
2answers
906 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 ...
12
votes
2answers
1k 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 ...
5
votes
3answers
562 views

Basepoints in the Canonical System of Algebraic Surfaces

Let X be an 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 ...