**4**

votes

**1**answer

499 views

### Del pezzo surfaces in positive characteristic

For me a Del Pezzo surface $X$ over an algebraically closed field of characteristic $p$ is an algebraic surface where the anticanonical bundle $\omega^{-1}_X$ or $-K_X$ is ample. (I prefer the second ...

**2**

votes

**0**answers

371 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

222 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 ...

**3**

votes

**1**answer

166 views

### Boundedness of $C.K$ on a surface with $-K$ pseudoeffective

Let $S$ be a projective surface with pseudoeffective anticanonical divisor $-K_S$. Is it true that if $C$ is an integral curve with $C^2<0$ and $C \cdot K_S >0$, then $\max_C (C \cdot K_S)$ is ...

**0**

votes

**1**answer

252 views

### signature of picard lattice

Let's assume $X$ is a K3 surface. Why the signature of the picard lattice is $(1,\rho(X)-1)$? We know that (by Hodge index theorem) if we have a curve $D$ in $X$ with self intersection $D^2>0$ then ...

**2**

votes

**0**answers

157 views

### selfintersection of curves inside

What are all the possibilities of the self intersection number of a smooth curve inside an Enriques surface?

**3**

votes

**1**answer

476 views

### Two basic questions concerning geometrically ruled surfaces. [closed]

Good Morning,
These questions are a result of trying to understand the proof of Proposition III.18 in Beauville's book 'Complex Algebraic Surfaces'.
Here is the setup - everything is smooth, ...

**1**

vote

**0**answers

172 views

### signature of $Pic(X)$

Let $X$ be a $K3$ surface and $\sigma$ an antisymplectic involution on $X$ and so $X$ is algebraic. 1.Why the signature of $Pic(X)$ is $(1,\rho -1)$? (This is well known but I cant find any direct ...

**4**

votes

**2**answers

485 views

### every involution of an Enriques surface is

Is it true that every involution $\sigma$ (i.e., $\sigma^2=identity$) of an Enriques surface $X$ acts trivially on $K_X^{\otimes 2}$ i.e., for any $\omega\in K_X^{\otimes 2}$ we have $\sigma^* ...

**2**

votes

**2**answers

1k 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**

votes

**1**answer

414 views

### Minimal resolution of Log del Pezzo surfaces

Suppose $X$ is a log del pezzo projective surface of index $l$. As far as I understand it will have a finite number of singular points all of which can be resolved by sucessive blow-ups.
Let $E_i$ be ...

**2**

votes

**1**answer

371 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 ...

**17**

votes

**3**answers

845 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 ...

**6**

votes

**1**answer

1k views

### Hirzebruch Surfaces

Good Morning,
I'm trying to prove that two different definitions of the Hirzebruch Surfaces coincide, and am having problems. Let $a \geq 0$. My first definition for the $a^{th}$ surface is
$X_a= ...

**2**

votes

**2**answers

335 views

### Global sections of a linear system

Recently, following Beauville's book (exercises iv.(1),(2)) I have been working on Hirzebruch surfaces (from the algebraic geometry point of view) and I had to compute the space of global sections of ...

**2**

votes

**1**answer

428 views

### Looking for an inequality between Chern and Todd classes (something in style of Bogomolov-Miyaoka-Yau)

Consider a smooth projective surface $S\subset\Bbb P^N_{\Bbb C}$ which is a complete intersection of hypersurfaces of degrees $(d_1,..,d_{k\ge2})$ with $d_i\ge2$ for all i. Is it true that for such ...

**1**

vote

**0**answers

243 views

### Singular conics on certain algebraic surfaces

Let S be an algebraic surface in 3-dimensional complex projective space. Suppose that:
The degree of S is either 5 or 6;
The generic plane section of S is a curve of genus 1.
(Equivalently, the ...

**2**

votes

**2**answers

599 views

### Elementary transformations of ruled surfaces as maps of vector bundles

This comes as a question in Beauville's Algebraic surfaces book (III.24 (2)). We work over $\mathbb{C}$.
All geometrically ruled surfaces (grs) $p:S\longrightarrow C$ over a curve $C$ can be seen as ...

**1**

vote

**1**answer

597 views

### Hirzebruch surfaces

I am sorry for too naive and stupid question,
How can I express the 2nd Hirzebruch surface, $F_{2}$ in terms of $SO(3)$. Can F_{2} be realizable as the total space of a bundle over $\mathbb{R}_{+}$ ...

**1**

vote

**2**answers

697 views

### Names of certain surfaces

Are there any generally used names for the following algebraic and nonalgebraic surfaces? Any references to literature where the surfaces are studied are also appreciated.
Surface I. Implicit ...

**-1**

votes

**0**answers

192 views

### Does there exist a fibration of genus two over P^1 with only 3 singular fibres but two are semi-stable fibers for algebraic surfaces?

We will work over the complex numbers C.
there exist a fibration of genus two over P^1 with only 3 singular fibres but one is semi-stable fiber.
there not exist a fibration of genus two over P^1 ...

**3**

votes

**0**answers

301 views

### Does there exist a non-isotrivial fibration of genus two over P^1 with only 3 singular fibres of general type surfaces?

We will work over the complex numbers C.
This question is based on Beauville's article :
there exist a non-isotrivial fibration of genus 2 over P^1 with only 3 singular fibres.
but not know for ...

**13**

votes

**2**answers

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 ...

**2**

votes

**2**answers

760 views

### Isolated conics on a del Pezzo surface

Is there anything known about isolated conics in a del Pezzo surface: their number, arrangement, and the corresponding elements of the class group of surface's minimal desingularization? (Isolated ...

**3**

votes

**1**answer

450 views

### Mordell-Weil Group of Elliptic Surface

Suppose $ X \to \mathbb{P}^1 $ is an elliptic surface with section, with Weierstrass model defined over $ \mathbb{Q} $. If $ \sigma: \mathbb{P}^1 \to X $ is a torsion section with order $n$, then for ...

**2**

votes

**0**answers

210 views

### On base locus of canoncal linear system on surfaces

Let $S$ be a minimal surface of general type over $\mathbb{C}$ with $p_g=h^0(K_S)>1$. As a convention, we can write $|K_S|=|M|+F$ such that $F$ is the fixed part. We know that $K_SM \le K_S^2$.
...

**6**

votes

**2**answers

618 views

### Embedding of algebraic surfaces

If I am not mistaken there is a theorem that says any curve $C$ can be embedded in $\mathbf{P}^3$. What can be said about surfaces? Do we have a theorem like:
All surfaces can be embedded in ...

**9**

votes

**3**answers

2k views

### Rational curves on varieties of general type

Let $S$ be a complex surface of general type. Are there infinitely many smooth rational curves on $S$? And more general, what if $V$ is a variety of general type?

**5**

votes

**3**answers

701 views

### Are there (-2)-curves on an Enriques surface?

Let $X$ be an Enriques surface. A $(-2)$-curve is an irriducible rational curve on X such that $C^2 = -2$. By Proposition [VIII,16.1] from Barth-Peters-Van de Ven, we have that if $D^2 = -2$, then it ...

**8**

votes

**1**answer

417 views

### Are there non-projective normal surfaces which are rational?

Every non-singular complete surface is projective. On the other hand, there are non-projective complete surfaces (see e.g. Excercise II.7.13 of Hartshorne) - and there are such examples where the ...

**21**

votes

**0**answers

580 views

### bound on the genus of a fiber of the Albanese map of a surface with $h^1({\mathcal O})=1$?

This is maybe more an open problem than a question, since I have seriously thought about it and asked several people working on algebraic surfaces with no success. I hope somebody here can ...

**13**

votes

**4**answers

1k 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 ...

**2**

votes

**1**answer

338 views

### Quotient of an abelian surface by a finite group, irreducible components

Given an abelian surface $A$ as the product of two non isogenous elliptic curves $E_1 $ and $E_2$ over $\mathbb{C}$. We also have a smooth 2 dimensional moduli space $M$ of sheaves on $A$ with some ...

**5**

votes

**3**answers

611 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 ...

**4**

votes

**1**answer

336 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 ...

**5**

votes

**1**answer

669 views

### an exercise about elliptic surface in Beauville's book

In Beauville's "Complex Algebraic Surfaces", given an elliptic surface $f : X \to C$ with a generic fiber $E$. Then either $\text{Alb}(X) \cong \text{Jac}(C)$ or there is an exact sequence of abelian ...

**20**

votes

**4**answers

2k views

### Abundance for algebraic surfaces

I am currently teaching a course in algebraic geometry where one of the aims is to give an overview of the Enriques-Kodaira classification of surfaces. I am trying to throw in some modern aspects so I ...

**5**

votes

**2**answers

713 views

### Ramification in morphisms of surfaces

The question can be generalized, but we might as well restrict to this case.
Let $X \rightarrow Y$ be a morphism between nonsingular surfaces (say over $\mathbb{C}$). Let $R_1$ be an irreducible ...

**2**

votes

**0**answers

255 views

### quasi-trigonal curves

I have read in the literature about quasi-trigonal curves. Such a curve C is a hyperelliptic curve X with two points p,q identified (basically a pinch). They seem to be pretty important in the theory ...

**8**

votes

**1**answer

415 views

### How do branched coverings of complex surfaces “fit” with branched coverings of curves?

Since I'm used to working with algebraic $\pi_1$'s, which don't work well with surfaces, I find myself lacking geometric intuition when I attempt to do these types of purely geometric arguments. I'm ...

**35**

votes

**5**answers

2k views

### Surfaces in $\mathbb{P}^3$ with isolated singularities

It is classically known that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only ordinary singularities, i.e. a curve $C$ of double points, ...

**3**

votes

**1**answer

634 views

### Variation of the Albanese map

Let $S$ be an irregular surface of general type over $\mathbb{C}$ and $a \colon S \to A:=\textrm{Alb}(S)$ be its Albanese map. Let Def($S$) and Def($A$) be the bases of the Kuranishi family of $S$ and ...

**12**

votes

**3**answers

1k views

### A nontrivial surface on which any two curves intersect

One interesting property of the projective plane is that any two plane curves intersect. (More generally, if $V$ and $W$ are subvarieties of any projective space, and codim $V$ + codim $W \geq 0$, ...

**4**

votes

**2**answers

328 views

### Shrinking Fano surfaces to a point in Calabi-Yau 3-folds

Let X be a Calabi-Yau 3-fold, and $D\subset X$, smooth,Fano divisor.
Since $K_X=0$ we have $N_D^X=K_D$. I have seen the following fact in many papers:
By deforming X within Kahler moduli, we can ...

**13**

votes

**3**answers

2k views

### Vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$

I have a question about vector bundles on the algebraic surface $\mathbb{P}^1\times\mathbb{P}^1$. My motivation is the splitting theorem of Grothendieck, which says that every algebraic vector bundle ...

**7**

votes

**1**answer

372 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 ...

**5**

votes

**3**answers

2k views

### Cone of curves and Mori Theorem for algebraic Surfaces

In describing part of the geometry of the cone of curves for an algebraic surface $S$, we need to find $(-1)$ curves within $S$. Once we've done that, then we can say that the "negative" part of the ...

**17**

votes

**5**answers

4k views

### blowing up, -1 curves, effective and ample divisors

Lets say we're on a smooth surface, and we blow up at a point.
Is there a simple explicit computation that shows to me the fact that the exceptional divisor E has self intersection -1 ? I don't ...

**2**

votes

**1**answer

2k views

### Algebraic equivalence VS Numerical Equivalence - An Example.

This question is arose from the question
Difference between equivalence relations on algebraic cycles
and the example 3 in lecture 1 in Mumford's book Lectures on curves on an algebraic surface.
...

**6**

votes

**4**answers

580 views

### Interaction of topology and the Picard group of Algebraic surfaces

It is well known that a smooth cubic surface $X\subset \mathbb{P}^3$ has exactly 27 lines in it. Furthermore, it is easy to check that Picard group $$Pic(X)\cong \mathbb{Z}^7$$ Here the generators are ...