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3
votes
1answer
237 views

Hilbert scheme of 2 points on an elliptic curve

The Hilbert scheme of 2 points on an elliptic curve $C$, $Hilb^2(C)$, has a natural structure of ruled surface, given by the map $f:Hilb^2(C) \to C$ such that $f(P,Q)=P+Q$. What can we say about the ...
2
votes
1answer
222 views

octic K3s inside cubic 4-folds

From the Thesis of B.Hassett I seem to understand that a smooth cubic 4-fold $X$ containing a $\mathbb{P}^2$ should contain also a octic K3, but I cannot see a natural way by which this K3 octic could ...
1
vote
1answer
169 views

Constructing a curve with good reduction over a function field

Let $K$ be the function field of a smooth projective connected curve $B$ over $\mathbf{C}$. Let $g\geq 0$ be an integer. Does there exist an nonsingular integral $\mathbf{C}$-scheme $X$ with a ...
3
votes
2answers
684 views

Bertini's Theorem small print

Suppose $S\subset \mathbb{P}^n$ is a smooth del Pezzo surface and $C$ is an irreducible smooth curve (you can make it rational if it simplifies the setting) such that $\mathcal{L}=\vert -K_S-C\vert $ ...
1
vote
2answers
295 views

How to construct Enriques surface from Fermat K3

Let $x_1^4+x_2^4+x_3^4+x_4^4=0 \subset \mathbb{P}^4$ be the Fermat K3 surface. Is it possible to start from some involution on $\mathbb{P}^3$, do blow-ups to get rid of fixed points and then quotient ...
4
votes
1answer
278 views

Invariant lattice of algebraic surface.

Given an algebraic surface $S$ with action of a finite group $G$. Is it true that the invariant lattice $H^2(X,\mathbb{Z})^G$ is generated by elements pulled back from the $H^2(X/G,\mathbb{Z})$ (or ...
3
votes
2answers
447 views

blowing up general k points on the plane

Del Pezzo surfaces are obtained by blowing up $1 \leq k \leq 8$ points on general position in $\mathbb{P}^2$. What does it happen when the number of points is larger than nine? In this sense, ...
4
votes
1answer
260 views

Divisorial contraction: when is the image an algebraic space or a stack?

Let $X$ be a smooth projective surface (in the category of varieties, or schemes), and let $C\subset X$ be a curve (a priori not irreducible, but the irreducible case in itself is already ...
2
votes
1answer
269 views

Question on Ball Quotients

Let $X$ be a compact Kahler surface which is a ball quotient. Can such $X$ contain a torus $T$ such that the fudamental class of $T$ is non-trivial? I expect this is false as $\pi_{1}(X)$ is a ...
3
votes
1answer
318 views

Algebraic surfaces of general type

Question. Are there smooth complex surfaces of general type with an irregularity $q = 1$ and Euler characteristic $3$. If the answer is yes, what is known about the geometry of such surfaces? Are ...
2
votes
0answers
163 views

On $\pi_1$ of surface of general type

Let $X$ be an algebraic surface of general type. Assume $K_X$ is an integer multiple of another class $A$, and the class $A$ can be represented by a symplectic submanifold $S$ of $X$ with non-negative ...
2
votes
1answer
302 views

Genus two pencil in K3 surface

It is known that smooth $K3$ surface can be obtained as two fold branched cover of rational elliptic surface $E(1) = \mathbb{CP}^2 9 \bar{{\mathbb{{CP}^2}}}$ along the smooth divisor $2F_{E(1)} = 6H - ...
3
votes
2answers
547 views

Question on K3 Surface

Is it possible to realize $K3$ surface as a ramified double cover of rational elliptic surface? If so, is there way to see an elliptic fibration structure on $K3$ from such cover? It seems to me one ...
1
vote
0answers
120 views

Classification of fibres in pencils of curves of genus two

For the case of characteristic positive, there exist some clasification of families of curves of genus two over a elliptic curve?.
2
votes
0answers
300 views

The cohomology of the relative dualizing sheaf of a relative curve

Let $X\to S$ be a curve over $S=\mathrm{Spec} \ \mathbf{Z}$. Let $\omega$ be the relative dualizing sheaf of $X\to S$. Let $g$ be the genus of the generic fibre. Assume that $g\geq 2$. I know that ...
3
votes
0answers
185 views

An arithmetic analogue of the discriminant curve of a conic bundle threefold

I am looking for an "arithmetic" analogue of a well known result on threefolds with a conic bundle structure. The following result can be found in [Iskovskikh - On the rationality problem for conic ...
4
votes
2answers
478 views

Q-factorial and rational singularities on surfaces

Let $X$ be a normal surface. Is any rational singularity $\mathbf{Q}$-factorial? I've seen this somewhere for surfaces over fields, but what about the general case of an integral 2-dimensional ...
2
votes
1answer
269 views

Picard/cohomology lattice of surfaces of low degree in $\mathbb P^3$

Let $S_{d>3}\subset\mathbb{P}^3_{\mathbb{C}}$ be a smooth surface of degree $d$. What is known (where to read?) about the Picard/cohomology lattice for small d? e.g. for $d=4$ the cohomology ...
4
votes
1answer
607 views

Minimal Model Program for surfaces over algebraically closed fields of characteristic p

Let $k$ be an algebraically closed field of characteristic $p>0$. I have been trying to find out unsuccessfully if there is a mmp for algebraic surfaces over $k$. I know minimal surfaces are ...
2
votes
0answers
238 views

bielliptic surfaces

Definition: A surface is called $S$ bielliptic if $S \cong (E \times F) /G$, where $E,F$ are elliptic curves and $G$ is a finite group of translations of $E$ acting on $F$ such that $F /G \cong ...
2
votes
1answer
239 views

Sheaves with zero Chern classes on a $K3$ surface.

Let $S$ be a $K3$ surface. Is it true that any sheaf on $S$ with zero Chern classes is isomorphic to $\mathcal{O}_S^{\oplus n}$ for some $n$? If not, do you have any counterexample?
15
votes
2answers
645 views

When does the blow-up of $CP^2$ at N points embed in $CP^4$?

Write $X_N$ for this blow up. Place the N points in 'general position' as needed. Then $X_6$ embeds in $CP^2$ as a smooth cubic surface. (See, eg, Griffiths and Harris.) But there is no other ...
5
votes
2answers
437 views

On a result about genus two pencils

I am reading the paper "Canonical models of surfaces of general type" by E. Bombieri. In the last section of this paper, there is a statement saying that surfaces with $K^2=1$ and $p_g=0$ do not have ...
2
votes
3answers
621 views

Elliptic curves on abelian surface

Let $Y$ be an abelian surface. Is it true that for every general point $P \in Y$, there exists an elliptic curve passing through $P$?
4
votes
1answer
356 views

Pathologies of analytic (non-algebraic) varieties.

Note: By an "analytic non-algebraic" surface below I mean a two dimensional compact analytic variety $X$ (over $\mathbb{C}$) which is not an algebraic variety. A property of Nagata's example (see ...
2
votes
1answer
269 views

Log resolutions on surfaces and 3-folds in characteristic p

If $X$ is a normal projective variety and $D$ a divisor in it, we say that $\pi\colon (\widetilde X,\widetilde D)\rightarrow (X,D)$ is a log resolution if $\widetilde X$ is a resolution of $X$, the ...
8
votes
2answers
609 views

Is the square of a curve minus its diagonal affine?

Let $X$ be a smooth irreducible projective algebraic curve of genus $g\geq 1$ and $S=X^2$ the surface one obtains as the cartesian product of $X$ with itself. Let $\Delta$ be the diagonal in $S$, that ...
8
votes
0answers
688 views

Ample divisors on projective surfaces

Question: If $X$ is a projective surface and $U$ is an open affine subset of $X$, then is it true that $X \setminus U$ is the support of an (effective) ample divisor on $X$? Background: I was reading ...
4
votes
1answer
394 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
0answers
255 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
196 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
1answer
156 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
1answer
193 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
0answers
150 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
0answers
272 views

Two basic questions concerning geometrically ruled surfaces.

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
0answers
163 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
2answers
448 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
2answers
849 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
1answer
393 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
1answer
336 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
3answers
670 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 ...
4
votes
1answer
852 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
2answers
275 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
1answer
392 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
0answers
224 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
2answers
438 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
1answer
545 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
2answers
677 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 ...
0
votes
0answers
166 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
0answers
242 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 ...