# Questions tagged [algebraic-surfaces]

An algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold.

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### 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, ...
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### When do 27 lines lie on a cubic surface?

Consider $27$ (pairwise distinct!) lines in $\mathbb{P}^3$ whose intersection graph is that expected¹ of the $27$ lines on a smooth cubic surface. Question: Is there a simple necessary and sufficient ...
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### Enriques surfaces over $\mathbb Z$

Does there exist a smooth proper morphism $E \to \operatorname{Spec} \mathbb Z$ whose fibers are Enriques surfaces? By a theorem of, independently, Fontaine and Abrashkin, combined with the Enriques-...
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### 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 ...
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### Analogies between classical geometry on complex surfaces and Arakelov geometry

This is my first question on this wonderful site. The following question about Arakelov geometry is gonna be quite long and wide; to be clear one of that kind of questions that are usually ignored. ...
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### 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 ...
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### 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 ...
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### Del Pezzo surfaces and homotopy groups of spheres

A (complex) del Pezzo surface is a smooth projective complex surface with ample anticanonical line bundle. Such surface has a degree defined as the self intersection of the canonical divisor. It is ...
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### 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 ...
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### 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 ...
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### 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 $N$...
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### what is the cyclic cover trick?

What do people mean by the "cyclic cover trick"? I have found this expression a couple of times with no complete explanation, both talking about curves and surfaces...
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### A question on surfaces in $\mathbb{P}^4$

On surfaces in $\mathbb P^4$,Ellingsrud and Peskine has proved that There exists an integer $d_0$ such that for any integer $d>d_0$,any smooth surface of degree $d$ in $\mathbb P^4$ is of ...
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### 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 $S$...
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### Curves with negative self intersection in the product of two curves

I wonder if the following is known: Are there two compact curves C1 and C2 of genus>1 defined over complex numbers, such that their product contains infinite number of irreducible curves of negative ...
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### 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 $W \geq 0$, ...
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### K3 surfaces with good reduction away from finitely many places

Let S be a finite set of primes in Q. What, if anything, do we know about K3 surfaces over Q with good reduction away from S? (To be more precise, I suppose I mean schemes over Spec Z[1/S] whose ...
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### 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 ...
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### 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 ...
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### Curves on rational surfaces and Lang's conjecture for M_g

There are a group of related conjectures associated to Lang's name - for this question I'll consider only the weakest one, namely that rational curves in a projective variety of general type are not ...
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### Nonprojective Surface

Let k be an algebraically closed field. It's well known that every complete curve, period, is projective. Also, that every smooth surface is, and that there are smooth 3-folds which are not, and ...
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### level sets of multivariate polynomials

Let $p:\mathbb R^n \rightarrow \mathbb R$ be a polynomial of degree at most $d$. Restrict $p$ to the unit cube $Q=[0,1]^n\subset\mathbb R^n$. We assume that $p$ has mean value zero on the unit cube $Q$...
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### Motivation for birational geometry

I'm interested in how do people that work in birational geometry view their field — specifically, what are the kinds of geometric questions (as opposed to commutative-algebraic questions) that ...
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### 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 $\mathbb{P}^3$ ...
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### Euler Characteristic of Real Algebraic Surfaces

Given a (compact if needed) real smooth surface $V(f)$ defined by $f\in \mathbb{R}[X,Y,Z]$, in particular it is oriented. Is there a formula which gives the Euler character of $V(f)$ ? Thanks.
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### A property of varieties between unirational and retract rational

EDIT: The vague question Q1 below is partially answered, while the concrete question Q2 seems to be still open. Let $V$ be a geometrically integral variety over a field $K$. I consider the following ...
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### Moduli stacks of algebraic surfaces—obstructions to existence?

The moduli stack $\mathcal{M}_g$ of genus $g$ curves is one of the deepest objects in mathematics, so of course you wonder to what extent you can construct an (Artin?) stack parametrising algebraic ...
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### Surfaces with $q=2$ and generically finite Albanese map

I have a family of surfaces of general type $S$ with $q(S)=2$, and such that the Albanese map $$\alpha \colon S \longrightarrow A:=\mathrm{Alb}(S)$$ is generically finite of degree $n$. By a result of ...
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### Singular curve on an abelian surface

Let $C_2$ be a smooth genus $2$ curve and $J(C_2)$ its Jacobian. It is well known that the blow-up of $J(C_2)$ at the origin $o$ is isomorphic to the second symmetric product $\textrm{Sym}^2(C_2)$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Pull-back of an irreducible ample divisor via an isogeny of abelian varieties

In the (wonderful) book by C. Birkenhake and H. Lange Complex Abelian Varieties we can find the following result, see Corollary 4.3.4 page 77. It is stated in any dimension $g \geq 2$, but let us ...
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### Does $X\times Y$ have the resolution property if both $X$ and $Y$ have?

We say a complex manifold $X$ has the resolution property if every coherent sheaf $\mathcal{M}$ on $X$ admits a surjection $\mathcal{E}\twoheadrightarrow \mathcal{M}$ by some finite rank locally free ...
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### Dimension-specific phenomena in algebraic geometry

In differential topology, there are some funny phenomena that can only happen in dimension 4. For example, only in dimension 4 you can have a closed topological manifold admitting infinitely many ...
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### How many characteristics is a random surface unirational in?

Suppose I have a surface $X$ defined over $\mathbb{Z}$. I am interested in the set $S_X$ of primes $p$ such that $X_{\overline{\mathbb{F}}_p}$ is unirational. If I choose a "random" surface ...
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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 ...