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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7
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1answer
309 views

General conditions for normality of blow-up

Let $X$ be an integral, affine, normal complex surface. I am looking for conditions on zero-dimensional closed subschemes $Z$ in $X$ such that the reduced scheme associated to the blow-up of $X$ along ...
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3answers
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" ...
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1answer
125 views

Is the positive part of the Zariski decomposition of a big $\mathbb{R}$-divisor big?

I can't understand why the positive part of the Zariski decomposition of a big class is itself big. More concretely: let $X$ be a smooth projective surface over $\mathbb{C}$. Let $N^1_{\mathbb{R}}(X)$...
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0answers
222 views

Is the surface in $\mathbb{A}^3$ rational?

Consider the surface $$ (u_1^6 + 1)w^3 = (u_2^6 + 1) \subset \mathbb{A}^3 $$ over an algebraically closed field of characteristic $p \neq 2,3$. Is it rational, i.e., is there its proper ...
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2answers
141 views

Surface of type $(2,2)$ on the Segre cubic scroll $\mathbb{P}^1 \times \mathbb{P}^2 \subset \mathbb{P}^5$

Let $S=\mathbb{P}^1 \times \mathbb{P}^2 \subset \mathbb{P}^5$ embedded with the Segre embedding given by $\mathcal{O}_S(1,1)$. If we intersect $S$ with a general smooth quadric $Q \subset \mathbb{P}^5$...
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1answer
461 views

Restriction of a Cartier divisor

Let $X$ be a surface (so $2$-dimensional proper $k$-scheme) $D \subset X$ an effective Cartier divisor of $X$ which corresponding to an invertible sheaf $\mathcal{L}=O_X(D)$ and $C \subset X$ a closed ...
4
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1answer
115 views

Existence of perfect Morse functions on Fermat surfaces $x^n+y^n+z^n+w^n=0$

It seems that whether a simply connected 4 manifold needs 1-handles and 3-handles is still an open question, see Existence of Morse functions on simply connected manifolds. I am wondering if it is ...
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2answers
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 ...
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0answers
146 views

nonvanishing higher cohomology of a very ample divisor

I am looking for smooth projective varieties $X$, with $h^i(X, \mathcal{O}_X) = 0$ for $i > 0$, with a very ample line bundle $L$ with some nonvanishing higher cohomology. What is clear: (1) Curves ...
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1answer
258 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 $...
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104 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 ...
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104 views

Minimal model vs canonical model of a surface

When I have a projective surface $X$, for simplicity smooth, I can find a simpler smooth surface on its binational class. In this way we find in a finite number of steps the simplest surface $Y$, i.e. ...
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50 views

Strict transform does not modify the Normalization

I have a question about a reduction step in proof of Lemma 10.1.24 from Qing Liu's Algebraic Geometry and Arithmetic Curves on page 463: Firstly we reduce to $S$ local, but then we replace $\mathcal{...
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2answers
337 views

Smooth projective surface with geometrically integral reduction

Let $S$ be a geometrically connected smooth projective surface over $\mathbb{Q}_p$. Can it be put in a proper flat $\mathbb{Z}_p$-scheme with a geometrically integral special fiber?
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0answers
72 views

arithmetic del Pezzo surfaces in comparison with del Pezzo surfaces over a field

A del Pezzo surface is a smooth, 2-dimensional projective variety $X$ with ample anticanonical divisor, i.e. a 2-dimensional Fano variety. I am interested in the arithmetic analogue, a 2-dimensional ...
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1answer
193 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 ...
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1answer
387 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 ...
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1answer
180 views

On a quadratic diophantine equation

Given a quadratic diophantine equation in $\mathbb Z[x,y,z]$ of form $$ax^2+by^2+cx+dy+ez+f=0$$ are there standard methods to solve for it when $$\|(x^2,y^2,z)\|_\infty\leq e^{1/2}$$ $$\|(a,b,c,d,e,f)...
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0answers
190 views

Deformations of a blow up

My question is related to this question, but I'm looking for something a bit more explicit. Let $S$ be a smooth surface over $\mathbb C$, fix a point $s\in S$ and take the blow up $\beta \colon S' \...
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1answer
215 views

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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0answers
69 views

Nef and effective cone of minimal conic bundle

Let $\pi: S\to C$ be a minimal conic bundle over a field $k$ of characteristic zero. That is, $S$ is a geometrically irreducible smooth surface with Picard rank $2$ and $C$ a geometrically ...
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3answers
1k 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 ...
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0answers
84 views

Is there an infinite order $\mathbb{F}_{p}$-section for a certain elliptic surface $\mathcal{E}_n$?

Consider a natural number $n$, a finite field $\mathbb{F}_{p}$ (such that $p$ is prime, $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$, and $\sqrt[3]{2} \notin \mathbb{F}_p$), ...
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0answers
56 views

What conditions are sufficient for two points to be independent in the Mordell-Weil group?

Consider a finite field $\mathbb{F}_{q}$ and an elliptic surface $$ \mathcal{E}\!: y^2 + a_1(t)xy + a_3(t)y = x^3 + a_2(t)x^2 + a_4(t)x + a_6(t), $$ where $a_i(t) \in \mathbb{F}_{q}[t]$. I am mainly ...
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0answers
72 views

Is there a way to calculate the Picard $\mathbb{F}_q$-number of an (rational or K3) elliptic surface?

Consider a finite field $\mathbb{F}_{q}$ and an elliptic surface $$ \mathcal{E}\!: y^2 + a_1(t)xy + a_3(t)y = x^3 + a_2(t)x^2 + a_4(t)x + a_6, $$ where $a_i(t) \in \mathbb{F}_{q}[t]$. Is there a way ...
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0answers
70 views

Picard numbers of isogenous K3 surfaces over a non-closed field

Let $S_1, S_2$ be K3 surfaces defined over a field $k$ and $\phi\!: S_1 \dashrightarrow S_2$ a dominant rational $k$-map (so-called isogeny). It is known that $\rho(S_1) = \rho(S_2)$ for the complex ...
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0answers
328 views

When does Bogomolov's inequality become an equality?

The Bogomolov theorem says if $V$ is a rank 2 vector bundle on an algebraic surfaces $S$ is $H$-stable (in the sense of Mumford-Takemoto) for some ample divisor $H$, then $c_1^2(V) \leq 4c_2(V)$ holds....
29
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1answer
1k views

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-...
2
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1answer
154 views

Stable torsion free sheaf on smooth projective surface

Let $E$ be a torsion-free sheaf on a smooth projective variety $X$ over $\mathbb{C}$. Let $H$ be an ample line bundle on $X$. Then we say $E$ is stable if $\mu_{H}(F)<\mu_{H}(E),\,\forall 0\neq F \...
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0answers
110 views

Structure of the big cone and Seshadri constant on Fano manifolds

I would like to know something about the following two questions. Given $X$ Fano manifold and $L$ an ample line bundle on $X$, we define \begin{gather} \sigma(L,x):=\sup\{t>0\, :\, \mu^{*}L-tE \,\,...
2
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1answer
183 views

Picard group modulo codimension 2

Let $X$ be a normal (possibly singular) projective surface over $\mathbb{C}$. Consider the set $M_X$ of all coherent sheaves $F$ on $X$ such that there exists a finite subset $Y\subset X$ such that $F$...
0
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1answer
148 views

On the birational equivalent class of algebraic surfaces with Picard number $1$

An open subset $U$ of a projective surface $Z$ is big if $\mathrm{codim}_Z(Z\setminus U)\geq2$. Let $X$ and $Y$ be smooth complex projective surface. If there exists a birational map $f:X\...
1
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1answer
126 views

Fixed part of a line bundle on a K3 surface

This question comes from Huybrechts' lecture notes on K3 surfaces, more specifically, chapter 2. Let $ X $ be a K3 surface (over an algebraically closed field $ k $) and $ L $ a line bundle on $ X $. ...
2
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2answers
283 views

Why does a complex linear normalization of a real algebraic surface inherit a real structure?

Could you recommend any references to (some of) the following very basic assertions in algebraic geometry? (It seems unreasonable to reprove them in a research paper.) (1) Let a surface $X$ in $\...
3
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0answers
149 views

Explicit equations for rational elliptic surfaces (Halphen surfaces)

I am looking for explicit equations for rational elliptic surfaces in characteristic $2$. For me, a rational elliptic surface $X$ is a smooth projective surface $X$ which is rational and equipped with ...
2
votes
2answers
900 views

singularities of the dual variety of a surface

I am looking for a proof/reference of the following simple fact, which I think it holds true. Let $S\subset \mathbb{P}^n$ be a surface embedded by a very ample linear system. Then I know that the ...
5
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1answer
243 views

Field of definition for general type surfaces

In the survey paper https://arxiv.org/abs/1004.2583 of Bauer-Catanese-Pignatelli, they mention a question of Mumford: Can a computer classify all surfaces of general type with $p_g=0$? I've been ...
2
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1answer
136 views

Lines on a toric cubic surface with a line of nodes

Consider a cubic surface cut out by equations $x^2y - z^2w$ inside $\mathbb{P}^3$. This gives a cubic surface with a line of nodes, it is toric and has normalisation $\mathbb{F}_1$, a Hirzebruch ...
5
votes
1answer
207 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 ...
2
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0answers
103 views

The Weil restriction of an elliptic curve with respect to $\mathbb{F}_{p^2}/\mathbb{F}_{p}$

For a prime $p > 3$ consider the quadratic finite field extension $\mathbb{F}_{p^2}/\mathbb{F}_{p}$. Also, consider the elliptic curves $$ E\!: y_0^2 = x_0^3 + ax_0 + b,\qquad E^{(1)}\!: y_1^2 = ...
2
votes
1answer
142 views

Is there a way to find any $\mathbb{F}_2(t)$-point on the elliptic curve $\mathcal{E}$?

Consider the ordinary elliptic curves $$ E\!:y_1^2 + x_1y_1 = x_1^3 + 1,\qquad E^\prime\!: y_2^2 + x_2y_2 = x_2^3 + x_2^2 + 1 $$ over the field $\mathbb{F}_2$. They are quadratic twists to each other....
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0answers
110 views

Resolution of rational surfaces

Let $S$ be a rational singular complete algebraic surface over $\mathbb{C}$. Let $\phi:\tilde{S}\to S$ be a resolution of singularities with minimal possible Picard rank (i.e. minimal $\mathrm{dim}(...
4
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0answers
187 views

Del Pezzo surfaces and Picard--Lefschetz theory

Let $X$ be a del Pezzo surface, say of degree $3$ for concreteness. Then compare: the $27$ $(-1)$-curves form a lattice $E_6\subseteq H^2(X;\mathbf{Z})$; the Weyl group is generated by the simple ...
5
votes
3answers
422 views

Is there a way to find any non-trivial $\mathbb{F}_p(t)$-point on the given elliptic curve?

Consider a finite field $\mathbb{F}_p$ (where $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$) and the elliptic curve $$ E\!:y^2 = x^3 + (t^6 + 1)^2 $$ over the univariate ...
2
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0answers
134 views

Is there a way to explicitly find any rational $\mathbb{F}_p$-curve on the Kummer surface?

Consider a finite field $\mathbb{F}_p$ (where $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$), $\mathbb{F}_{p^2}$-isomorphic elliptic curves (of $j$-invariant $0$) $$ E\!:y_1^2 = ...
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0answers
119 views

Hypersurfaces with maximal Picard rank

Is it true that for any $d \ge 4$, there exists a smooth, degree $d$ surface $X$ in $\mathbb{P}^3$ with maximal Picard rank i.e., Picard rank of $X$ equals $h^{1,1}(X)$?
4
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0answers
162 views

Fibered surfaces degenerating to Frobenius

Let $R$ be a DVR with algebraically closed, positive characteristic residue field $k$. Let $X\rightarrow Spec(R)$ and $C\rightarrow Spec(R)$ be smooth projective morphisms of relative dimension 2 and ...
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0answers
123 views

Sheaf of Kähler Differentials is Invertible in Dense Open Subset

Let $f:S→B$ be an elliptic fibration from an integral surface $S$ to integral curve $B$ . Here I use following definitions: A surface (resp. curve) is a $2$ -dim (resp. $1$-dim) proper k scheme ...
2
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0answers
116 views

Rational curves on ruled surfaces

Let $S$ be a ruled surface (over an algebraically closed field) with an $\mathbb{P}^1$-bundle $\pi\!: S \to E$ onto an elliptic curve $E$. What is the classification of (possibly singular) irreducible ...
3
votes
1answer
441 views

Where to find “Families of curves on a surface of general type” (MR0457450)?

I am currently doing some research on surfaces of general type and I need some results from Bogomolov's paper: Bogomolov, F. A. Families of curves on a surface of general type. Dokl. Akad. Nauk SSSR ...

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