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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67 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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192 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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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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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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69 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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67 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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1answer
162 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 $F$...
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90 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 \,\,...
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1answer
136 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\...
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105 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 $. ...
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131 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 ...
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1answer
200 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 ...
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269 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 $\...
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1answer
127 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 ...
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1answer
203 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 ...
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90 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 = ...
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What are other elliptic $\mathbb{F}_2$-fibrations on the Kummer surface with an $\mathbb{F}_2$-section?

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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1answer
139 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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109 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}(...
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178 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 ...
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133 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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3answers
416 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 ...
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114 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 $h^{1,1}(X)$?
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154 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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118 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 ...
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109 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 ...
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1answer
308 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 ...
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1answer
280 views

Birational Invariants of regular surfaces

Let $X,Y$ surfaces (so $2$-dimensional proper $k$-schemes) which are regular (so the stalks are regular) and birational and denote by $f: X \dashrightarrow Y$ the corresponding rational birational ...
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1answer
91 views

Automorphism of ruled surfaces associated to stable vector bundles

Let $X$ be a compact Riemann surface, and let $P \rightarrow X$ be a holomorphic $\mathbb P^1$-bundle over $X$. Then we know that $P$ is of form $\mathbb P(E)$ for some vector bundle $E \rightarrow X$ ...
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1answer
131 views

Possible configurations of rational curves on a rational surface

Consider a set of smooth rational curves on a rational surface, say, with normal crossings between curves. Is anything known on what combinatorics of configurations are possible? Say, what ...
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1answer
333 views

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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1answer
199 views

elliptic fibration over $\mathbb{P}^1$ with exactly two fibres with monodromy of unipotency rank 1

Despite the apparent simplicity of the following question I couldn't find the answer so far. I am looking to construct an elliptic fibration $X \to \mathbb{P}^1$ with $X$ smooth, and exactly two ...
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2answers
392 views

Blow-up of the plane at $5$ points

If we Blow-up $\mathbb P^2_{\mathbb C}$ at $5$ points $T=\{p_1,\ldots,p_5\}$ we obtain a Del Pezzo surface $X$ of degree $4$. Now take another set of $5$ points $T'=\{q_1,\ldots,q_5\}$ ($T'\neq T$), ...
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1answer
405 views

Why only some del Pezzo are toric?

Let us define smooth del Pezzo surfaces $dP_r$ as the blowup of $r$ generic points in $\mathbb{CP}_2$. One can show that if we request $dP_r$ to be Fano, then $r=0,...,8$. In theoretical physics ...
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1answer
151 views

Infinitesimal deformation of strict transform

Let $X$ be an affine, complex surface with isolated singularities and $\pi:\widetilde{X} \to X$ be a resolution of singularities (not necessarily minimal) i.e., $\widetilde{X}$ is non-singular and $\...
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95 views

Arnold's theorem on small denominators and holomorphic tubular neighborhoods

By a theorem of Grauert, along a curve with negative self-intersection a complex surface is locally biholomorphic to a neighborhood of the zero section of that curve inside its normal bundle. For ...
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132 views

Is there a $\sum e_if_i=n$ in higher dimensions?

If $X\to Y$ is a finite map of connected proper algebraic curves over a field, then for any point $y\in Y$, the sum $\sum e_xf_x=n$ of ramification times inertia degrees over points $x$ mapping to $y$ ...
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1answer
215 views

Quotient of a smooth projective surface by an involution

Is the quotient of a smooth complex projective surface by an involution projective? Suppose the quotient happens to be smooth; does that change the situation?
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134 views

Are there three ordinary elliptic curves $E$, $E_1$, $E_2$ such that $E^2 \cong E_1 \!\times\! E_2$?

Consider the elliptic curve $E\!: y^2 = x^3 + 1$ of $j$-invariant $0$ over an algebraically closed field $k$ of characteristics $p$. Let me remind that $E$ is ordinary (i.e., non-supersingular) iff $p ...
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2answers
368 views

Show Fiber Product of Rational Elliptic Surfaces is Calabi-Yau

In a handful of contexts people study Calabi-Yau threefolds formed by taking the fiber product of two rational elliptic surfaces. I can't find any detailed explanation of why such geometries are ...
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0answers
108 views

Is the generalized Kummer threefold rational in characteristics 3?

Let $E_i\!: y_i^2 = x_i^3 - x_i$, $i = 1, 2, 3$ be three copies of the supersingular elliptic curve in characteristics $3$. Consider on $E_i$ the following automorphism of order $3$: $$ \sigma(x_i,...
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1answer
87 views

Existence of meromorphic 2-forms over normal surface singularities

Let $(X,o)$ be an isolated normal surface singularity. Denote by $U:=X\backslash \{o\}$. I am looking for conditions on $(X,o)$ under which there exists a holomorphic section $\omega \in H^0(U, \Omega^...
7
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1answer
396 views

Classification of smooth algebraic surfaces with a smooth morphism to $\Bbb P^1$

Let $k$ be an algebraically closed field, it is well known that $\mathbb P^1$ is simply connected, but how about smooth projective surfaces $X$ with a smooth morphism to $\Bbb P^1$? Except the case $...
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1answer
58 views

Minimal complex surfaces with pseudo-effective canonical bundles

A complex line bundle $L$ over a complex surface $X$ is said to be pseudo-effective if it admits a (possibly singular) Hermitian metric $h$ whose curvature is positive semi-definite in the sense of ...
6
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1answer
281 views

Breaking a morphism with generic fiber $\mathbb{F}_n$

Assume we are working over $\mathbb{C}$, and we have a projective morphism with connected fibers $f: X \rightarrow Z$ whose geometric generic fiber $X_\overline{\eta}$ is isomorphic to a Hirzebruch ...
4
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1answer
375 views

Intersection form in Algebraic Geometry/Topology

Let $S$ be a smooth complex projective surface. We let define an intersection form $(-)\cdot(-)$ on $\mathsf{Pic}(S)$ by setting $$D\cdot D':=\mathcal{O}_S(D)\cdot\mathcal{O}_S(D')$$ where the ...
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0answers
76 views

Thom-type isomorphism on sheaf cohomology

Let $X$ be a smooth, projective surface and $T$ a finite set of points in $X$ i.e., of codimension $2$ in $X$. Is it true that $H^i(\mathcal{O}_X)=H^i(\mathcal{O}_{X\backslash T})$ for $i \ge 1$?
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164 views

Very ample divisors on blow ups of the projective plane

Let $X$ be $\mathbb{P}^2$ blown up at $k$ points in general position. The Picard group of $X$ is just $\mathbb{Z}^{k+1}$ and one knows the intersection product explicitly. If $D$ is an ample divisor, ...
3
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2answers
202 views

Singularities of a central fibre of a flat family of smooth surfaces

Suppose I have a one parameter flat family of complex surfaces (regular, of general type) whose general fibre is smooth. Is it possible for the central fibre to have singularities which are not ...
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0answers
91 views

Fiber product of an elliptic surface

Let $f:S\to P^1$ be a smooth elliptic surface and let $X=S\times_{P^1} S$ be the fiber product. The threefold $X$ is singular in general (typically isolated ODPs). But is $X$ $\mathbb Q$-factorial? Or,...

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