# 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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### Equality case of the log-Bogomolov-Miyaoka-Yau inequality

The Bogomolov-Miyaoka-Yau inequality for sufaces says that if $X$ is a smooth projective minimal surface of general type then $c_1(X)^2 \le 3 c_2(X)$. It is a theorem of Yau (I think) that equality ...

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### Triple covers of $\mathbb{P}^2$ with Tschirnhausen module $\mathcal{O}(-1)\oplus\mathcal{O}(-1)$

Let $X$ be a surface as in the title. Rick Miranda said that $X$ is a Steiner cubic in $\mathbb{P}^4$, and the cover map is projection. Invariants of $X$ can be computed directly, $p_g(X)=0,K^2_X=8,e(...

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### Action on Enriques surface by sections of Jacobian fibration

A question about a statement in Shigeyuki Kondo's paper Enriques surfaces with finite automorphism groups:
The setup: Let $\pi: Y \to \mathbb{P}^1$ be a special elliptic pencil of complex Enriques ...

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### Multiplicity of irreducible component of a singular fiber of a $\mathbb{P}^1$-fibration

Let $X$ be a smooth projective surface and $f:X\to\mathbb{P}^1$ be a $\mathbb{P}^1$-fibration with a singular fiber consisting of a tree with three irreducible rational ($-2$)-curves $D_1$, $D_2$, $...

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### Formula for Pushforward of structure sheaf for branched coverings

I have some questions of same flavour about two following constructions in Daniel Huybrechts's notes on K3 surfaces.
Construction 1: Kummer surface (Example 1.3 (iii), page 8) Let $k$ be a field of $...

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### On the positivity of cotangent bundle of elliptic surfaces

I am working on the complex numbers field $\mathbb{C}$, for simplicity. However you can relax this assumption if you want.
Let $X$ be an elliptic surface, id est there is a proper morphism $\pi\colon ...

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### Surfaces with rational double points

Let $S\rightarrow \mathbb{P}^1$ a surface fibered in conics over a field. Assume that $S$ has a single non reduced fiber $F$ with two points of type $A_1$ on it.
Blowing-up the two points and ...

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### 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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### What are the finite étale coverings of a quasi-hyperelliptic surface?

Let $X$ be a quasi-hyperelliptic surface in characteristic 3 where the canonical bundle $K_X$ is trivial.
Question: Is there a finite étale covering $Y \rightarrow X$ such that
$Y$ is an abelian ...

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### (Non-)Rationality of a certain quotient of the symmetric square of the Fermat sextic (quartic) curve

Consider the Fermat sextic curve $F: x^6 + y^6 + 1 = 0$ over an algebraically closed field of characteristic $0$. It has the two order $3$ automorphisms $\omega_x(x,y) := (\omega x, y)$ and $\omega_y(...

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### Derived McKay correspondence between a weighted projective plane and a Hirzebruch surface

Let $k$ be an algebraically closed field of $\text{ch}(k) =0$.
Let $\mathbb{P}(1,1,2)$ be the weighted projective plane of weight $(1,1,2)$ as a stack.
Let $\mathbf{P}(1,1,2)$ be the weighted ...

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### Existence of elliptic curves on surfaces of general type

Let $X$ be a complex minimal surface of general type, id est $K_X$ is big and nef. It is well-known that $\displaystyle\int_X3c_2(X)-c_1(X)^2\geq0$, and the equality holds if and only if $X$ is ...

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### On surfaces of general type wich saturate the BMY-inequality

Let $\mathbb{K}$ an algebraically closed field of characteristic $0$, let $X$ be a smooth minimal surface of general type.
It is known that surfaces satisfy, among other thing, the (Bogomolov-Miayoka-...

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### 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 ...

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### Does there exist a simply connected surface with CM whose cotangent bundle is ample?

Does there exist a smooth projective complex surface $X$ such that,
(1) $\pi_1(X) = 0$
(2) $\Omega_X^1$ is ample
(3) the Mumford-Tate group of $H^2(X)$ is a torus
There exist examples with any two of ...

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### Can "fake rational surfaces" be simply-connected?

I say that a smooth projective complex algebraic surface $X$ is a "fake rational surface" if its Hodge diamond looks like:
and $X$ is of general type.
It is well-known that fake projective ...

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### Fundamental groups of Hirzebruch's line arrangement varities

Let $\Lambda$ be a line arrangement in $\mathbb{P}^2$ and $n > 0$ an integer. Then Hirzebruch defined a smooth projective surface $H(\Lambda, n)$ as the minimal desingularization of a covering $Y \...

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### Explict equations for unirational Enriques surface with a nonzero 1-form

I am hoping to write down very explicitly the equations for the following data:
an Enriques surface $X$ of type $\mathrm{Pic}^{\tau} = \mathbb{Z} / 2 \mathbb{Z}$ such that its canonical $\mu_2$-cover ...

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### Constructions of complex surfaces covered by the ball of $\mathbb{C}^2$

Let $S$ be a compact complex surface. It is well-known that the following two facts are equivalent
$c_1^2(S) = 3 c_2(S)$ and $S \neq \mathbb{CP}^2$
The universal cover of $S$ is biholomorphic to the ...

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### Minimal resolution of singularities of surfaces

Let $X$ be a normal projective irreducible surface over an algebraically closed field $k$. Let $\pi\colon Y\to X$ be a birational morphism, such that $Y$ is a smooth projective surface, and assume ...

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### 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 ...

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### $K3$ surfaces can't be uniruled

Let $S$ be a uniruled surface, ie admits a dominant map $ f:X \times \mathbb{P}^1$. Why then it's canonical divisor $\omega_X$ cannot be trivial? Motivation: I want to understand why $K3$ surfaces ...

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### A constructive proof of the theorem of the cube

Do you know a constructive proof of the theorem of the cube ? More precisely, let $X$, $Y$, $Z$ be projective varieties (e.g., over an algebraically closed field $k$) with points $x$, $y$, $z$ ...

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### BMY inequality for surfaces of general type in characteristic 0

Let $X$ be a smooth, complex, projective, minimal surface of general type, i.e. the canonical (line) bundle $K_X$ is big and nef.
It is known that $3c_2\geq c_1^2$ (the Bogomolov-Miyaoka-Yau ...

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### Families of torsion-free sheaves whose length jumps

For a long time, I had a false belief that the space/stack $\text{Coh}^{tf}_{c_1,c_2}S$ of torsion-free sheaves $\mathcal{E}$ on a smooth algebraic surface $S$ was not connected, since if you take its ...

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### Confusion with the genus of a curve $Y$ in a ruled surface $X\to C$ such that $Y\to C$ is inseparable

This was originally posted on MSE, but after a fair amount of time and a bounty it got no response. Unfortunately I have not yet resolved my doubts.
There's a fair bit of setup here, but you don't ...

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### On the positivity of the second Segre class of ample vector bundles

Let $E$ be an ample rank $r\geq2$ vector bundle over a smooth projective surface $X$ defined on an algebraically closed field $\mathbb{K}$ of characteristic $0$.
In Kleiman S. L. - Ample Vector ...

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### Classification of quartic surfaces

Let $k$ be a field of characteristic zero (non necessarily algebraically closed, we may assume for instance that $k = \mathbb{C}(t)$). Does there exist a classification of degree four surfaces $S\...

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### Geometry of contracted divisors

Let $f:\mathbb{P}^3\dashrightarrow\mathbb{P}^2$ be a dominant rational map defined over a field $k$ (not necessarily algebraically closed) of characteristic zero.
Consider a resolution $\widetilde{f}:...

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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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### Fundamental group of a smoothing of a complex surface

Let $X_0$ be a compact complex algebraic surface with an isolated singularity and let $X_t$ be a smoothing of $X_0$ over the disc. How can we compute the fundamental group of $X_t$ say in terms of the ...

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### 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 ...

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### Global generation of $S^n \Omega_X$ for a fake projective plane

Let $X$ be a fake projective plane, namely, a compact complex surface with
$$p_g(X)=q(X)=0, \quad K_X^2=9$$ and $K_X$ ample.
Since $K_X^2=9 \chi(\mathcal{O}_X)$, Yau's celebrated proof of the Calabi ...

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### An elliptic threefold and the Mordell–Weil lattices of its reductions

Let $T\!: y^2 = x^3 + a(t, s)x + b(t, s)$ be an elliptic threefold over a finite field $\mathbb{F}_q$ of characteristic $p > 3$. In other words, we have an elliptic curve over the function field $\...

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### What is the surface area of the finite part of the Cayley nodal cubic surface?

The Cayley nodal surface is defined by the equation $x^2+y^2+z^2-2xyz=1$. The finite part of the surface is the tetrahedral part bounded by the 4 nodes $(1,1,1)$, $(1,-1,-1)$, $(-1,1,-1)$, $(-1,-1,1)$....

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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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### Surface with $\Omega_X$ globally generated and singular Albanese image

This question is inspired by abx's comment to my previous question MO430933.
Let $X$ be a complex surface of general type, and denote by $$a \colon X \to \operatorname{Alb}(X)$$ the Albanese map of $X$...

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### Diffeomorphism induced by small perturbation

Consider the surface $S_{\epsilon}$ defined as:
\begin{align}
%S&=\{\vec x \in \mathbf{R}^3: x=0\}, \\
S_{\epsilon}&=\{\vec x \in \mathbf{R}^3:\epsilon (x^2 + y^2 + z^2 - 1) + x=0\}.
\end{...

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### Topology change induced by small perturbation

Consider the surface $S_{\epsilon}$ defined as:
\begin{align}
%S&=\{\vec x \in \mathbf{R}^3: x=0\}, \\
S_{\epsilon}&=\{\vec x \in \mathbf{R}^3:f_{\epsilon}(\vec x)\equiv\epsilon ((x^2 + y^2 - ...

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### Lower bound for $h^0(X, \operatorname{Sym}^n \Omega_X)$

This is a weaker version of my previous (unanswered) question MO429574.
Let us start with a smooth, ample divisor $X$ in an abelian threefold $A$. It is a surface of general type such that $\Omega_X$ ...

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### Basepoints in the canonical system of algebraic surfaces

Let $X$ be a 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 ...

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### Different algebraic structures on complements to divisors

Complements to square-zero curves in projective surfaces sometimes have several non-isomorphic algebraic structures. Serre’s example is possibly the most famous illustration of this phenomenon (see f....

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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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### Computation of $H^i(X, \, \operatorname{Sym}^n \Omega_X)$ for a surface of general type $(i=0, \, 1)$

Let $X$ be a smooth, complex surface of general type such that $\Omega_X$ is globally generated, and let $n \geq 2$ be a natural number.
Question. Is there a way to compute $h^i(X, \, \operatorname{...

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### Abelian subvarieties corresponding to vector subspaces

Let $S$ be a connected smooth projective surface.
Let $C$ a smooth curve on $S$
In page 9 of the paper "https://arxiv.org/abs/1704.04187v1" a read the following:
Let
\begin{equation*}
r: ...

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### Some questions about the (projectivized cotangent bundle of the) symmetric square of a genus $3$ curve

Let $C$ be a smooth, non-hyperelliptic curve of genus $3$ and $X:= \mathrm{Sym}^2{C}$ its symmetric square. Then $X$ is a smooth, minimal surface of general type with $p_g=q=3, \, K^2=6$.
Calling $\...

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### Ample divisor of degree two on a blow-up of $\mathbb P^2$ at nine points

Let $\pi:S \rightarrow \mathbb P^2$ be a blow-up at nine points in general position.
I am finding an ample divisor $L$ on $S$ of degree two ($L^2=2$).
Since $Pic(S) = \mathbb Z h \oplus \mathbb Z e_1 \...

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### A K3 cover over a Del Pezzo surface

Let $V \rightarrow \mathbb P^2$ be the blow-up at two distinct points. ($V$ is a Del Pezzo surface of degree 7.)
Choose a smooth curve $C$ from the linear system $|-2K_V|$ and let $S \rightarrow V$ be ...

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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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### Singular Del Pezzo of degree 2

Throughout, singular Del Pezzo means a surface with only isolated singularities and ample anti-canonical divisor.
Suppose $X$ is a singular Del Pezzo of degree 2 over a field $k$ where $\text{char}(k)\...