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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Fundamental group of cyclic branched cover of affine plane

Let $f\in \mathbb{C}[x,y]$ be an irreducible polynomial. Let $n>0$ be an integer such that the hypersurface $S:=\{ (x,y,z)\in \mathbb{C}^3|z^n=f(x,y) \}$ is a connected complex submanifold of $\...
Doug Liu's user avatar
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One question about Manetti surface

I am reading Ascher-Devleming-Liu's paper "Wall crossing for K-moduli spaces of plane curves" theorem 5.2 ADL19 and l have some confusions about the proof. Theorem 5.2 states that fixed a ...
RedLH's user avatar
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239 views

Question about surface singularities

Throughout, $X$ will be a projective surface. I am looking for examples of the following surface singularities, I) A rational singularity that is not quotient. Obviously, it has to be non-Gorenstein, ...
Rio's user avatar
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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 ...
Ben C's user avatar
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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 ...
Armando j18eos's user avatar
1 vote
0 answers
186 views

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 ...
user267839's user avatar
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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 ...
LeechLattice's user avatar
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2 votes
1 answer
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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(...
Dimitri Koshelev's user avatar
1 vote
1 answer
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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 ...
Z.N's user avatar
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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 ...
Armando j18eos's user avatar
1 vote
0 answers
87 views

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-...
Armando j18eos's user avatar
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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 ...
Ben C's user avatar
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7 votes
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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 ...
Ben C's user avatar
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4 votes
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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 \...
Ben C's user avatar
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1 vote
0 answers
68 views

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 ...
Ben C's user avatar
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0 votes
1 answer
243 views

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(...
Mobius's user avatar
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3 votes
1 answer
272 views

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 ...
Jérémy Blanc's user avatar
2 votes
1 answer
356 views

$K3$ 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 ...
user267839's user avatar
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0 votes
1 answer
164 views

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 ...
Armando j18eos's user avatar
1 vote
1 answer
114 views

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 ...
Pulcinella's user avatar
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1 vote
0 answers
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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 ...
Hank Scorpio's user avatar
0 votes
0 answers
109 views

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 ...
Armando j18eos's user avatar
2 votes
1 answer
225 views

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\...
Puzzled's user avatar
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1 vote
1 answer
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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}:...
Puzzled's user avatar
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8 votes
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260 views

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 ...
Dori Bejleri's user avatar
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1 vote
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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 $\...
Dimitri Koshelev's user avatar
5 votes
1 answer
231 views

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 $(1,1,1)$, $(1,-1,-1)$, $(-1,1,-1)$, $(-1,-1,1)$....
LeechLattice's user avatar
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7 votes
0 answers
201 views

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 ...
Francesco Polizzi's user avatar
1 vote
0 answers
69 views

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{...
dennis's user avatar
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4 votes
0 answers
151 views

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 $X$...
Francesco Polizzi's user avatar
6 votes
1 answer
473 views

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 - ...
dennis's user avatar
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6 votes
0 answers
171 views

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 $\Omega_X$ ...
Francesco Polizzi's user avatar
7 votes
1 answer
242 views

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....
cll's user avatar
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5 votes
0 answers
229 views

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{...
Francesco Polizzi's user avatar
1 vote
0 answers
114 views

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: ...
Roxana's user avatar
  • 519
3 votes
2 answers
265 views

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 $D_1$, $D_2$, $...
tota's user avatar
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5 votes
1 answer
281 views

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 $\...
Francesco Polizzi's user avatar
3 votes
0 answers
242 views

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 \...
Basics's user avatar
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3 votes
0 answers
266 views

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 ...
Basics's user avatar
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2 votes
0 answers
152 views

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)\...
H U's user avatar
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4 votes
2 answers
191 views

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 ...
Selim G's user avatar
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11 votes
0 answers
362 views

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 ...
Pulcinella's user avatar
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2 votes
0 answers
155 views

Automorphisms of finite order on $K3$ surfaces

Is there a $K3$ surface (algebraic, complex) that has infinitely many automorphisms of finite order? Many K3 surfaces have infinite automorphism groups. In particular, all K3 surfaces of Picard ...
Basics's user avatar
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2 votes
0 answers
173 views

Automorphisms of a K3 surface

I was studying the following algebraic surface in $\mathbb{P}^5$ defined by the following three quadrics: \begin{cases} x^2 + xy + y^2=w^2\\ x^2 + 3xz + z^2=t^2\\ y^2 + 5yz + z^2=s^2. \...
did's user avatar
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2 votes
0 answers
256 views

Example of a K3 surface with two non-symplectic involutions

$\DeclareMathOperator\Pic{Pic}$Let $X$ be a K3 surface (algebraic, complex). An involution $\sigma:X\rightarrow X$ is called non-symplectic if it acts as multiplication by $-1$ on $H^{2,0}(X)=\Bbb{C}\...
Basics's user avatar
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7 votes
1 answer
528 views

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 $x$, $y$, $z$ ...
Dimitri Koshelev's user avatar
1 vote
1 answer
219 views

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 ...
Puzzled's user avatar
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3 votes
1 answer
170 views

Singularities of surfaces fibered in rational curves

Let $S$ be a projective surface with a morphism $S\rightarrow\mathbb{P}^1$ whose fibers are either smooth $\mathbb{P}^1$'s or the union of two smooth $\mathbb{P}^1$'s intersecting in a point. ...
Puzzled's user avatar
  • 8,852
4 votes
1 answer
164 views

Tri-homogenous polynomials of tridegree $(3,3,3)$ to add three points on an elliptic curve

Consider an elliptic curve $E \subset \mathbb{P}^2$ with the zero point $\mathcal{O}$. There are classical articles about complete systems of addition laws on $E$ (see Lange and Ruppert - Complete ...
Dimitri Koshelev's user avatar

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