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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1 answer
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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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2 votes
1 answer
140 views

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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8 votes
1 answer
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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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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 $\...
5 votes
1 answer
174 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 $(1,1,1)$, $(1,-1,-1)$, $(-1,1,-1)$, $(-1,-1,1)$....
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7 votes
0 answers
185 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 ...
1 vote
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59 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{...
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4 votes
0 answers
130 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 $X$...
6 votes
1 answer
454 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 - ...
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6 votes
0 answers
159 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 $\Omega_X$ ...
7 votes
1 answer
188 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....
5 votes
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224 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{...
1 vote
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106 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: ...
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3 votes
1 answer
136 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 $D_1$, $D_2$, $...
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4 votes
1 answer
227 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 $\...
4 votes
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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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4 votes
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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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2 votes
0 answers
101 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)\...
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4 votes
1 answer
148 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 ...
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11 votes
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329 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 ...
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3 votes
0 answers
128 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 ...
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2 votes
0 answers
158 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. \...
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3 votes
0 answers
221 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}\...
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5 votes
1 answer
345 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 $x$, $y$, $z$ ...
2 votes
1 answer
115 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 ...
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4 votes
1 answer
144 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. ...
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4 votes
1 answer
140 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 ...
7 votes
1 answer
998 views

Relating the holomorphic Euler characteristic of a family of algebraic varieties to properties of the base and fibers

Let $f : X\rightarrow Y$ be a proper flat morphism (of schemes) with connected fibers over a smooth projective curve $Y$ over $\mathbb{C}$. Let $X_{y_0}$ denote a smooth fiber over $y_0\in Y$. If $f$ ...
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2 votes
0 answers
161 views

rational curves over K3 surfaces over $\mathbb{Q}$

There are many partial results towards the following conjecture: Every projective K3 surface over an algebraically closed field contains infinitely many integral rational curves. My question is: is ...
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3 votes
1 answer
103 views

du Val singularities in Magma

Is there any way to decide whether a singularity of a surface embedded in $\mathbb{P}^5(\mathbb{Q})$ is a du Val/rational double point in Magma? Any help is much appreciated.
8 votes
3 answers
863 views

Seeking concrete examples of "generic" elliptic fibrations of K3 surfaces

For me a K3 surface will be a smooth complex projective variety of dimension 2 that is simply-connected and has trivial canonical bundle. Given a K3 surface $X$, an elliptic fibration $\pi \colon X \...
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6 votes
0 answers
140 views

Weyl group and Galois action on cubic surfaces

Let $X$ be a smooth cubic surface over a field $k$. Denote by $\bar{k}$ the separable closure of $k$ and $\bar{X}:=X\times_{k}\bar{k}$. Then it is well know that there exists a homomorphism $$ \phi:\...
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5 votes
1 answer
278 views

K3 surfaces with small Picard number and symmetry

I am looking for examples of K3 surfaces that have a low Picard rank and at least one holomorphic involution. Here, low is no mathematically precise concept. I want to do computations with Monad ...
1 vote
1 answer
187 views

Characterization of an Abelian surface

I have a smooth projective surface $X$, and two flat family of elliptic curves on it: $E_{1,t}$ and $E_{2,t}$, (I don't know what either $t$ runs through!) such that (1), for any i={1,2}, the closed ...
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2 votes
0 answers
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Rational points on surfaces

Let $k$ be a field of characteristic zero. In the affine space $\mathbb{A}_{x,y,t}^3$ consider a surface $S$ of the form $$ S = \{a_0(t)x^2+a_1(t)xy+a_2(t)x+a_3(t)y^2+a_4(t)y+a_5(t) = 0\} $$ where $...
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9 votes
1 answer
299 views

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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3 votes
1 answer
456 views

Cohomology of singular projective cubic surface

Let $X\subset \mathbb{P}_{\mathbb{C}}^3$ be a projective singular cubic surface with two singular points. Is the rationalcohomology of such objects known? As an example of the type of surfaces I'd be ...
3 votes
2 answers
275 views

Abelian varieties corresponding to Hodge substructures

In an exercise of Voisin book, says: Let $j:C\rightarrow S$ the inclusion of a smooth curve on a smooth connected projective surface. Set $H=ker(j_*:H^1(C,\mathbb{Z})\rightarrow H^3(S,\mathbb{Z}))$. ...
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3 votes
0 answers
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Articles of Casnati on algebraic varieties

I am attempting to track down online copies of the following two algebraic geometry articles. Is there some repository where these might be found? If necessary I could use the first few pages of each ...
7 votes
2 answers
312 views

Nef divisors on surfaces

Let $X$ be a smooth projective rational surface over an algebraically closed field of characteristic zero, and $D$ a divisor on $X$ such that $D$ is nef and $D^2 = 0$ with the following properties: $...
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4 votes
2 answers
182 views

Newton polygon notation for algebraic surface singularities

In various sources (e.g. here, Theorem 1.1 and here, Theorem 2.1 (3)), a certain notation which uses a fraction followed by a tuple is used to describe surface singularities. For example, the first ...
2 votes
1 answer
272 views

Behavior of canonical divisor under a finite group quotient

Given a smooth algebraic surface $X$, and a group $G$ acting on it and letting $Y := X / G$, how can we compute $K_Y^2$ from from $K_X^2$? Current progress: In Borisov and Fatighenti - New explicit ...
11 votes
2 answers
1k views

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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1 vote
0 answers
132 views

Rational classes of $(-2)$-curves in a minimal surface of general type

Let $X$ be a minimal surface of general type over $\mathbb{C}$. One can show that if for any set of $(-2)$-curves $C_1,\cdots,C_l$ on $X$, there exists $k$, $1\le k\le l$ such that $$\sum_{i=1}^k\...
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1 vote
1 answer
223 views

Pseudoeffective divisors on surfaces

Consider a minimal smooth conic bundle $S$ of dimension two. Assume that there are two curves $C,F$ on $S$ such that $C^2 < 0$ and $F^2 = 0$. Let $D$ be a pseudoeffective divisor on $S$ such that $...
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1 vote
0 answers
61 views

Elliptic fibrations on some Kummer surface in characteristic $2$

In the question I ask about one elliptic fibration on the surface $$ K\!: y^2 + x_1x_2y = (x_1x_2)^2(x_1 + x_2 + 1) + (x_1 + x_2)^2. $$ over a finite field $\mathbb{F}_q$ of characteristic $2$ such ...
3 votes
0 answers
137 views

Log canonical surface with an elliptic singularity

I would like to know if there is an example as follows: $X$ is a log canonical surface and $x \in X$ is an elliptic singularity such that The minimal resolution of $x$ is a circle of rational curves (...
2 votes
0 answers
101 views

Semi-stable sheaves on quadric surface

https://downloads.hindawi.com/journals/tswj/2014/346126.pdf In this paper, Stable sheaves on a smooth quadric surface with linear Hilbert bipolynomials(E. Ballico and S.Huh), I have a question. On the ...
2 votes
1 answer
277 views

When does the Hirzebruch surface have a nef anticanonical divisor?

Let $\mathcal H_r=\mathbb P (\mathcal O_{\mathbb P^1}\oplus \mathcal O_{\mathbb P^1}(r))$ be a Hirzebruch surface for some $r\in\mathbb Z$. As a toric variety, the fan structure is spanned by $(-1,0)$,...
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1 vote
0 answers
215 views

Rational and rationally chain connected surfaces

A projective variety $X$ over the complex numbers is rationally connected if two general points of $X$ can be joined by a rational curve in $X$, and rationally chain connected if two general points of ...
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