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.
468 questions
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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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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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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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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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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 $X$...
- 66.3k
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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 - ...
- 521
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0
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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 $\Omega_X$ ...
- 66.3k
7
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1
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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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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{...
- 66.3k
1
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0
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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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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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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 $\...
- 66.3k
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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 \...
- 1,841
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0
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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 ...
- 1,841
2
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0
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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)\...
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2
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196
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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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0
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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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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 ...
- 1,841
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0
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182
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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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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}\...
- 1,841
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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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1
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1
answer
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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 ...
- 8,998
3
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1
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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.
...
- 8,998
4
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1
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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 ...
- 1,833
7
votes
1
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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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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
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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.
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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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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
$$
\phi:\...
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5
votes
1
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K3 surfaces with small Picard number and symmetry
I am looking for examples of K3 surfaces that have a low Picard rank and at least one holomorphic involution.
Here, low is no mathematically precise concept.
I want to do computations with Monad ...
- 577
1
vote
1
answer
241
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Characterization of an Abelian surface
I have a smooth projective surface $X$, and two flat family of elliptic curves on it: $E_{1,t}$ and $E_{2,t}$, (I don't know what either $t$ runs through!) such that
(1), for any i={1，2}, the closed ...
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0
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Rational points on surfaces
Let $k$ be a field of characteristic zero. In the affine space $\mathbb{A}_{x,y,t}^3$ consider a surface $S$ of the form
$$
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 $...
user168611
9
votes
1
answer
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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 ...
- 9,019
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votes
1
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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 ...
- 1,061
3
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2
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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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5
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1
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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 ...
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6
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2
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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:
$...
- 8,998
4
votes
2
answers
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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 ...
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1
answer
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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 ...
- 29
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2
answers
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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 ...
- 251
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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
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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 $...
- 8,998
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vote
0
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76
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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 ...
- 1,833
3
votes
0
answers
176
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 (...
- 393
2
votes
0
answers
113
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 ...
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votes
1
answer
520
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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
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0
answers
341
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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 ...
user91659
3
votes
0
answers
221
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Historical proof of Leschetz Hyperplane Theorem
I browse in Phillip Griffiths' Slides
on historical development of
Hodge-theory and these include a sketch of the original approach
with Lefschetz used to study complex surfaces in his famous
...
- 6,028
6
votes
2
answers
445
views
Representability of flat cohomology by a group scheme
In his paper "Supersingular K3 surfaces", Artin states the following theorem (Theorem 3.1) without proof:
Let $\pi:X \to S = \mathrm{Spec}(k)$ be a smooth proper surface with $k$ an ...
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