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.
390
questions
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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
$$
\phi:\...
5
votes
1answer
211 views
K3 surfaces with small Picard number and symmetry
I am looking for examples of K3 surfaces that have a low Picard rank and at least one holomorphic involution.
Here, low is no mathematically precise concept.
I want to do computations with Monad ...
1
vote
1answer
171 views
Characterization of an Abelian surface
I have a smooth projective surface $X$, and two flat family of elliptic curves on it: $E_{1,t}$ and $E_{2,t}$, (I don't know what either $t$ runs through!) such that
(1), for any i={1，2}, the closed ...
3
votes
0answers
218 views
Rational points on surfaces
Let $k$ be a field of characteristic zero. In the affine space $\mathbb{A}_{x,y,t}^3$ consider a surface $S$ of the form
$$
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 $...
0
votes
1answer
672 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 closed ...
13
votes
3answers
2k views
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$, ...
9
votes
1answer
257 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 ...
7
votes
2answers
267 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:
$...
5
votes
0answers
148 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 ...
2
votes
1answer
313 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
2answers
238 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}))$.
...
3
votes
0answers
177 views
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 ...
10
votes
2answers
869 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 ...
4
votes
2answers
162 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
1answer
182 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 ...
1
vote
0answers
122 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\...
2
votes
1answer
187 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 $...
1
vote
0answers
57 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
0answers
122 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
0answers
94 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
1answer
165 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 $(-1,0)$,...
0
votes
0answers
176 views
$0$-cycles and $0$-cycles of degree $0$
$\DeclareMathOperator\Rat{Rat}$Let $S$ be a surface, $\Rat_0(S)$ the set of $0$-cycles rationally equivalent to zero on $S$, and $\Rat_0^0(S)$ the set of $0$-cycles of degree $0$ rationally equivalent ...
2
votes
0answers
160 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 ...
3
votes
0answers
207 views
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
votes
2answers
326 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 ...
5
votes
1answer
267 views
Volume of a divisor on a smooth projective surface
Let $X$ be a smooth projective surface (over complex numbers). Let $D$ be a divisor on $X$. Then we know that its volume is defined as $$\text{vol}_X(D):= \lim \sup_{m \rightarrow \infty} \frac{h^0(X,...
2
votes
1answer
174 views
Is the positive part of the Zariski decomposition of a big $\mathbb{R}$-divisor big?
I can't understand why the positive part of the Zariski decomposition of a big class is itself big. More concretely:
let $X$ be a smooth projective surface over $\mathbb{C}$. Let $N^1_{\mathbb{R}}(X)$...
4
votes
1answer
167 views
Jacobians $\mathbb{F}_q$-isogenous to the direct square of an ordinary elliptic $\mathbb{F}_q$-curve of $j$-invariant $0$
Consider an ordinary elliptic curve $E_b\!: y^2 = x^3 + b$, of $j$-invariant $0$ over a finite field $\mathbb{F}_q$, such that $\sqrt{b} \not\in \mathbb{F}_q$.
Question. What are some examples of ...
3
votes
0answers
134 views
Singular del Pezzo surfaces and degeneration of root systems
Let $S$ be a smooth del Pezzo surface of degree $d$ and $K_S^*$ the anticanonical class. It is well known that the set of classes
$$R(S)=\{\alpha\in H^2(S,\mathbb Z)|\alpha^2=-2,\alpha\cdot K_S^*=0\},$...
2
votes
0answers
164 views
description of very ample bundle of Hirzebruch surface
I learned some basic properties of Hirzebruch surface mainly from Vakil's notes "the rising sea", section 20.2.9. the Hirzebruch surface is defined as $\mathbb{F}_n:=\operatorname{Proj} (\...
1
vote
0answers
123 views
Surfaces of general type with $h^1(-K_X)\neq 0$
By a result of Ekedahl, in characteristic 2 one may have minimal surfaces of general type such that $h^1(X,-K_X)\neq 0$ and $X$ is birational to an inseparable double cover of a rational surface. How ...
3
votes
0answers
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"Simplification" of the map constructed at the proof of Castelnuovo's contractibility theorem
I'm reading the proof of the Castelnuovo's contractibility criterion in Beauville's book(Theorem II.17), and I guess I could understand all its affirmations. But I still has one question.
For those ...
0
votes
1answer
39 views
Intersection of the tautological bundle with a fiber of a geometrically ruled surface
I'm reading Beauville's book, Complex Algebraic Surfaces, and I'm trying to understand an affirmation in a proposition that characterizes the Picard group of a geometrically ruled surface.
First, let $...
9
votes
0answers
211 views
How many characteristics is a random surface unirational in?
Suppose I have a surface $X$ defined over $\mathbb{Z}$. I am interested in the set $S_X$ of primes $p$ such that $X_{\overline{\mathbb{F}}_p}$ is unirational. If I choose a "random" surface ...
7
votes
1answer
343 views
General conditions for normality of blow-up
Let $X$ be an integral, affine, normal complex surface. I am looking for conditions on zero-dimensional closed subschemes $Z$ in $X$ such that the reduced scheme associated to the blow-up of $X$ along ...
8
votes
3answers
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Cone of curves and Mori theorem for algebraic surfaces
In describing part of the geometry of the cone of curves for an algebraic surface $S$, we need to find $(-1)$ curves within $S$. Once we've done that, then we can say that the "negative" ...
3
votes
0answers
231 views
Is the surface in $\mathbb{A}^3$ rational?
Consider the surface
$$
(u_1^6 + 1)w^3 = (u_2^6 + 1) \subset \mathbb{A}^3
$$
over an algebraically closed field of characteristic $p \neq 2,3$. Is it rational, i.e., is there its proper ...
5
votes
2answers
185 views
Surface of type $(2,2)$ on the Segre cubic scroll $\mathbb{P}^1 \times \mathbb{P}^2 \subset \mathbb{P}^5$
Let $S=\mathbb{P}^1 \times \mathbb{P}^2 \subset \mathbb{P}^5$ embedded with the Segre embedding given by $\mathcal{O}_S(1,1)$.
If we intersect $S$ with a general smooth quadric $Q \subset \mathbb{P}^5$...
4
votes
1answer
128 views
Existence of perfect Morse functions on Fermat surfaces $x^n+y^n+z^n+w^n=0$
It seems that whether a simply connected 4 manifold needs 1-handles and 3-handles is still an open question, see Existence of Morse functions on simply connected manifolds.
I am wondering if it is ...
5
votes
2answers
2k views
Generalisations of Riemann-Roch for surfaces
Let $X$ be a smooth projective algebraic surface (over $\mathbb{C}$ ). For all $L\in \mathrm{Pic}(X)$, we have
$$\chi(L)=\chi(\mathcal{O}_X)+\frac{1}{2}(L^2-L\cdot \omega_X).$$
This is the famous ...
6
votes
0answers
228 views
nonvanishing higher cohomology of a very ample divisor
I am looking for smooth projective varieties $X$, with $h^i(X, \mathcal{O}_X) = 0$ for $i > 0$, with a very ample line bundle $L$ with some nonvanishing higher cohomology.
What is clear:
(1) Curves ...
13
votes
1answer
299 views
Are any of these complex surfaces ever projective?
Let $C$ and $T$ be compact connected Riemann surfaces (or: smooth projective connected curves over $\mathbb{C}$) of genus at least two and let $X:=C\times T$. Let $(c,t)$ be a point of $X$, and let $...
4
votes
0answers
126 views
Reducible surface as a degeneration
I am interested in the following situation. If $S_1\cup_D S_2$ is a union of two irreducible smooth projective surfaces over $k=\overline{k}$(over $k=\mathbb{C}$ is enough, if it's relevant) glued ...
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vote
0answers
183 views
Minimal model vs canonical model of a surface
When I have a projective surface $X$, for simplicity smooth, I can find a simpler smooth surface on its binational class. In this way we find in a finite number of steps the simplest surface $Y$, i.e. ...
6
votes
2answers
393 views
Smooth projective surface with geometrically integral reduction
Let $S$ be a geometrically connected smooth projective surface over $\mathbb{Q}_p$. Can it be put in a proper flat $\mathbb{Z}_p$-scheme with a geometrically integral special fiber?
2
votes
0answers
80 views
arithmetic del Pezzo surfaces in comparison with del Pezzo surfaces over a field
A del Pezzo surface is a smooth, 2-dimensional projective variety $X$ with ample anticanonical divisor, i.e. a 2-dimensional Fano variety.
I am interested in the arithmetic analogue, a 2-dimensional ...
1
vote
1answer
203 views
Non-isotrival fiber bundle over compact Riemann surface
In this paper, Kodaira constructed a fiber bundle $\Phi:M_{m,n}\to S$ from a compact complex surface $M_{m,n}$ to a compact Rieman surface $S$ of genus $>0$. In particular, (on p.212) for any point ...
10
votes
1answer
402 views
A diffeomorphism of complex surfaces mapping subvarieties to subvarieties
Let $X$ and $Y$ be smooth projective complex surfaces. If a diffeomorphism from $X$ to $Y$ maps subvarieties to subvarieties does it have to be holomorphic or antiholomorphic? Can we at least verify ...
1
vote
1answer
196 views
On a quadratic diophantine equation
Given a quadratic diophantine equation in $\mathbb Z[x,y,z]$ of form
$$ax^2+by^2+cx+dy+ez+f=0$$ are there standard methods to solve for it when $$\|(x^2,y^2,z)\|_\infty\leq e^{1/2}$$
$$\|(a,b,c,d,e,f)...
5
votes
0answers
214 views
Deformations of a blow up
My question is related to this question, but I'm looking for something a bit more explicit.
Let $S$ be a smooth surface over $\mathbb C$, fix a point $s\in S$ and take the blow up $\beta \colon S' \...
