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
3
votes
0
answers
398
views
What is the most useful rationality criterion of surfaces?
The motivation for this question is that I would like to extract some information from derived category of surfaces to conclude the rationality of surface. There is a well known rationality criterion ...
- 1,982
4
votes
0
answers
198
views
Can Kummer surfaces coming from the same abelian surface be Cremona equivalent / isomorphic?
Assume we are given a simple abelian surface $A$ which has 2 non-equivalent principal polarizations $D_1$ and $D_2$ in $NS(A)$ (up to isomorphism), thus giving rise to two non-isomorphic smooth ...
- 1,025
4
votes
0
answers
217
views
Finding Rational Curves on a Surface
Let the field of rational numbers be our base field $k$. I hope to find all rational curves on the following surface $S$ defined by $f$. You can find the motivation in the end.
$f= (x^2y^2)z^3 + (5x^...
- 417
3
votes
1
answer
203
views
"Direct" calculation of $K_0$ for surfaces, 3-folds
I apologize in advance for the vagueness of my question, but I am looking for sources (if they exist) where $K_0$ (the Grothendieck group of coherent sheaves) is computed "by hand" for some low-...
- 528
3
votes
1
answer
177
views
Intersection graph of $(-1)$-class divisors on surface of general type
Let $X$ be a rational surface, say $X$ is del Pezzo surface. Let $D$ be $(-1)$-class divisor, i.e: $D^2=-1$ and $D^2+D.K_X=-2$. It is easy to show that on del Pezzo surface any $(-1)$-class divisor is ...
- 1,982
5
votes
0
answers
326
views
Max Noether's theorem for algebraic surfaces
The well-known Max Noether's theorem for curves [see Arbarello-Cornalba-Griffiths-Harris Geometry of Algebraic Curves vol.1, p. 117] states that, if a smooth curve $C$ is non-hyperelliptic, then the ...
- 66.3k
4
votes
1
answer
164
views
Hyperplane section through normal surface singularity
Let $(X,p)\subset \mathbb{C}^N$ be the germ of a normal surface singularity, is it true that for a general hyperplane section $H$ passing through $p$ the curve $X∩H$ does not have an embedded ...
- 405
3
votes
1
answer
382
views
Surface in $\mathbb{P}^N$ covered by rational normal curves
Assume that for some $n \in \mathbb{N}$ I have a (possibly singular) irreducible, non-degenerate complex surface $X_n \subset \mathbb{P}^{N}$ with the following properties:
for all $p \in X_n$ there ...
- 66.3k
6
votes
1
answer
535
views
Algebraic surfaces with no deformations
Is very well known that the only algebraic curve which admits no deformations is the projective line.
Q. What are "rigid" smooth algebraic surfaces? Is there a sensible classification?
- 6,891
2
votes
0
answers
196
views
How to show that $X$ is covered by rational curves?
I am reading a paper, there is a statement like this:
Maybe not a good question for MO.
Let $X$ be a smooth projective surface, let $|D|$ be a linear system having moving component $M$. Assume that ...
- 1,982
9
votes
1
answer
513
views
Pull-back of an irreducible ample divisor via an isogeny of abelian varieties
In the (wonderful) book by C. Birkenhake and H. Lange Complex Abelian Varieties we can find the following result, see Corollary 4.3.4 page 77. It is stated in any dimension $g \geq 2$, but let us ...
- 66.3k
3
votes
0
answers
287
views
How much information is encoded in the Jacobian-Kummer K3 surface of a curve of genus two?
Assume we work over $\mathbb{C}$.
Let $S\subset \mathbb{P}^3$ be a quartic surfaces with 16 nodes (ordinary double points). Then there is a simple principally polarized abelian surface $(A,\theta)$ ...
- 1,025
3
votes
1
answer
481
views
geometric genus of curves and generically finite morphism of surfaces
For a generically finite morphism $f:X\rightarrow Y$ of smooth projective surfaces over $\mathbb{C}$. Fix any integer $g$, denote $\mathcal{A}$ to be the set of smooth irreducible curve $C$ in $Y$ ...
- 1,081
2
votes
2
answers
232
views
Effectivity of $(-1)$-class on smooth projective surface
Let $X$ be a smooth projective surface such that $\chi(O_X)=1$ and $K_X^2\geq 3$(or just $K_X^2>0$). Let $D$ be $(-1)$-class, i.e: $D^2=-1,D^2+D.K_X=-2$(equivalently, $D^2=-1, \chi(-D)=0$). I ...
- 1,982
5
votes
0
answers
254
views
How to prove a statement on weak del pezzo surface?
Let $X$ be a weak del pezzo surface($-K_X$ is nef and $K_X^2>0$) with $3\leq K_X^2\leq 5$. Let $H$ be a $1$-class on $X$ such that $H.C\geq 1$ for any $C\in I^{irr}(X)$. Then $2H+K_X$ is effective ...
- 1,982
2
votes
1
answer
714
views
Castelnuovo and Artin contractibility criteria for families
In the course of a proof, I need some version of Castelnuovo and Artin's contractibility criteria for a family of surfaces. Say I have a family (flat is probably needed, in order to compare ...
- 625
4
votes
1
answer
735
views
Infinitely many exceptional curves on ruled surfaces
Take an algebraically closed field $k$. Let $C$ be a smooth projective variety of dimension $1$ over $k$(a curve). Consider a geometrically ruled surface $S :=C\times\mathbb{P}^1$. Does there exists a ...
- 215
6
votes
0
answers
168
views
How does the "todd class operator" commute with Nakajima's q operators on Hilbert schemes of points on surfaces
Let $S$ be a surface, and $S^{[n]}$ the Hilbert scheme of $n$ points on $S$. One defines
$$
\mathbb H = \bigoplus_n H^*(S^{[n]}).
$$
One has Nakajima's operators $\mathfrak q_n(\alpha)$ (with $\alpha \...
- 1,509
2
votes
1
answer
271
views
Does rational surface have exceptional collection of maximal length but not full?
Let $X$ be a rational surface. Let $\mathbb{E}:=(E_1,\ldots,E_n)$ be strong exceptional collection of line bundles of maximal length $l=rk Pic(X)+2$ in $D^b(coh(X))$, Is there any example that such ...
- 1,982
8
votes
1
answer
318
views
rational effective implies effective?
Let $X$ be a weak del pezzo surface, I Wonder whether the following statment is true:
Let $L$ be a line bundle on $X$, then $h^0(L)=0$ implies $h^0(nL)=0$ for all $n\geq 1$.
- 1,982
6
votes
1
answer
342
views
What is Mumford's example of a normal complex algebraic surface $X$ with non-torsion elements in $H^2_{et}(X,\mathbb{G}_m)$?
I have heard that Mumford has constructed an example of a normal complex algebraic surface $X$ such that $H^2_{et}(X,\mathbb{G}_m)$ contains a non-torsion element. But I cannot find the reference.
...
- 9,019
4
votes
0
answers
172
views
How close is $h^0(mD)$ to be a polynomial?
Let $X$ be a normal (or smooth if it helps) projective variety over an algebraically closed field $k$. Fix a Cartier divisor $D$: I am interested in knowing how $h^0(mD)$ behaves as $m$ varies.
At ...
- 625
7
votes
2
answers
557
views
Algebraic Surfaces
Is there any example of a smooth, projective surface $S$ over $\mathbb{C}$, with Picard group $\mathbb{Z}$ and such that $H^1(S, L)$ is not zero for some ample line bundle $L$ ?
3
votes
1
answer
211
views
Existence of a universal $0$-cycle
For a smooth (complex) projective surface $S$ with non trivial Albanese variety, are there simple conditions for the existence a universal codimension $2$ cycle i.e. a $Z\in \mathrm{CH}^2(Alb(S)\times ...
- 1,463
2
votes
1
answer
259
views
Isogeny from kernel in higher dimensional abelian varieties
Is there any kind of generalization of Vélu formulae for Jacobians?
The question technically is:
Given a hyperelliptic curve $H/\overline{\mathbb{F}}_q$ and its jacobian $J_H$ of dimension $2$, if $...
1
vote
0
answers
59
views
$N$ general points and family of curves forming a linear system of dimension $\geq N$
Let $N \in \mathbf{Z}$. Suppose $S$ be a projective surface and $C \subset S$ be a curve on $S$. Assume that the linear system $|C|$ has dimension $\geq N$. Suppose we are given $N$ general points on $...
- 11
3
votes
0
answers
255
views
Intuition behind results in Mumford's “Lectures on curves on an algebraic surface”, II
NOTE: This is a followup to my question here.
These are some questions concerning Mumford's "Lectures on curves on an algebraic surface".
We concern ourselves with questions of the Picard variety $P$...
user97565
38
votes
1
answer
2k
views
When do 27 lines lie on a cubic surface?
Consider $27$ (pairwise distinct!) lines in $\mathbb{P}^3$ whose intersection graph is that expected¹ of the $27$ lines on a smooth cubic surface. Question: Is there a simple necessary and sufficient ...
- 32.5k
6
votes
1
answer
1k
views
Intuition behind results in Mumford's "Lectures on curves on an algebraic surface", I
These are some questions concerning Mumford's "Lectures on curves on an algebraic surface".
We concern ourselves with questions of the Picard variety $P$, and its dimension, of a complete nonsingular ...
user97565
2
votes
1
answer
216
views
Principal elliptic bundles over curve with Kähler total space
I wonder what could be a Kähler surface, which is a total space of principal elliptic bundle over a curve. I believe that there is a classification and that it must be pretty simple, but I cannot find ...
- 1,170
6
votes
1
answer
646
views
Additivity of Kodaira dimension for a nice fibration
Consider a surjective holomorphic map between two complex projective manifolds $\pi :X \rightarrow Y$. Iitaka conjectured the subadditivity of Kodaira dimensions: $\kappa(X)\geqslant\kappa(Y)+\kappa(...
user60772
4
votes
1
answer
255
views
Are pseudo-isomorphism between normal surfaces isomorphisms?
Let $X,Y$ be two normal algebraic surfaces (for instance projective) and let $\varphi\colon X\dashrightarrow Y$ be a birational map which restricts to an isomorphism $(X\setminus F)\to (Y\setminus G)$ ...
- 7,722
1
vote
0
answers
125
views
Torsion $0$-cycle as difference of two points
If $S$ is an abelian surface over $\mathbb C$, we can always find points $x,y\in S$ such that $x-y\neq 0$ but $2x-2y=0$ in $\mathrm{CH}_0(S)$. Hence my question: for any surface $S$ (or for which ...
- 1,463
27
votes
1
answer
1k
views
Analogies between classical geometry on complex surfaces and Arakelov geometry
This is my first question on this wonderful site. The following question about Arakelov geometry is gonna be quite long and wide; to be clear one of that kind of questions that are usually ignored. ...
- 529
2
votes
0
answers
200
views
Top intersections on the Hilbert scheme of points on a surface
The Picard group of $S^{[n]}$ is generated by the Picard group of $S$ (via a map $L \mapsto L_n$) and $E$, where $E = -\frac{B}{2}$, where $B$ is the exceptional divisor of the Hilbert Chow morphism.
...
- 1,509
4
votes
1
answer
322
views
Albanese of Siegel modular variety $\mathcal{A}_2$
Jacobian varieties of Shimura curves are very interesting objects. For one thing they provide a geometric relation between elliptic curves and modular forms of weight 2 (say we are over $\mathbb{Q}$). ...
- 845
6
votes
1
answer
1k
views
Smoothness of the branch divisor and ramification on surfaces
Let $f \colon X \longrightarrow Y$ be a finite morphism of degree $n \geq 2$ between smooth compact, complex surfaces.
Let $B \subset Y$ be the branch divisor of $f$ and assume that the ...
- 66.3k
2
votes
0
answers
185
views
How can I find a basis of $H^0(Y, -K_Y)$ for the del Pezzo surface $Y$ of degree 5?
Consider the surface $X \subset \mathbb{P}^2_{x, y, z}\times\mathbb{P}^1_{t, s}$:
$$
s^3y^2 + t^3yz = (t+s)x^2 + tsxz + t(t^2 +s^2)z^2
$$
over a finite field $k = \mathbb{F}_{2^d}$, $\mathrm{gcd}(d, 6)...
- 1,833
4
votes
0
answers
173
views
Are there methods to compute the induced action of Frobenius map on the Neron-Severi group of a supersingular abelian surface over a finite field?
Let $A$ be a supersingular abelian surface over a finite field $\mathbb{F}_q$. In that case the Neron-Severi group $NS(A\otimes\overline{\mathbb{F}_q})$ is the lattice of rank $6$. Are there methods ...
- 1,833
1
vote
1
answer
493
views
Fundamental group of Log del Pezzo surfaces
A Log del Pezzo surface is a normal complex surface with ample anti-canonical bundle and at worst quotient singularities.
It is known that such surfaces are rational. This is proven, for example, ...
- 14.3k
3
votes
1
answer
414
views
Modern reference for the theory of correspondences for curves
The classic theory of correspondences between smooth algebraic curves can be found in André Weil's Foundations of algebraic geometry. However, this reference works in a pre-modern algebraic geometry ...
- 2,871
1
vote
1
answer
269
views
The definition of the face of a convex set by a nonnegative affine linear polynomial
My question comes from the paper: https://arxiv.org/abs/0911.2750 (p.2~p.3)
For $n\in \mathbf{N}$
Let $X = (X_1,\ldots,X_n)$ be an $n$-tuple of variables.
Let $\mathbf{R}[X]$ denote the ...
- 345
2
votes
2
answers
321
views
What are sufficient and necessary conditions to be a generalized Zariski surface over a finite field?
Let $X$ be an absolutely irreducible reduced surface over a finite field $k$ of characteristic $p$. What are sufficient and necessary conditions for $X$ to be a generalized Zariski surface over $k$ (...
- 1,833
2
votes
0
answers
147
views
Purely inseparable $k$-rational dominant maps between an absolutely irreducible $k$-surface and $\mathbb{P}^2$
Let $X$ be an absolutely irreducible surface over an algebraically closed or finite field $k$ of characteristic $p$. Is it true that the following conditions are equivalent?
There is a purely ...
- 1,833
2
votes
1
answer
229
views
Is there a regular surjective map $\psi\!: \mathbb{P}^2 \to X$ over $k$?
Assume there is $\varphi\!: \mathbb{P}^2 \to X$, a purely inseparable rational dominant map over a finite field $k$, where $X$ is an absolutely irreducible smooth surface over $k$. Is there a regular ...
- 1,833
2
votes
0
answers
137
views
Is a supersingular Kummer surface $k$-unirational in characteristic 2?
Let $k$ be a perfect field of even characteristic. Consider the simplest example of a supersingular genus 2 curve, i.e.,
$$
C\!: y^2 + y = x^5.
$$
By the article of J. S. Müller "Explicit Kummer ...
- 1,833
5
votes
1
answer
461
views
Homeomorphism between del Pezzo surfaces
Let $X$ and $Y$ be smooth del Pezzo surfaces of the same degree $K_X^2=K_Y^2$.
Are the sets $X(\mathbb{C})$ and $Y(\mathbb{C})$ homeomorphic, or at least homotopy equivalent?
- 53
11
votes
0
answers
310
views
Surfaces with $q=2$ and generically finite Albanese map
I have a family of surfaces of general type $S$ with $q(S)=2$, and such that the Albanese map $$\alpha \colon S \longrightarrow A:=\mathrm{Alb}(S)$$ is generically finite of degree $n$. By a result of ...
- 66.3k
1
vote
1
answer
659
views
Birational morphism and invariance of arithmetic genus
Let $f:X \to Y$ be a birational morphism between projective, irreducible surfaces. Assume $X$ is non-singular and $Y$ is a hypersurface in $\mathbb{P}^3$ (not necessarily smooth). Is the arithmetic ...
- 1,981
0
votes
0
answers
265
views
Explicit adjunction formula and local top form
I am reading the section 4.2 in Kollar-Mori, where they provide the explicit equations for Du Val Singularities. In the course of the proof, they reduce to studying the equation $x^2+f(x,y)=0$ in a ...
- 625