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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Stable torsion free sheaf on smooth projective surface
Let $E$ be a torsion-free sheaf on a smooth projective variety $X$ over $\mathbb{C}$. Let $H$ be an ample line bundle on $X$. Then we say $E$ is stable if $\mu_{H}(F)<\mu_{H}(E),\,\forall 0\neq F \...
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0
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Curves in a non-normal surface
We are studying the behavior of families of curves inside stable families of surfaces. The non-existance of the following configurations of curves in a non-normal surface would be sufficient to prove ...
2
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1
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Where to find "Families of curves on a surface of general type" (MR0457450)?
I am currently doing some research on surfaces of general type and I need some results from Bogomolov's paper:
Bogomolov, F. A.
Families of curves on a surface of general type.
Dokl. Akad. Nauk SSSR ...
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0
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The specific elliptic fibration on the Kummer surface of the superspecial abelian surface
Consider two copies $E_1$, $E_2$ of the supersingular elliptic curve
$$
y^2 = x^3 - 1\qquad (y^2 = x^4 - 1)
$$
over a finite field $\mathbb{F}_{p^2}$ of odd characteristics $p$ such that
$$p
\...
2
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0
answers
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The quotient of a superspecial abelian surface by the involution
Let $E_i\!: y_i^2 = f(x_i)$ be two copies of a supersingular elliptic curve over a field of odd characteristics. Consider the involution
$$
i\!: E_1\times E_2 \to E_1\times E_2,\qquad (x_1, y_1, x_2, ...
2
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Elliptic fibrations on the Fermat quartic surface
Consider the Fermat quartic surface
$$
x^4 + y^4 + z^4 + t^4 = 0
$$
over an algebraically closed field $k$ of characteristics $p$, where $p \equiv 3$ ($\mathrm{mod}$ $4$).
Is there the full ...
2
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0
answers
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Is the Fermat quartic surface a generalized Zariski surface?
Consider the Fermat quartic surface $$F\!: x^4 + y^4 + z^4 + t^4 = 0$$ over an algebraically closed field $k$ of odd characterstics $p$. Shioda proved that for $p=3$ this surface is a generalized ...
3
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0
answers
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Under what conditions are superspecial abelian surfaces isomorphic over a finite field?
Let $E_1$, $E_2$, $E_3$, $E_4$ be supersingular elliptic curves over a finite field $\mathbb{F}_{p^2}$, where $p$ is an odd prime. There is a well known theorem stating that over the algebraic closure ...
2
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2
answers
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Projective surfaces with vanishing first cohomology
Let $X$ be a smooth, projective surface in $\mathbb{P}^3$ and $D$ an effective divisor in $X$. Is it true that $H^1(\mathcal{O}_{X\backslash D})=0$ (we know that $H^1(\mathcal{O}_X)=0$)? If not true ...
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A question on surfaces in $\mathbb{P}^4$
On surfaces in $\mathbb P^4$,Ellingsrud and Peskine has proved that
There exists an integer $d_0$ such that for any integer $d>d_0$,any smooth surface of degree $d$ in $\mathbb P^4$ is of ...
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1
answer
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Is there a purely inseparable covering $\mathbb{A}^2 \to K$ of a Kummer surface $K$ over $\mathbb{F}_{p^2}$?
Let $E_i\!: y_i^2 = x_i^3 + a_4x_i + a_6$ be two copies ($i = 1$, $2$) of a supersingular elliptic curve over a finite field $\mathbb{F}_{p^2}$, for odd prime $p > 3$. Consider the Kummer surface $...
5
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1
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surface with rational curve in the double locus
I am interested in the existence of a surface $X$ over $\mathbb{C}$ with the following properties (or a reason for why one cannot exist):
$X$ is slc (and not-normal)
There is rational curve $C \...
2
votes
1
answer
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Explicit families of elliptic curves
I am interested in finding families $X\to C$ of elliptic curves over a projective curve $C$. The fibers are not all isomorphic, not all smooth, and the singular fibers should be nodal stable curves. ...
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0
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If there is no more than $k$ smooth rational curves on algebraic surface, what is the minimal value of k
Let $X$ be a smooth surface of general type with $q=p_g=0$, define a set $A=\{C\subset X|$ C is smooth rational and $C^2<-1\}$, let $k=|A|$ which is cardinality of $A$.
What is the minimal value ...
5
votes
1
answer
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Curves in del Pezzo surfaces satisfying certain intersection inequality
Let $X$ be a del Pezzo surface (over $\mathbb{C}$), which is obtained by a blow up $\pi: X \rightarrow \mathbb{P}^{2}$ in a collection of points. Let $H$ be the hyperplane class of $\mathbb{P}^{2}$.
...
1
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0
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202
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on bilinear form on free abelian group
Let $P$ be a free abelian group of rank $n-2$ with an integral symmetric bilinear form $\left<,\right>$. A sequence of elements $A_1,\ldots,A_n$ in $P$ is called an abstract toric system iff it ...
3
votes
1
answer
236
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Linear homogenous polynomials that generates several quadratic polynomials
This is a generalization of this question.
Let $P_1, \ldots, P_m$, $Q_1, \ldots, Q_k \in \mathbb{C}[x_0,\ldots,x_n]$ be linear homogenous polynomials. Let $f_1, \ldots, f_s$ be a homogenous ...
2
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3
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Linear homogenous polynomials that generates one quadratic polynomial
Let $P_1, \ldots, P_m$, $Q_1, \ldots, Q_k \in \mathbb{C}[x_0,\ldots,x_n]$ be linear homogenous polynomials. Let $f$ be a homogenous quadratic polynomial of degree $2$.
Assume that for every $i$ and ...
1
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0
answers
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Section ring $R(X,D)$ of $D$ is finitely generated if $\kappa (X,D) \leq 1$
In the remark on the bottom of page 5 of this paper, the author states that
It is well-known fact that the algebra of a Divisor $D$ with $\kappa (X,D) \leq 1$ is finitely generated over $k$. In ...
5
votes
0
answers
230
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Map associated to linear system onto curve is morphism
In Mumford's first paper on Surfaces in char $p$ [1], part 2 Step (II), he wants to show that, given an indecomposable curve of canonical type $D$ on a smooth projective surface $F$ with $p_g(F)=0, ...
6
votes
1
answer
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Resolution of Gorenstein rational singularities on a surface
I am reading Artin's notes "Lipman's Proof of Resolution of Singularities for Surfaces" from the book "Arithmetic Geometry". I am very confused by the proof of Lemma $6.5.$ (I am formulating it below ...
2
votes
0
answers
351
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Number of rational points of a singular cubic surface over a finite field
I have a (geometrically irreducible) cubic surface defined over a finite field $F_q$ with three non-$F_q$-rational singularities (defined over the cubic extension of $F_q$).
Counting the number of $...
3
votes
0
answers
264
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Are unirational K3 surfaces defined over finite fields?
Is every supersingular (thus unirational for ${\rm char }\ k = p\geq 5$, from Liedtke) $K3$ surface defined over a finite field? I guess this is true for Kummer surfaces, for example, since ...
1
vote
1
answer
2k
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Intersection number of divisors with its pull back and its push forward
I am in an ideal situation but I would appreciate a hint. First here is the scenario.
Let $\mathcal{J}$ be an the abelian variety obtained from the Jacobian of a genus $2$ curve $\mathcal{H}/k$ ...
3
votes
0
answers
385
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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 ...
4
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0
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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 ...
4
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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^...
3
votes
1
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"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-...
3
votes
1
answer
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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 ...
5
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0
answers
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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 ...
4
votes
1
answer
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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 ...
3
votes
1
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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 ...
6
votes
1
answer
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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?
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0
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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 ...
9
votes
1
answer
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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 ...
3
votes
0
answers
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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 $(A,\theta)$ ...
3
votes
1
answer
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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 $Y$ ...
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2
answers
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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 ...
5
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0
answers
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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 ...
2
votes
1
answer
655
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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 ...
4
votes
1
answer
636
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 ...
6
votes
0
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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 \...
2
votes
1
answer
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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 ...
8
votes
1
answer
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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 $n\geq 1$.
6
votes
1
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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.
...
4
votes
0
answers
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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 ...
7
votes
2
answers
526
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 $L$ ?
2
votes
0
answers
111
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
vote
1
answer
236
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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
57
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 $...