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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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Quotient of a smooth projective surface by an involution

Is the quotient of a smooth complex projective surface by an involution projective? Suppose the quotient happens to be smooth; does that change the situation?
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Are there three ordinary elliptic curves $E$, $E_1$, $E_2$ such that $E^2 \cong E_1 \!\times\! E_2$?

Consider the elliptic curve $E\!: y^2 = x^3 + 1$ of $j$-invariant $0$ over an algebraically closed field $k$ of characteristics $p$. Let me remind that $E$ is ordinary (i.e., non-supersingular) iff $p ...
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2answers
324 views

Show Fiber Product of Rational Elliptic Surfaces is Calabi-Yau

In a handful of contexts people study Calabi-Yau threefolds formed by taking the fiber product of two rational elliptic surfaces. I can't find any detailed explanation of why such geometries are ...
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Is the generalized Kummer threefold rational in characteristics 3?

Let $E_i\!: y_i^2 = x_i^3 - x_i$, $i = 1, 2, 3$ be three copies of the supersingular elliptic curve in characteristics $3$. Consider on $E_i$ the following automorphism of order $3$: $$ \sigma(x_i,...
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1answer
80 views

Existence of meromorphic 2-forms over normal surface singularities

Let $(X,o)$ be an isolated normal surface singularity. Denote by $U:=X\backslash \{o\}$. I am looking for conditions on $(X,o)$ under which there exists a holomorphic section $\omega \in H^0(U, \Omega^...
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1answer
356 views

Classification of smooth algebraic surfaces with a smooth morphism to $\Bbb P^1$

Let $k$ be an algebraically closed field, it is well known that $\mathbb P^1$ is simply connected, but how about smooth projective surfaces $X$ with a smooth morphism to $\Bbb P^1$? Except the case $...
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1answer
53 views

Minimal complex surfaces with pseudo-effective canonical bundles

A complex line bundle $L$ over a complex surface $X$ is said to be pseudo-effective if it admits a (possibly singular) Hermitian metric $h$ whose curvature is positive semi-definite in the sense of ...
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1answer
263 views

Breaking a morphism with generic fiber $\mathbb{F}_n$

Assume we are working over $\mathbb{C}$, and we have a projective morphism with connected fibers $f: X \rightarrow Z$ whose geometric generic fiber $X_\overline{\eta}$ is isomorphic to a Hirzebruch ...
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1answer
332 views

Intersection form in Algebraic Geometry/Topology

Let $S$ be a smooth complex projective surface. We let define an intersection form $(-)\cdot(-)$ on $\mathsf{Pic}(S)$ by setting $$D\cdot D':=\mathcal{O}_S(D)\cdot\mathcal{O}_S(D')$$ where the ...
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68 views

Thom-type isomorphism on sheaf cohomology

Let $X$ be a smooth, projective surface and $T$ a finite set of points in $X$ i.e., of codimension $2$ in $X$. Is it true that $H^i(\mathcal{O}_X)=H^i(\mathcal{O}_{X\backslash T})$ for $i \ge 1$?
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Very ample divisors on blow ups of the projective plane

Let $X$ be $\mathbb{P}^2$ blown up at $k$ points in general position. The Picard group of $X$ is just $\mathbb{Z}^{k+1}$ and one knows the intersection product explicitly. If $D$ is an ample divisor, ...
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2answers
161 views

Singularities of a central fibre of a flat family of smooth surfaces

Suppose I have a one parameter flat family of complex surfaces (regular, of general type) whose general fibre is smooth. Is it possible for the central fibre to have singularities which are not ...
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Fiber product of an elliptic surface

Let $f:S\to P^1$ be a smooth elliptic surface and let $X=S\times_{P^1} S$ be the fiber product. The threefold $X$ is singular in general (typically isolated ODPs). But is $X$ $\mathbb Q$-factorial? Or,...
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Can the base of an elliptically fibered Calabi-Yau threefold be an Enriques surface?

For this question, a Calabi-Yau manifold or variety of dimension $n$ is defined as a non-singular projective variety with trivial canonical bundle and $h^{i,0} = 0$ unless $i = 0$ or $i = n$. If ...
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148 views

Curves on rational surfaces and Lang's conjecture for M_g

There are a group of related conjectures associated to Lang's name - for this question I'll consider only the weakest one, namely that rational curves in a projective variety of general type are not ...
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106 views

Pushforward of structure sheaf on quotient surface singularity

Let $f:\mathbb{C}^2 \to \mathbb{C}^2/G$ be a quotient surface singularity. What properties should $G$ have such that the pushforward $f_*\mathcal{O}_{\mathbb{C}^2}$, of the structure sheaf $\mathcal{O}...
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1answer
181 views

Properly elliptic surface with no multiple fibers and without a section

I am aware that if an elliptic surface contains multiple fibers, then it has no section. Is the converse false? In particular, I am looking for an example of a projective, properly elliptic surface (...
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81 views

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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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 ...
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1answer
267 views

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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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 \...
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56 views

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, ...
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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 ...
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173 views

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 ...
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56 views

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 ...
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2answers
219 views

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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256 views

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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1answer
297 views

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 $...
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1answer
322 views

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 \...
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1answer
218 views

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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0answers
221 views

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 ...
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1answer
150 views

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 $\mathbb{P}^{2}$. ...
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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 ...
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1answer
170 views

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 ...
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3answers
280 views

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 ...
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0answers
129 views

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 ...
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137 views

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, ...
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1answer
358 views

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 ...
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0answers
190 views

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 $...
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130 views

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 ...
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1answer
290 views

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 $\mathcal{H}/k$ ...
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0answers
249 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 ...
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0answers
130 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 ...
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0answers
139 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^...
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1answer
153 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-...
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1answer
151 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 ...
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175 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 ...
3
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1answer
86 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 ...
3
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1answer
233 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 ...
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1answer
492 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?