The algebraic-surfaces tag has no usage guidance.

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### Can an ample class be written as a sum of two classes which are not even nef?

Let $X$ be a smooth projective surface over $\mathbb{C}$. We know that nef line bundles form a closed convex cone whose interior is the ample cone. My doubt is the other direction, is it possible to ...

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190 views

### Globally generated, nef and big line bundles which are not ample on a K3 surface

Let $X$ be a K3 surface over $\mathbb{C}$. On a $K3$ surface we know that $Pic(X)\cong Num(X)\cong NS(X)$. A class $L\in Num(X)$ is called movable if $L.C\geq 0$ for every curve $C$ in $X$. It just ...

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119 views

### Relation between curves in a complete linear system contained in another

Let $X$ be a projective surface over $\mathbb{C}$, let $x\in X$ be the only singular point of $X$. Let $L$ be an ample line bundle on $X$. Consider the blow up $Y$ of $X$ along $x$, ...

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113 views

### Quartic symmetroids and 10-points sets

A quartic surface in $\mathbb{P}^3$ is said to be a "symmetroid" if its equation is obtained as the determinant of a 4x4 symmetric matrix of linear forms. It is well known that the general symmetroid ...

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141 views

### Self intersection and deformations

Suppose I have a smooth projective surface $S$ and a curve $C\subset S$. The self-intersection of $C$ is by definition the degree of the restriction to $C$ of the normal bundle of $C$ inside $S$.
By ...

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83 views

### Dimension of Quot scheme of zero dimensional quotients of a locally free sheaf

Given a locally free sheaf $E$ of rank $r$ on a (smooth, projective, algebraic) surface, I want to know the dimension of the scheme parametrizing the zero-dimensional (meaning they have zero ...

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227 views

### A question on young persons guide to canonical singularities

In the Corollary at pag 407 of Young persons guide to canonical singularities there is a formula to compute the contributions $c_q(D)$ to Riemann-Roch of a divisor $D$ passing through a point $q\in ...

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**1**answer

195 views

### On surfaces with $p_g=0$, $q=1$, and $K^2=-3$ [closed]

I am having a trouble in understanding the Example 4.7 (pages 65-66), the genus two fibrations with $p_g=0$, $q=1$, and $K^2 = -3$, in "Surfaces fibrées en courbes de genre deux", Lecture Notes in ...

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215 views

### Euler Characteristic of Real Algebraic Surfaces

Given a (compact if needed) real smooth surface $V(f)$ defined by $f\in \mathbb{R}[X,Y,Z]$, in particular it is oriented. Is there a formula which gives the Euler character of $V(f)$ ?
Thanks.

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### Does there exist a holomorphic fibration of genus two over $\mathbb{P}^{1}$ with $7$ nodal singularities?

This is a problem about the holomorphic fibration on a complex manifold.
Does there exist a holomorphic fibration of genus two over $\mathbb{CP}^{1}$ with 7 nodal singularities?
If you are aware of ...

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**1**answer

342 views

### Del Pezzo surfaces and homotopy groups of spheres

A (complex) del Pezzo surface is a smooth projective complex surface with ample anticanonical line bundle. Such surface has a degree defined as the self intersection of the canonical divisor. It is ...

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139 views

### Does the Bombieri-Lang conjecture imply severe restrictions on rational points on twists of hyperelliptic curves?

According to Silverman, the Bombieri-Lang conjecture implies
that the rational points of surface on general type lie on
finite set of curves, except for a finite set of points.
Let $f$ be univariate ...

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**1**answer

152 views

### What is the type of the surfaces $x^5 - y^5 + z^2 + x=0$ and $x^5 - y^5 + z^2 + x+1=0$?

Crossposted from MSE.
I am interested what is the type of the surfaces over the
rationals
$$ x^5 - y^5 + z^2 + x=0$$
and
$$ x^5 - y^5 + z^2 + x+1=0$$
Magma's ...

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100 views

### How to check with a CAS if a surface is of general type?

The main question is:
How to check with a CAS if a surface is of general type?
Magma's function KodairaEnriquesType is close to this,
but doesn't always work.
...

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272 views

### Is it normal surface of general type to have infinitely many positive rank elliptic curves?

Cross-posted from MSE.
I am not good at algebraic geometry and almost surely am
misunderstanding something.
Got an alleged argument against Bombieri-Lang conjecture and
would like to know what the ...

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**1**answer

200 views

### Automorphisms of del Pezzo surfaces

Let $S$ be a del Pezzo surface of degree six over $\mathbb{C}$. Then $S$ is the blow-up of $\mathbb{P}^2$ in three general points $p_1,p_2,p_3$.
Is it true that its automorphism group is ...

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149 views

### Morphisms contracting a family of curves

Let $f:X\rightarrow Y$ be a morphism of normal projective varieties. Let $S\subseteq X$ be a surface admitting a morphism $g:S\rightarrow C$ to a curve $C$ such that any fiber of $g$ is a curve.
...

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160 views

### Göttsche's formula for non-compact complex surfaces?

Is the Göttsche's formula (Eq (2.1) of this paper) expressing the Poincare polynomial (or the Euler char version) of the Hilbert scheme of points on a projective surface valid for non-compact complex ...

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212 views

### Automorphisms of surfaces

Let $X$ be a projective surface with a morphism $f:X\rightarrow\mathbb{P}^1$. Assume that $f^{-1}(t)\cong\mathbb{P}^1$ for any $t\neq 0$ but $f^{-1}(0)$ is the union of two $\mathbb{P}^1$'s ...

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201 views

### Infinitesimal deformations of a singular projective surface

Let $X$ be a normal projective surface with just two singular points $x_1,x_2\in X$, where $X$ has rational quotient singularities.
Assume that both the singularities in $x_1$ and in $x_2$ admit a ...

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109 views

### Endomorphism algebras of abelian surfaces with real multiplication

Given an abelian variety $A$ over a field $F$, one may consider the ring of endomorphisms $End(A)$, the ring of $F$-rational maps $A \to A$ respecting the group structure on $A$. We may also consider ...

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181 views

### On Abelian Galois Covering

Consider a complete quadrangle $\Delta$ in $\mathbb{CP}^2$ (i.e. the union of the six
lines through points $P_1$, $P_2$, $P_3$ and $P_4$ in general position). Let $f: Y := \hat{\mathbb{CP}^2}(P_1, ...

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126 views

### Enriques classification of algebraic surfaces in characteristic zero

I am searching for a reference about the classification of algebraic surfaces over an arbitrary algebraically closed field of characteristic zero. In the 1949 book "le superficie algebriche" by ...

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60 views

### Is a general extension of general stable sheaves on $\mathbb P^2$ stable?

Theorem 2 in this paper by Bhosle gives a nice condition on slopes for when a general extension of general stable bundles on curves is stable. Does anyone know whether there is an analogous result for ...

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### Real algebraic surface

Let $f(u,v,z)\in \mathbb{Q}[u,v,z]$ be a polynomial in three variables such that $X_{\mathbb{R}}\subset \mathbb{R}^{3} $ (the associated surface of real solution) is smooth. Suppose that the set of ...

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141 views

### Why does this vector bundle on the surface sit in this exact sequence?

Let $X$ be a K3 surface. Let $E$ be a semistable rank 3 vector bundle. Now suppose $0 = E_0\subset E_1\cdots\subset E_s=E$ be the Harder-Narasimhan filtration. Suppose $E_1$ is $\mu$-stable and rank ...

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### Are all these K3 surfaces supersingular?

Consider all the smooth K3 surfaces given by $X^4+W^2X^2+XW^3 = f(Y,Z,W)$ or $X^4+XW^3 = g(Y,Z,W)$ over $\mathbb F_{2}$ with $f$ or $g$ homogenous of degree 4. There are a lot of choices for $f$ and ...

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163 views

### What is the co-kernel of the morphism of vector bundles?

Let $X$ be a surface, and $i:C\subset X$ be a smooth curve. Let $A$ be a line bundle on $C$, and $E$ be a vector bundle of rank $r$ on $X$.
Suppose there is a surjection: $E\longrightarrow ...

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562 views

### Enriques surfaces over $\mathbb Z$

Does there exist a smooth proper morphism $E \to \operatorname{Spec} \mathbb Z$ whose fibers are Enriques surfaces?
By a theorem of, independently, Fontaine and Abrashkin, combined with the ...

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196 views

### Smoothing transverse intersections

Let $S$ be a complex surface with ample canonical class. Let $C_1$ and $C_2$ be smooth complex curves in $S$ that intersect transversally at $n $ points. Furthermore, assume that the self-intersection ...

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### Disjoint curves in an algebraic surface

Let $X$ be an algebraic surface (over the complex) with $p_g=q=0$. Is it possible to have disjoint curves $C_1,\ldots, C_b$, of positive genus, spanning $H_2(X,{\mathbb Q})$, $b=b_2(X)$?
(When $X$ is ...

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310 views

### A proper smooth surface is projective

My question is a reference request for the following fact: if $k$ is a field and $X$ a proper smooth surface over $k$, then $X \rightarrow \mathrm{Spec}\, k$ is projective. Where is this well-known ...

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### Adjunction map for projective surfaces

Before stating my question, let me recall (part of) the classical result on the adjunction map for complex projective surfaces, due in this modern form to Beltrametti and Sommese:
Adjunction ...

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### Pic^0 of the surface of bitangents of a quartic

Let $S$ be a generic quartic surface in $\mathbf{P}^3$.
Let $T$ be the surface of the lines bitangent to $S$.
What can we say about $Pic^0(T)$?

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176 views

### Computing Euler Characteristics of Line Bundles on the Hilbert Scheme of n points

Let $S^{[n]}$ be the Hilbert scheme of $n$ points on a smooth projective surface (actually, right now I am particularly interested in del Pezzo surfaces). Let $B$ be the exceptional divisor of the ...

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108 views

### Blow-up of $\mathbb{P}^4$ along a smooth surface

Let $\pi \colon X\to \mathbb{P}^4$ be the blow-up of a smooth surface $S\subset \mathbb{P}^4$. Is there a formula to compute $(K_X)^4$ ? (which should be dependent on invariants of $S$).
In dimension ...

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338 views

### Is there a description of the moduli space of elliptic surfaces?

In this question elliptic surface means a smooth projective complex surface $X$, such that there is an elliptic fibration $\pi \colon X \to C$. (I.e., there is a curve $C$ and a proper map $\pi$, such ...

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221 views

### Hypersurface with singularities

I heard once about one open problem. That was about existing a hypersurface of a small degree (5? or 6?) passing through some number (5? 6?) of 3-fold points and 2-fold lines (3 lines?).
It was said ...

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### Stable Lefschetz fibrations

Let $S$ be a non-singular complex projective surface, then construct the Lefschetz fibration $\pi:\widetilde S\longrightarrow\mathbb P^1$ associated to $S$ (we have a birational morphism ...

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### Topological/numerical constraints for the existence of more than one pencil

A famous theorem of Castelnuovo and de Franchis tells us that for $S$ a smooth projective complex algebraic surface that for $b \geq 2$, pencils $f : S \to B$ of genus $b :=g(B)$ are in bijective ...

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### Isotrivial fibrations over $\mathbb P^1$

First of all I want to say that algebraic geometry is not "my field of research" so I apologize if the notation is not standard.
$S$ is a smooth complex projective surface with a fibration $f$ over ...

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286 views

### Kummer Coverings

Let $L_1$, $\cdots$, $L_k$ be homogenous linear forms in three variables $z_0$, $z_1$, $z_2$ defining $k$ lines in $\mathbb{P}_{2}$. Consider the abelian extension $K
((L_2/L_1)^{1/n}, \cdots, ...

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### Computing Euler Charactistics of Line bundles on Hilbert Schemes of points on Surfaces

Let $S^{[2]}$ be the Hilbert scheme of two points on a smooth projective surface (actually, right now I am particularly interested in del Pezzo surfaces). Let $B$ be the exceptional divisor of the ...

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202 views

### Number of minimal models of a surface

I would like to know if the following statement is true or false:
Given a non-singular complex projective surface $S$, it has at most a countable number of minimal models (up to isomorphism).
...

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### discriminant of smooth quartic del Pezzo surface in $\mathbb{P}^4$

I can't understand the proof of Lemma3.3 in Stability of genus 5 canonical curves.
Let $C$ be a complete intersection of three quadrics in $\mathbb{P}^4$ and let $\Lambda$ be the net of quadrics ...

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### Understanding a proof of a lemma in elliptic surfaces

In the following paper , The Kahler-Ricci flow on surfaces of positive Kodaira dimension (Sung and Tian) in page 621, I am trying to understand the proof of part 1 of Lemma 3.4.
In the part 1 of ...

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270 views

### Is each rationally chain connected surface rational?

Consider a 2-dimensional smooth projective algebraic surface S over complex numbers. Could you recommend any exact references to the proofs of the following assertions (of course, if they are true):
...

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### Degree of a smooth curve in an abelian variety

Let $A$ be an abelian variety, $g$ be a positive integer and $\mathcal{L}$ be an ample line bundle on $A$.
Question : Is there a real $r>0$ such that, for all smooth curve $C$ of genus $g$ in ...

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### vector bundles on $\mathbb{C}[x,y,z]/(x y - z^k)$

Let $A = \mathbb{C}[x,y,z]/(x y - z^k)$. In fact $A$ is the ring of $\mu_k$ invariants: $A = \mathbb{C}[u,v]^{\mu_k}$ where $g \in \mu_k$ acts by $g(u,v) = (g u, g^{-1} v)$.
This allows one to ...

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145 views

### Divisors on an abelian surface

Let $A$ be an abelian surface given by the quotient of a product of two generic elliptic curves $E_1 \times E_2$ by the product $T_1 \times T_2$ of two translations by $2$-torsion points. Then $A$ ...