# Tagged Questions

**2**

votes

**1**answer

95 views

### 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 ...

**2**

votes

**1**answer

177 views

### 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 ...

**1**

vote

**1**answer

125 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 $A$ ...

**5**

votes

**1**answer

135 views

### Linear systems on bielliptic surfaces

A bielliptic surface is a surface of type $S=E_1 \times E_2/G$ where $E_1, E_2$ are elliptic curves and $G$ is a finite group of translations of $E_1$ acting on $E_2$ such that $E_2/G=\mathbf{P}^1$.
...

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**1**answer

131 views

### Conjugate surfaces: informations about the orbits

Consider a complex algebraic variety $X$ (namely a $\mathbb C$-scheme, of finite type, geometrically integral and separated); if $\sigma\in\textrm{aut}(\mathbb C)$, then is well defined the complex ...

**5**

votes

**1**answer

199 views

### Is the Tate conjecture known for etale covers of products of curves

Let $X$ be a (smooth projective geometrically connected) surface over a finitely generated field $k$. The Tate conjecture predicts that, for $l$ a prime number invertible in $k$, the Chern class map ...

**1**

vote

**1**answer

93 views

### Boundedness of the number of curves negative on a varying big divisor

For a divisor $D$ on a smooth complex projective surface $X$, the stable fixed part is the maximal effective divisor $E$ which, for every $n \in \mathbb{N}$, is contained in every memeber of the ...

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votes

**1**answer

94 views

### Cohomology of a fibered surface

Let $R$ be a complete Henselian discrete valuation ring, $\pi:X \to \mathrm{Spec} (R)$ be a smooth, proper, integral, flat $\mathrm{Spec} (R)$-scheme of dimension $2$. Assume that the genus of the ...

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votes

**2**answers

146 views

### First cohomological support locus of a fibration

I have a fibration $f \colon S \longrightarrow E$, where $S$ is a compact, complex surface of general type belonging to a special class I'm studying and $E$ is an elliptic curve.
I computed the ...

**4**

votes

**1**answer

119 views

### Castelnuovo's rationality criterion on singular surfaces?

Let $S$ be a projective surface over an algebraically closed field. Suppose that $q(S)=h^1(\mathcal O_S)=0$ and $P_2(S)=h^0(\mathcal O_S(2K_S))=0$. If $S$ is smooth, Castelnuovo's rationality ...

**3**

votes

**0**answers

106 views

### Non-vanishing of $H^0$ on rational surface

Let $X$ be a smooth rational surface that admits a proper birational morphism to $\mathbb{P}^2$ and $D$ a simple normal crossing divisor on $X$ such that $K_X+D$ is big, is it true that ...

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votes

**1**answer

124 views

### Linear system on an abelian surface

On a K3 surface $S$, a linear system $|C|$ is said to be hyperelliptic if the corresponding map is of degree 2 and the image is of degree $g_a(C)-1$ in $\mathbb P^{g_a}$.
For $g_a(C) > 2$, if ...

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vote

**1**answer

131 views

### Intrinsically proving a singularity is rational

In general, how to prove a variety has rational singularities intrinsically? i.e., don't use the Artin's criterion concerning the exceptional locus. And what kinds of varieties have only rational ...

**2**

votes

**1**answer

202 views

### Surfaces singular along a curve

Let $C\subset\mathbb{P}^3$ be a smooth curve a degree $d$ and genus $g$. Let $\mathcal{S}$ be the system of surfaces of degree $k$ in $\mathbb{P}^3$ containing $C$ with multiplicity $\beta$.
What is ...

**0**

votes

**1**answer

180 views

### Kleiman's and Nakai-Moishezon's ampleness criteria

I would like to work out a simple example to understand the relation between Kleiman ampleness criterion and Nakai-Moishezon ampleness criterion.
Namely, let $X$ be the blow-up of $\mathbb{P}^{2}$ at ...

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votes

**3**answers

255 views

### Automorphisms of a smooth quadric surface $Q\subset\mathbb{P}^{3}$

Let $Q\cong\mathbb{P^{1}_{1}}\times\mathbb{P^{1}_{2}}\subset\mathbb{P}^{3}$ be a smooth quadric surface. We have the following two actions on $Q$:
$$S_2\times Q\rightarrow Q,\; ...

**7**

votes

**1**answer

208 views

### Finite morphisms to projective space

Let $X$ be a projective variety of dimension n. Then there exists a finite surjective morphism $X \to \mathbf P^n$. Let $d$ be the minimal degree of such a finite surjective morphism.
Let $d^\prime ...

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votes

**1**answer

203 views

### Does $h^1(D)=0$ imply numerical connectedness on K3 surfaces?

Let $X$ be a complex K3 surface and $D$ an effective divisor on $X$.
We shall say: $D$ is connected if its support is connected. $D$ is numerically connected if for any non-trivial effective ...

**3**

votes

**1**answer

188 views

### K3 surface with $D_{14}$ singular fiber

Let $X$ be an elliptic K3 surface with $D_{14}$ singular fiber. Do you know an explicit equation for such $X$? Also, how many disjoint sections such fibration admits? Any reference would be greatly ...

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**1**answer

182 views

### On the dualizing sheaf of a curve

Let $X$ be a smooth projective surface in $\mathbb{P}^n$ and $C$ be an effective curve. I know that the dualizing sheaf, $\omega_C$ of $C$ is ...

**1**

vote

**1**answer

110 views

### A question on very ample line bundle on smooth projective surfaces

I had been reading a couple of texts by J.P. Demailly, one of them titled "Effective bounds for very ample line bundles". In the introduction the author mentions a result due to I. Reider (stated in ...

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votes

**1**answer

215 views

### Torsion of the Picard group for surfaces isogenous to a product

We say that a complex surface $S$ is isogenous to an (unmixed) product if there exists a finite group $G$, acting faithfully on
two smooth projective curves $C_1$ and $C_2$ and freely on their
product ...

**3**

votes

**1**answer

145 views

### Blowing up rational singularities

Let $X$ be a projective surface embedded into $\mathbb{P}^n_{\mathbb{C}}$ having at most rational singularities. Let $\tilde{X} \to X$ be the minimal resolution of $X$. Is it possible to embed ...

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votes

**1**answer

118 views

### Locally trivial deformations of surfaces with quotient singularities

Let us consider the surface $\mathbb{A}^{2}/\mu_{6}$ where the action is given by
$$
\begin{array}{ccc}
\mu_{6}\times\mathbb{A}^{2} & \longrightarrow & \mathbb{A}^{2}\\
(\epsilon,x_{1},x_{2}) ...

**8**

votes

**1**answer

358 views

### Why can you deform singularities in two dimensions but not in higher dimensions?

I've been trying to read this paper to understand deformations of surface quotient singularities. I'm particularly interested in when one can deform certain cyclic quotient singularities into other ...

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**0**answers

120 views

### birational invariants of projective surfaces

I am studying Castelnuovo's rationality criterion for surfaces.
Let $S$ be a projective surface and $K$ a canonical divisor on $S$.
Let's use the notation ...

**0**

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**0**answers

162 views

### Hodge structure of abelian surfaces

In my case, I have an abelian surface $A$ of (2,8)-polarization, and I have some finite group (may not be abelian group) $G$ acting on $A$ without fixed point. I want to understand when there is a ...

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votes

**2**answers

158 views

### Points of a linear system on a cubic surface

Let $S$ be a generic cubic surface and let $C$ be its intersection with a generic quadric surface.
In the linear system of hyperplane sections of $S$, how many points represent the planes $H$ tangent ...

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vote

**1**answer

128 views

### Smoothness of the quotient surface by an involution with nice fixed locus

Let $X$ be a (smooth complex algebraic) surface. Suppose $\theta$ is an automorphism of order $2$ of $X$, such that its fixed locus is a disjoint union of smooth curves. I am trying to prove that the ...

**4**

votes

**1**answer

175 views

### Kodaira classification and the McKay correspondence

Kodaira's table of singular fibers has a singular fiber
for each of $\tilde{A}_n$, $\tilde{D}_n$, $\tilde{E}_6$, $\tilde{E}_7$, and $\tilde{E}_8$; these are chains or cycles of (-2)-curves connected ...

**3**

votes

**1**answer

243 views

### Do there exist double points on an algebraic surface in $\mathbb{P}_{\mathbb{C}}^3$ that are not rational?

The title explains it all.
I'm familiar with the du val singularities on surfaces, also known as rational double points. In http://homepages.warwick.ac.uk/~masda/surf/more/DuVal.pdf, 2.1, they are ...

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vote

**0**answers

69 views

### invariance of the dimension of severi varieties of surfaces

Suppose I have a smooth projective surface $S\subset P^n$ embedded by a very ample linear system $|L|$. Consider now the generalized Severi varieties that parametrize curves on $S$ belonging to $|L|$, ...

**0**

votes

**1**answer

208 views

### singularities of the dual variety of a surface

I am looking for a proof/reference of the following simple fact, which I think it holds true.
Let $S\subset \mathbb{P}^n$ be a surface embedded by a very ample linear system. Then I know that the ...

**2**

votes

**1**answer

109 views

### Dual of a Complex 2-Torus

Is a complex torus $A$ of dimension 2 always isomorphic to its dual torus (i.e. the torus obtained by taking the dual lattice), or are there counterexamples to this?

**4**

votes

**2**answers

301 views

### Reference for Automorphisms of K3 surfaces

I am looking for some introductory reference concerning Automorphisms (of finite order) on K3 surfaces. Any suggestion?

**1**

vote

**1**answer

167 views

### Complete Linear system on Del Pezzo surfaces

Is there always a reducible curve (EDIT: with exactly two irreducible components intersecting in at least 2 points) in a complete linear system (EDIT: of dimension at least 2 with curves of genus at ...

**2**

votes

**1**answer

278 views

### On complex surfaces with Kodaira dimension 1

Let $S$ be a complex surface of Kodaira dimension $1$ and $\pi_{1}(S) \neq 1 $.
What is known on possible diffeomorphism types of such $complex$ surfaces with a given fundamental group? Is it true ...

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votes

**0**answers

491 views

### Bezout Theorem in $\mathbb P^3$

For a $\mathbb C P^2$ is known a result: if through the generic points $p_1,p_2,\dots p_n$ with multiplicities $m_1,m_2\dots, m_n$ correspondingly a degree $d$ curve passes then $d^2\geq ...

**8**

votes

**2**answers

317 views

### Uniformization of Kodaira fibered surfaces

Consider a Kodaira fibration. i.e. a smooth non-isotrivial fibration $X\rightarrow C$ with $X$ a smooth complex surface and $C$ a smooth complex curve, such that both the genus of $C$ and genus of the ...

**7**

votes

**1**answer

196 views

### A question on an elliptic fibration of the Enriques surface

Let $S$ be an Enriques surface over complex numbers. It is known that $S$ admits an elliptic fibration over $\mathbb{P}^1$ with $12$ nodal singular fibers and $2$ double fibers. How can I see this ...

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votes

**1**answer

641 views

### what is the cyclic cover trick?

What do people mean by the "cyclic cover trick"? I have found this expression a couple of times with no complete explaination, both talking about curves and surfaces...

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**0**answers

127 views

### What does Hodge theory tell us about simply connected surfaces of general type

Let $X$ be a smooth complex projective variety. We know that $\Omega^1_X$ has a non-zero section if and only if the abelianization of the fundamental group of X is infinite. This follows from Hodge ...

**3**

votes

**1**answer

186 views

### A question on existence of degeneration of Enriques surface.

Let $S$ be an Enriques surface, i.e. a quotient of a K3 surface by a free involution. Enriques surfaces arise as elliptic fibrations $S\rightarrow \mathbb{P}^1$ with 12 singular fibers and 2 double ...

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votes

**0**answers

99 views

### The central fiber of this family of surfaces?

I have a question on a description of a central fiber of the following family of surfaces.
Let $B,C \subset \mathbb{P}^2$ be smooth curve of degree $2d$ and $d$ respectively. Let's consider the ...

**3**

votes

**1**answer

112 views

### What can a quartic surface in $\mathbb{P}^3$ with an ordinary quadruple point look like?

All varieties will be projective and over $\mathbb{C}$.
If $S$ is any surface in $\mathbb{P}^3$ of degree 2 that posseses an ordinary double point, it follows easily that $S$ is projectively ...

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votes

**3**answers

349 views

### Surfaces ruled over elliptic curves

Ground field $\Bbb{C}$. Algebraic category. Elliptic surfaces are those surfaces endowed with a morphism onto some smooth curve, with generic fiber an elliptic curve.
Suppose $E$ is an elliptic curve ...

**4**

votes

**1**answer

193 views

### Basics of minimal Elliptic Surfaces [following Beauville]

I am reading Beauville's chapter IX on Elliptic surfaces.
Let $S$ be a minimal elliptic surface with $\kappa=1$ and $p:S\rightarrow C$ be the elliptic fibration.
We know $K^2=0$. Suppose the ...

**5**

votes

**1**answer

206 views

### Embeddings of smooth projective surfaces

Let $X$ be a smooth projective surface not contained in $\mathbb{P}^3$.
Is there any known condition on $X$ under which I can embed it into $\mathbb{P}^3$ such that the its image contains at most ...

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votes

**0**answers

171 views

### pencils on varieties of general type

I was wondering about a generalization of the following property of surfaces of general type.
Let $X$ be a smooth projective surface of general type. Then there is no pencil of rational or elliptic ...

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votes

**2**answers

329 views

### Existence of smooth surfaces containing a curve

Let $C$ be a curve in $\mathbb{P}^3$, possibly non-reduced. Assume, there exists a smooth surface in $\mathbb{P}^3$ containing $C$. Is it true that for $d \gg 0$, a generic element of $I_d(C)$ defines ...