**3**

votes

**1**answer

221 views

### What's $H^*(X - \{x_1,\ldots,x_n\},\mathcal{O})$, when $X$ is a projective smooth surface?

Let $X$ be a smooth projective surface over a field $k$. Is there a way to compute $H^1(X - \{x\},\mathcal{O}_{X-\{x\}})$ in terms of similar invariants for $X$? Actually I'd like to remove even ...

**10**

votes

**0**answers

269 views

### Singular curve on an abelian surface

Let $C_2$ be a smooth genus $2$ curve and $J(C_2)$ its Jacobian. It is well known that the blow-up of $J(C_2)$ at the origin $o$ is isomorphic to the second symmetric product $\textrm{Sym}^2(C_2)$, ...

**6**

votes

**1**answer

191 views

### Del Pezzo surfaces of degree $2$

I'm trying to understand the relationship between the different models of del Pezzo surfaces of degree $2$.
Let $k$ be a field of characteristic not equal to $2$. Usually, del Pezzo surfaces of ...

**3**

votes

**1**answer

204 views

### Are rational surface singularities $\mathbb{Q}$-Gorenstein?

I know that, in general, rational singularities are not necessarily $\mathbb{Q}$-Gorenstein. So I ask:
is there any positive result in this direction known for surfaces?

**5**

votes

**0**answers

90 views

### Is a Kummer surface over an finite field $\mathbb{F}_q$ supersingular iff $\mathbb{F}_q$-unirational?

Let $A$ be an abelian surface over an finite field $\mathbb{F}_q$. In particular, I am interested in the case when $A$ is a Jacobian variety. Is the Kummer surface $K_A/\mathbb{F}_q$ Shioda-...

**2**

votes

**1**answer

213 views

### Normal bundle to fibers of a rational morphism

Let $f:X\dashrightarrow C$ be a rational fibration from a 3-dimensional variety $X$ to a curve $C$ such that generic fiber is smooth and different fibers intrsect in smooth curves. Take $S$ to be a ...

**4**

votes

**1**answer

156 views

### Triple covers of $\mathbb{P}^2$ with fixed branch locus

Let us consider a smooth (complex) cubic surface $X \subset \mathbb{P}^3$ and a general point $p \notin X$. Then it is classically well-known that linear the projection $$\pi_p \colon X \...

**2**

votes

**1**answer

149 views

### Does $\omega_C\simeq N_{C/S}$ always happen on Enriques surfaces?

Let $S$ be an Enriques surface and $C\subset S$ a smooth irreducible curve of genus $g$.
Consider the condition $$\omega_C\simeq N_{C/S}$$
For example, when $g=1$ then $\omega_C=\mathcal{O}_C$ and ...

**3**

votes

**0**answers

118 views

### What is the integral cohomology of an Enriques surface over a finite field?

Probably this is well-known, but I could not find it. I would like to understand the integral $2$-adic etale cohomology of an Enriques surface over $\mathbb{F}_q$ in dimension 2: $H_{et}^2(X, \mathbb{...

**3**

votes

**1**answer

207 views

### Polars of algebraic curves and surfaces

I asked this on Math.StackExchange, but received no response, so trying here ...
A paper I'm reading says the following ...
With homogeneous coordinates $\mathbf{x} = [x,y,z,w]$, let $F(\mathbf{x}...

**2**

votes

**0**answers

77 views

### What is $\mathrm{Num}(X)$ for the canonical cover $X$ of a bielliptic surface $S$?

A bielliptic surface $S$ is a smooth projective complex surface of Kodaira dimension 0 with $h^1(\mathcal O_S)=1$ and $h^2(\mathcal O_S)=0$. It is well known that $S=(A\times B)/G$, where $G$ is a ...

**7**

votes

**1**answer

231 views

### Restriction of the Picard group of a surface to a curve

In a paper by Griffiths and Harris on the Noether-Lefschetz theorem, they use the following fact which they don't comment as if it is obvious:
For a general (smooth) surface $S$ in $\mathbb{P}^3$ ...

**1**

vote

**1**answer

255 views

### Global sections of higher direct image sheaf

Let $X$ be a regular projective (complex) surface and $S$ be a finite set of closed points on $X$. Denote by $j:X\backslash S \to X$ the open immersion. Assume further that $H^1(\mathcal{O}_X)=0$. Is $...

**7**

votes

**1**answer

275 views

### When is a general sheaf (on the projective plane) globally generated?

Let $v$ be a chern character on $\mathbb P^2$ so that the moduli of sheaves of chern character $v$ is non-empty of the expected dimension. When is it true that the general sheaf in moduli is globally ...

**4**

votes

**0**answers

120 views

### Obstruction to lifting of global sections of invertible sheaves

Are there known examples of smooth projective hypersurface in $\mathbb{P}^3$, say $X$ and an invertible sheaf $L$ on $X$ with $H^0(X,L)>0$ satisfying the following property: There exists an ...

**2**

votes

**0**answers

99 views

### Elliptic fibration arising from a higher genus linear system

Let $H$ be a very ample linear system on a smooth compact complex surface $X$ whose Kodaira dimension is $\geq 0$. A general element of $H$ is smooth and has genus $\geq 2$.
Let $L\subset H$ be a ...

**1**

vote

**0**answers

127 views

### Gromov-Witten invariants for arithmetic surfaces counting sections passing through points

Suppose we are given an arithmetic surface, $X\to \text{Spec}\mathbb{Z}[1/N]$ smooth and quasi-projective, and a finite set of closed points all in different vertical fibers.
Can we count the number ...

**1**

vote

**1**answer

154 views

### Elliptic fibrations with few singular fibers

It is known that non-isotrivial fibrations of genus $g>0$ curves over the projective line have a bunch of singular fibers. There are at least three of them.
It is not difficult to prove that an ...

**2**

votes

**1**answer

120 views

### Pencils in very ample linear systems without curve in its base locus

If $L$ is a very ample line bundle over a smooth complex projective surface $X$ and $s_0, \dots, s_n$ is a basis of the global sections of $L$, is there some choice of $i,j$ such that the pencil ...

**2**

votes

**3**answers

177 views

### 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 ...

**2**

votes

**2**answers

280 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 ...

**2**

votes

**1**answer

126 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 $x$, $f:Y\...

**3**

votes

**2**answers

135 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 ...

**4**

votes

**1**answer

155 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 ...

**2**

votes

**1**answer

101 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 ...

**5**

votes

**1**answer

270 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 $q\in X$,...

**2**

votes

**1**answer

203 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 ...

**10**

votes

**1**answer

239 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.

**3**

votes

**3**answers

327 views

### 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 ...

**18**

votes

**1**answer

373 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 ...

**3**

votes

**0**answers

157 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 ...

**3**

votes

**1**answer

184 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 ...

**1**

vote

**0**answers

102 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.
...

**3**

votes

**2**answers

291 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 ...

**8**

votes

**1**answer

274 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 $((\mathbb{C}...

**1**

vote

**1**answer

163 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.
...

**1**

vote

**1**answer

177 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 ...

**3**

votes

**1**answer

230 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 ...

**6**

votes

**1**answer

222 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 ...

**7**

votes

**1**answer

153 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 ...

**2**

votes

**0**answers

187 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, \...

**3**

votes

**1**answer

130 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 ...

**2**

votes

**1**answer

121 views

### 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 ...

**0**

votes

**1**answer

147 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 $...

**5**

votes

**0**answers

148 views

### 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 $...

**1**

vote

**0**answers

167 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 i_*A\...

**24**

votes

**0**answers

631 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 Enriques-...

**2**

votes

**1**answer

207 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 ...

**5**

votes

**2**answers

246 views

### 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 ...

**8**

votes

**1**answer

374 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 ...