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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How can I describe explicitely a nonsingular model of the elliptic surface?

Consider the surface $\mathcal{E} = Q_1 \cap Q_2 \subset \mathbb{P}_k^3\times \mathbb{P}_k^1$ with homogenous coordinates $x$, $x^\prime$, $y$, $z$ and $t$, $s$ respectively and a field $k$ of even ...
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self-intersection number of rational curves in smooth projective surfaces

Given a smooth projective surface $X/\mathbb{C}$, denote the $Hom_1(\mathbb{P}^1, X)$ to be the set of all degree 1 morphisms from $\mathbb{P}^1$ to $X$. We know that we can regard $Hom_1(\mathbb{P}^1,...
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Kähler classes for surfaces of general type with $c_1^2=3c_2$

Given a smooth, compact complex surface with ample canonical bundle satisfying $c_1^2=3c_2$, is it true that every Kahler class is a multiple of $c_1$? This seems to be the case for fake projective ...
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Is a Kummer surface unirational over a sufficiently large finite field of characteristic 2?

Let $A$ be a supersingular abelian surface over a sufficiently large finite field $\mathbb{F}_q$ of characteristic $2$ and let $K_A = A/(-1)$ be the Kummer surface. Shioda ("Kummer surfaces in ...
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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 ...
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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 $\textrm{Sym}^2(C_2)$, ...
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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 ...
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1answer
215 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?
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1answer
218 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 ...
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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-...
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Rational curves on varieties of general type

Let $S$ be a complex surface of general type. Are there infinitely many smooth rational curves on $S$? And more general, what if $V$ is a variety of general type?
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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 \...
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1answer
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 ...
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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{...
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209 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}...
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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 ...
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242 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 $\mathbb{P}^3$ ...
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1answer
262 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 $...
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level sets of multivariate polynomials

Let $p:\mathbb R^n \rightarrow \mathbb R$ be a polynomial of degree at most $d$. Restrict $p$ to the unit cube $Q=[0,1]^n\subset\mathbb R^n$. We assume that $p$ has mean value zero on the unit cube $Q$...
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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 ...
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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 ...
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155 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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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 ...
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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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130 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 ...
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1answer
158 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 ...
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1answer
122 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 ...
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Surfaces in $\mathbb{P}^3$ with isolated singularities

It is classically known that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only ordinary singularities, i.e. a curve $C$ of double points, ...
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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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2answers
292 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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1answer
127 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\...
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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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1answer
157 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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1answer
102 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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280 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 $q\in X$,...
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Two basic questions concerning geometrically ruled surfaces. [closed]

Good Morning, These questions are a result of trying to understand the proof of Proposition III.18 in Beauville's book 'Complex Algebraic Surfaces'. Here is the setup - everything is smooth, ...
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1answer
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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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1answer
242 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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3answers
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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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1answer
374 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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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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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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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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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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1answer
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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}...
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1answer
223 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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1answer
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. ...
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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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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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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 ...