# 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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### Fundamental group of a smoothing of a complex surface

Let $X_0$ be a compact complex algebraic surface with an isolated singularity and let $X_t$ be a smoothing of $X_0$ over the disc. How can we compute the fundamental group of $X_t$ say in terms of the ...
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### Abelian subvarieties corresponding to vector subspaces

Let $S$ be a connected smooth projective surface. Let $C$ a smooth curve on $S$ In page 9 of the paper "https://arxiv.org/abs/1704.04187v1" a read the following: Let \begin{equation*} r: ...
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### Constructions of complex surfaces covered by the ball of $\mathbb{C}^2$

Let $S$ be a compact complex surface. It is well-known that the following two facts are equivalent $c_1^2(S) = 3 c_2(S)$ and $S \neq \mathbb{CP}^2$ The universal cover of $S$ is biholomorphic to the ...
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### Moduli stacks of algebraic surfaces—obstructions to existence?

The moduli stack $\mathcal{M}_g$ of genus $g$ curves is one of the deepest objects in mathematics, so of course you wonder to what extent you can construct an (Artin?) stack parametrising algebraic ...
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### Automorphisms of finite order on $K3$ surfaces

Is there a $K3$ surface (algebraic, complex) that has infinitely many automorphisms of finite order? Many K3 surfaces have infinite automorphism groups. In particular, all K3 surfaces of Picard ...
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### Automorphisms of a K3 surface

I was studying the following algebraic surface in $\mathbb{P}^5$ defined by the following three quadrics: \begin{cases} x^2 + xy + y^2=w^2\\ x^2 + 3xz + z^2=t^2\\ y^2 + 5yz + z^2=s^2. \...
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Consider a minimal smooth conic bundle $S$ of dimension two. Assume that there are two curves $C,F$ on $S$ such that $C^2 < 0$ and $F^2 = 0$. Let $D$ be a pseudoeffective divisor on $S$ such that $... • 351 1 vote 0 answers 61 views ### Elliptic fibrations on some Kummer surface in characteristic$2$In the question I ask about one elliptic fibration on the surface $$K\!: y^2 + x_1x_2y = (x_1x_2)^2(x_1 + x_2 + 1) + (x_1 + x_2)^2.$$ over a finite field$\mathbb{F}_q$of characteristic$2$such ... • 1,539 3 votes 0 answers 137 views ### Log canonical surface with an elliptic singularity I would like to know if there is an example as follows:$X$is a log canonical surface and$x \in X$is an elliptic singularity such that The minimal resolution of$x$is a circle of rational curves (... • 353 2 votes 0 answers 101 views ### Semi-stable sheaves on quadric surface https://downloads.hindawi.com/journals/tswj/2014/346126.pdf In this paper, Stable sheaves on a smooth quadric surface with linear Hilbert bipolynomials(E. Ballico and S.Huh), I have a question. On the ... • 63 2 votes 1 answer 277 views ### When does the Hirzebruch surface have a nef anticanonical divisor? Let$\mathcal H_r=\mathbb P (\mathcal O_{\mathbb P^1}\oplus \mathcal O_{\mathbb P^1}(r))$be a Hirzebruch surface for some$r\in\mathbb Z$. As a toric variety, the fan structure is spanned by$(-1,0)$,... • 2,641 1 vote 0 answers 215 views ### Rational and rationally chain connected surfaces A projective variety$X$over the complex numbers is rationally connected if two general points of$X$can be joined by a rational curve in$X\$, and rationally chain connected if two general points of ... 