# Tagged Questions

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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### 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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### 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 ...
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### 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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### 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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### 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-...