A Superficial Working Guide to Deformations and Moduli

This is the first part of a guide to deformations and moduli, especially viewed from the perspective of algebraic surfaces (the simplest higher dimensional varieties). It contains also new results, regarding the question of local homeomorphism between Kuranishi and Teichmueller space, and a survey of new results with Ingrid Bauer, concerning the discrepancy between the deformation of the action of a group G on a minimal models S, respectively the deformation of the action of G on the canonical model X. Here Def(S) maps properly onto Def(X), but the same does not hold for pairs: Def(S,G) does not map properly onto Def(X,G). Indeed the connected components of Def(S), in the case of tertiary Burniat surfaces, only map to locally closed sets. The last section contains anew result on some surfaces whise Albanese map has generic degree equal to 2.Comment: 56 pages, revision to appear in the Handbook of Moduli, in honour of David Mumford, to be published by International press, editors Gavril Farkas and Ian Morrison. The former theorem 29 on moduli spaces for minimal surfaces has been correcte

Deformations of elliptic Calabi--Yau manifolds

The aim of this note is to investigate characterizations and deformations of elliptic Calabi--Yau manifolds, building on earlier works of Wilson and Oguiso. Version 2: References updated and small changes. Version 3: Smoothness conditions removed from several theorems, plus some reorganization. Version 4: Several references added

On the Moduli space of diffeomorphic algebraic surfaces

It is proved that the number of deformation types of complex structures on a fixed oriented smooth four-manifold can be arbitrarily large. The considered examples are locally simple abelian covers of rational surfaces.Comment: Plain Tex, 41page

On the Hilbert scheme of curves in higher-dimensional projective space

In this paper we prove that, for any $n\ge 3$, there exist infinitely many $r\in \N$ and for each of them a smooth, connected curve $C_r$ in $\P^r$ such that $C_r$ lies on exactly $n$ irreducible components of the Hilbert scheme \hilb(\P^r). This is proven by reducing the problem to an analogous statement for the moduli of surfaces of general type.Comment: latex, 12 pages, no figure