Related papers: A connected component of the moduli space of surfa…

For a two-dimensional surface in the four-dimensional Euclidean space we introduce an invariant linear map of Weingarten type in the tangent space of the surface, which generates two invariants k and kappa. The condition k = kappa = 0…

We give infinite series of groups Gamma and of compact complex surfaces of general type S with fundamental group Gamma such that 1) Any surface S' with the same Euler number as S, and fundamental group Gamma, is diffeomorphic to S. 2) The…

It is proved that the moduli space of all connected compact orientable embedded minimal affine Lagrangian submanifolds of a complex equiaffine space constitutes an infinite dimensional Frechet manifold (if it is not the empty set). The…

The Eisenstein-Picard modular surface $M$ is the quotient space of the complex hyperbolic plane by the modular group $\rm PU(2,1; \mathbb{Z}[\omega])$. We determine the global topology of $M$ as a 4-orbifold.

We count the connected components in the moduli space of PU(p,q)-representations of the fundamental group for a closed oriented surface. The components are labelled by pairs of integers which arise as topological invariants of the flat…

We construct from a general del Pezzo surface of degree 1 a Gorenstein stable surfaces $X$ with $K_X^2=1$ and $p_g(X)=q(X)=0$. These surfaces are not smoothable but give an open subset of an irreducible component of the moduli space of…

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…

In this paper we construct a new family of simply connected minimal complex surfaces of general type with $p_g=1$, $q=0$, and $K^2=3, 4, 5, 6, 8$ using a $\mathbb{Q}$-Gorenstein smoothing theory. We also reconstruct minimal complex surfaces…

We give many examples in which there exist infinitely many divisorial conditions on the moduli space of polarized K3 surfaces $(S,H)$ of degree $H^2=2g-2$, $g \geq 3$, and Picard number $rk N(S)=\rho(S)=2$ such that for a general K3 surface…

Let $S$ be a smooth minimal surface of general type with a (rational) pencil of hyperelliptic curves of minimal genus $g$. We prove that if $K_S^2<4\chi(\mathcal O_S)-6,$ then $g$ is bounded. The surface $S$ is determined by the branch…

We use function field analytic number theory to establish the irreducibility and dimension of the moduli space that parameterises morphisms of fixed degree from $\mathbb{P}^2$ to an arbitrary smooth hypersurface of sufficiently small…

We show that a family of minimal surfaces of general type with p_g = 0, K^2=7, constructed by Inoue in 1994, is indeed a connected component of the moduli space: indeed that any surface which is homotopically equivalent to an Inoue surface…

We prove that the bicanonical system on a surface of general type with K^2=4 has no base components and describe clusters contracted by 4K_X for a numerical Godeaux surface and 3K_X for a numerical Campedelli surface.

The moduli space $\mathscr{M}$ of surfaces of general type with $p_g=q=1, K^2=g=3$ (where $g$ is the genus of the Albanese fibration) was constructed by Catanese and Ciliberto in \cite{CaCi93}. In this paper we characterize the subvariety…

Numerical Campedelli surfaces are minimal surfaces of general type with p_g=0 (and so q=0) and K^2=2. Although they have been studied by several authors, their complete classification is not known. In this paper we classify numerical…

We prove that the Gauss curvature and the curvature of the normal connection of any minimal surface in the four dimensional Euclidean space satisfy an inequality, which generates two classes of minimal surfaces: minimal surfaces of general…

We prove that the bicanonical map of the Cartwright-Steger surface is an embedding. We also discuss two minimal surfaces of general type, both covered by the Cartwright-Steger surface. One has $K^2=2$, $p_g=1$, $\pi_1=\{1\}$ and the other…

In this paper we construct a simply connected, minimal, complex surface of general type with p_g=0 and K^2=2 using a rational blow-down surgery and Q-Gorenstein smoothing theory.

We investigate the topological structures of Galois covers of surfaces of minimal degree (i.e., degree n) in n+1 dimensional complex projective space. We prove that for n is greater than or equal to 5, the Galois covers of any surfaces of…

We construct a new family of simply connected minimal complex surfaces with $p_g=1$, $q=0$, and $K^2=8$ using a $\mathbb{Q}$-Gorenstein smoothing theory.