On surfaces of general type with pg=q=1,K2=3p_g=q=1, K^2=3

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 $\mathscr{M}_2 \subset \mathscr{M}$ corresponding to surfaces containing a genus 2 pencil, and moreover we show that there exists a non-empty, dense subset $\mathscr{M}^0 \subset \mathscr{M}$ which parametrizes isomorphism classes of surfaces with birational bicanonical map.Comment: 35 pages. To appear in Collectanea Mathematic

Surfaces of general type with pg=q=1,K2=8p_g=q=1, K^2=8 and bicanonical map of degree 2

We classify the minimal algebraic surfaces of general type with $p_g=q=1, K^2=8$ and bicanonical map of degree 2. It will turn out that they are isogenous to a product of curves, so that if $S$ is such a surface then there exist two smooth curves $C, F$ and a finite group $G$ acting freely on $C \times F$ such that $S = (C \times F)/G$. We describe the $C, F$ and $G$ that occur. In particular the curve $C$ is a hyperelliptic-bielliptic curve of genus 3, and the bicanonical map $\phi$ of $S$ is composed with the involution $\sigma$ induced on $S$ by $\tau \times id: C \times F \longrightarrow C \times F$, where $\tau$ is the hyperelliptic involution of $C$. In this way we obtain three families of surfaces with $p_g=q=1, K^2=8$ which yield the first known examples of surfaces with these invariants. We compute their dimension, and we show that they are three smooth and irreducible components of the moduli space $\mathcal{M}$ of surfaces with $p_g=q=1, K^2=8$. For each of these families, an alternative description as a double cover of the plane is also given, and the index of the paracanonical system is computed.Comment: 36 pages. To appear in Transactions of the American Mathematical Societ

Representations of braid groups and construction of projective surfaces

Braid groups are an important and flexible tool used in several areas of science, such as Knot Theory (Alexander's theorem), Mathematical Physics (Yang-Baxter's equation) and Algebraic Geometry (monodromy invariants). In this note we will focus on their algebraic-geometric aspects, explaining how the representation theory of higher genus braid groups can be used to produce interesting examples of projective surfaces defined over the field of complex numbers.Comment: Note written for the Proceedings of the Conference "Group 32 - The 32nd International Colloquium on Group Theoretical Methods in Physics", held on Czech Technical University (Prague) on July 9-13, 201

On surfaces with p_g=q=2, K^2=5 and Albanese map of degree 3

We construct a connected, irreducible component of the moduli space of minimal surfaces of general type with $p_g=q=2$ and $K^2=5$, which contains both examples given by Chen-Hacon and the first author. This component is generically smooth of dimension 4, and all its points parametrize surfaces whose Albanese map is a generically finite triple cover.Comment: 35 pages, 2 figures. Final version, to appear in the Osaka Journal of Mathematic

A pair of rigid surfaces with pg=q=2p_g=q=2 and K2=8K^2=8 whose universal cover is not the bidisk

We construct two complex-conjugated rigid surfaces with $p_g=q=2$ and $K^2=8$ whose universal cover is not biholomorphic to the bidisk. We show that these are the unique surfaces with these invariants and Albanese map of degree $2$, apart the family of product-quotient surfaces constructed by Penegini. This completes the classification of surfaces with $p_g=q=2, K^2=8$ and Albanese map of degree $2$.Comment: Final version. To appear in IMR

On factoriality of threefolds with isolated singularities

We investigate the existence of complete intersection threefolds $X \subset \mathbb{P}^n$ with only isolated, ordinary multiple points and we provide some sufficient conditions for their factoriality.Comment: 18 pages. To appear in the Michigan Mathematical Journa

Finite quotients of surface braid groups and double Kodaira fibrations

Let $\Sigma_b$ be a closed Riemann surface of genus $b$. We give an account of some results obtained in the recent papers \cite{CaPol19, Pol20, PolSab21} and concerning what we call here \emph{pure braid quotients},namely non-abelian finite groups appearing as quotients of the pure braid group on two strands $\mathsf{P}_2(\Sigma_b)$. We also explain how these groups can be used in order to provide new constructions of double Kodaira fibrations.Comment: 23 pages, 3 figures. To appear in "The Art of Doing Algebraic Geometry", a Springer volume dedicated to Ciro Ciliberto. arXiv admin note: text overlap with arXiv:2102.04963, arXiv:2002.01363, arXiv:1905.0317