arXiv subjects
Publications and source records attributed to Francesco Polizzi.
We give a survey of our previous work on relatively minimal isotrivial fibrations $α\colon X \to C$, where $X$ is a smooth, projective surface and $C$ is a curve. In particular, we consider two inequalities involving the numerical invariants $K_X^2$ and $χ(\mathcal{O}_X)$ and we illustrate them by means of several examples and counterexamples.
We consider the deformation spaces of some singular product-quotient surfaces $X=(C_1 \times C_2)/G$, where the curves $C_i$ have genus 3 and the group $G$ is isomorphic to $\mathbb{Z}_4$. As a by-product, we give a new construction of Todorov surfaces with $p_g=1$, $q=0$ and $2\le K^2\le 8$ by using $\mathbb{Q}$-Gorenstein smoothings.
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.
We classify minimal surfaces $S$ of general type with $p_g=q=2$ and $K_S^2=6$ whose Albanese map is a generically finite double cover. We show that the corresponding moduli space is the disjoint union of three generically smooth, irreducible components $\mathcal{M}_{Ia}$, $\mathcal{M}_{Ib}$, $\mathcal{M}_{II}$ of dimension 4, 4, 3, respectively. The general surface $S$ contains a smooth elliptic curve $Z$ such that $Z^2=-2$, which is contracted by the Albanese map and which is preserved by any first-order deformation.
We consider minimal surfaces of general type with $p_g = 2$, $q = 1$ and $K^2 = 5$. We provide a stratification of the corresponding moduli space and we give some bounds for the number and the dimensions of its irreducible components.
Two proper polynomial maps $f_1, \,f_2 \colon \mC^n \lr \mC^n$ are said to be \emph{equivalent} if there exist $Φ_1,\, Φ_2 \in \textrm{Aut}(\mC^n)$ such that $f_2=Φ_2 \circ f_1 \circ Φ_1$. In this article we investigate proper polynomial maps of topological degree $d \geq 2$ up to equivalence. In particular we describe some of our recent results in the case $n=2$ and we partially extend them in higher dimension.
In this paper we investigate the numerical properties of relatively minimal isotrivial fibrations $φ\colon X \lr C$, where $X$ is a smooth, projective surface and $C$ is a curve. In particular we prove that, if $g(C) \geq 1$ and $X$ is neither ruled nor isomorphic to a quasi-bundle, then $K_X^2 \leq 8 χ(\mO_X)-2$; this inequality is sharp and if equality holds then $X$ is a minimal surface of general type whose canonical model has precisely two ordinary double points as singularities. Under the further assumption that $K_X$ is ample, we obtain $K_X^2 \leq 8 χ(\mO_X)-5$ and the inequality is also sharp. This improves previous results of Serrano and Tan.
Two proper polynomial maps $f_1, f_2 \colon \mathbb{C}^2 \longrightarrow \mathbb{C}^2$ are said to be \emph{equivalent} if there exist $Φ_1, Φ_2 \in \textrm{Aut}(\mathbb{C}^2)$ such that $f_2=Φ_2 \circ f_1 \circ Φ_1$. We investigate proper polynomial maps of arbitrary topological degree $d \geq 2$ up to equivalence. Under the further assumption that the maps are Galois coverings we also provide the complete description of equivalence classes. This widely extends previous results obtained by Lamy in the case $d=2$.
A smooth, projective surface $S$ is called a $\emph{standard isotrivial fibration}$ if there exists a finite group $G$ which acts faithfully on two smooth projective curves $C$ and $F$ so that $S$ is isomorphic to the minimal desingularization of $T:=(C \times F)/G$. Standard isotrivial fibrations of general type with $p_g=q=1$ have been classified in \cite{Pol07} under the assumption that $T$ has only Rational Double Points as singularities. In the present paper we extend this result, classifying all cases where $S$ is a minimal model. As a by-product, we provide the first examples of minimal surfaces of general type with $p_g=q=1$, $K_S^2=5$ and Albanese fibration of genus 3. Finally, we show with explicit examples that the case where $S$ is not minimal actually occurs.
A smooth, projective surface $S$ of general type is said to be a \emph{standard isotrivial fibration} if there exist a finite group $G$ which acts faithfully on two smooth projective curves $C$ and $F$ so that $S$ is isomorphic to the minimal desingularization of $T:=(C \times F)/G$. If $T$ is smooth then $S=T$ is called a $\emph{quasi-bundle}$. In this paper we classify the standard isotrivial fibrations with $p_g=q=1$ which are not quasi-bundles, assuming that all the singularities of $T$ are rational double points. As a by-product, we provide several new examples of minimal surfaces of general type with $p_g=q=1$ and $K_S^2=4,6$.
A smooth algebraic surface $S$ is said to be \emph{isogenous to a product of unmixed type} if there exist two smooth curves $C, F$ and a finite group $G$, acting faithfully on both $C$ and $F$ and freely on their product, so that $S=(C \times F)/G$. In this paper we classify the surfaces of general type with $p_g=q=1$ which are isogenous to an unmixed product, assuming that the group $G$ is abelian. It turns out that they belong to four families, that we call surfaces of type $I, II, III, IV$. The moduli spaces $\mathfrak{M}_{I}, \mathfrak{M}_{II}, \mathfrak{M}_{IV}$ are irreducible, whereas $\mathfrak{M}_{III}$ is the disjoint union of two irreducible components. In the last section we start the analysis of the case where $G$ is not abelian, by constructing several examples.
A projective surface S is said to be isogenous to a product if there exist two smooth curves C, F and a finite group G acting freely on C \times F so that S=(C \times F)/G. In this paper we classify all surfaces with p_g=q=1 which are isogenous to a product.
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.
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 $ϕ$ of $S$ is composed with the involution $σ$ induced on $S$ by $τ\times id: C \times F \longrightarrow C \times F$, where $τ$ 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.