We introduce the notion of a simple fibration in $(1,2)$ -surfaces – that is, a hypersurface insi... more We introduce the notion of a simple fibration in $(1,2)$ -surfaces – that is, a hypersurface inside a certain weighted projective space bundle over a curve such that the general fibre is a minimal surface of general type with $p_g=2$ and $K^2=1$ . We prove that almost all Gorenstein simple fibrations over the projective line with at worst canonical singularities are canonical threefolds ‘on the Noether line’ with $K^3=\frac 43 p_g-\frac {10}3$ , and we classify them. Among them, we find all the canonical threefolds on the Noether line that have previously appeared in the literature. The Gorenstein simple fibrations over ${\mathbb {P}}^1$ are Cartier divisors in a toric $4$ -fold. This allows to us to show, among other things, that the previously known canonical threefolds on the Noether line form an open subset of the moduli space of canonical threefolds, that the general element of this component is a Mori Dream Space and that there is a second component when the geometric genus is...
In this work we classify all Riemann surfaces having a quasi large abelian group of automorphisms... more In this work we classify all Riemann surfaces having a quasi large abelian group of automorphisms, i.e. having an abelian group of automorphisms of order strictly bigger than 2(g−1), where g denotes the genus of the Riemann surface.
We settle the first step for the classification of surfaces of general type with K^2 = 8, p_g = 4... more We settle the first step for the classification of surfaces of general type with K^2 = 8, p_g = 4 and q = 0, classifying the even surfaces (K is 2-divisible). The first even surfaces of general type with K^2=8, p_g=4 and q=0 were found by Oliverio as complete intersections of bidegree (6,6) in a weighted projective space P(1,1,2,3,3). In this article we prove that the moduli space of even surfaces of general type with K^2 = 8, p_g = 4 and q = 0 consists of two 35 -dimensional irreducible components intersecting in a codimension one subset (the first of these components is the closure of the open set considered by Oliverio). For the surfaces in the second component the canonical models are always singular, hence we get a new example of generically nonreduced moduli spaces. Our result gives a posteriori a complete description of the half-canonical rings of the above even surfaces. The method of proof is, we believe, the most interesting part of the paper. After describing the graded r...
A mixed quasi-\'etale quotient is the quotient of the product of a curve of genus at least 2 ... more A mixed quasi-\'etale quotient is the quotient of the product of a curve of genus at least 2 with itself by the action of a group which exchanges the two factors and acts freely out of a finite subset. A mixed quasi-\'etale surface is the minimal resolution of its singularities. We produce an algorithm computing all mixed quasi-\'etale surfaces with given geometric genus, irregularity, and self-intersection of the canonical class. We prove that all irregular mixed quasi-\'etale surfaces of general type are minimal. As application, we classify all irregular mixed quasi \'etale surfaces of general type with genus equal to the irregularity, and all the regular ones with K^2>0, thus constructing new examples of surfaces of general type with \chi=1. We mention the first example of a minimal surface of general type with p_g=q=1 and Albanese fibre of genus bigger than K^2.
We classify the subgroups of the automorphism group of the product of 4 projective lines admittin... more We classify the subgroups of the automorphism group of the product of 4 projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi-Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is non-trivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K^2=3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type.
We study a family of surfaces of general type with p_g=q=2 and K^2=7, originally constructed by C... more We study a family of surfaces of general type with p_g=q=2 and K^2=7, originally constructed by Cancian and Frapporti by using the Computer Algebra System MAGMA. We provide an alternative, computer-free construction of these surfaces, that allows us to describe their Albanese map and their moduli space.
Chapters : Old and new inequalities; Surfaces with χ=1 and the bicanonical map; Surfaces with p_g... more Chapters : Old and new inequalities; Surfaces with χ=1 and the bicanonical map; Surfaces with p_g=4; Surfaces isogeneous to a product, Beauville surfaces and the absolute Galois group;Lefschetz pencils and braid monodromies;DEF, DIFF and other equivalence relations.
In this short note we construct unbounded families of minimal surfaces of general type with canon... more In this short note we construct unbounded families of minimal surfaces of general type with canonical map of degree 4 such that the limits of the slopes assume countably many different values among 6+2/3 and 8.
We introduce the notion of a simple fibration in $(1,2)$ -surfaces – that is, a hypersurface insi... more We introduce the notion of a simple fibration in $(1,2)$ -surfaces – that is, a hypersurface inside a certain weighted projective space bundle over a curve such that the general fibre is a minimal surface of general type with $p_g=2$ and $K^2=1$ . We prove that almost all Gorenstein simple fibrations over the projective line with at worst canonical singularities are canonical threefolds ‘on the Noether line’ with $K^3=\frac 43 p_g-\frac {10}3$ , and we classify them. Among them, we find all the canonical threefolds on the Noether line that have previously appeared in the literature. The Gorenstein simple fibrations over ${\mathbb {P}}^1$ are Cartier divisors in a toric $4$ -fold. This allows to us to show, among other things, that the previously known canonical threefolds on the Noether line form an open subset of the moduli space of canonical threefolds, that the general element of this component is a Mori Dream Space and that there is a second component when the geometric genus is...
In this work we classify all Riemann surfaces having a quasi large abelian group of automorphisms... more In this work we classify all Riemann surfaces having a quasi large abelian group of automorphisms, i.e. having an abelian group of automorphisms of order strictly bigger than 2(g−1), where g denotes the genus of the Riemann surface.
We settle the first step for the classification of surfaces of general type with K^2 = 8, p_g = 4... more We settle the first step for the classification of surfaces of general type with K^2 = 8, p_g = 4 and q = 0, classifying the even surfaces (K is 2-divisible). The first even surfaces of general type with K^2=8, p_g=4 and q=0 were found by Oliverio as complete intersections of bidegree (6,6) in a weighted projective space P(1,1,2,3,3). In this article we prove that the moduli space of even surfaces of general type with K^2 = 8, p_g = 4 and q = 0 consists of two 35 -dimensional irreducible components intersecting in a codimension one subset (the first of these components is the closure of the open set considered by Oliverio). For the surfaces in the second component the canonical models are always singular, hence we get a new example of generically nonreduced moduli spaces. Our result gives a posteriori a complete description of the half-canonical rings of the above even surfaces. The method of proof is, we believe, the most interesting part of the paper. After describing the graded r...
A mixed quasi-\'etale quotient is the quotient of the product of a curve of genus at least 2 ... more A mixed quasi-\'etale quotient is the quotient of the product of a curve of genus at least 2 with itself by the action of a group which exchanges the two factors and acts freely out of a finite subset. A mixed quasi-\'etale surface is the minimal resolution of its singularities. We produce an algorithm computing all mixed quasi-\'etale surfaces with given geometric genus, irregularity, and self-intersection of the canonical class. We prove that all irregular mixed quasi-\'etale surfaces of general type are minimal. As application, we classify all irregular mixed quasi \'etale surfaces of general type with genus equal to the irregularity, and all the regular ones with K^2>0, thus constructing new examples of surfaces of general type with \chi=1. We mention the first example of a minimal surface of general type with p_g=q=1 and Albanese fibre of genus bigger than K^2.
We classify the subgroups of the automorphism group of the product of 4 projective lines admittin... more We classify the subgroups of the automorphism group of the product of 4 projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi-Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is non-trivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K^2=3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type.
We study a family of surfaces of general type with p_g=q=2 and K^2=7, originally constructed by C... more We study a family of surfaces of general type with p_g=q=2 and K^2=7, originally constructed by Cancian and Frapporti by using the Computer Algebra System MAGMA. We provide an alternative, computer-free construction of these surfaces, that allows us to describe their Albanese map and their moduli space.
Chapters : Old and new inequalities; Surfaces with χ=1 and the bicanonical map; Surfaces with p_g... more Chapters : Old and new inequalities; Surfaces with χ=1 and the bicanonical map; Surfaces with p_g=4; Surfaces isogeneous to a product, Beauville surfaces and the absolute Galois group;Lefschetz pencils and braid monodromies;DEF, DIFF and other equivalence relations.
In this short note we construct unbounded families of minimal surfaces of general type with canon... more In this short note we construct unbounded families of minimal surfaces of general type with canonical map of degree 4 such that the limits of the slopes assume countably many different values among 6+2/3 and 8.
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Papers by Roberto Pignatelli