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Fabrizio Catanese

Publications and source records attributed to Fabrizio Catanese.

At least 37 records · Page 2Linked to original sources
arXiv2023

On the components of the Main Stream of the moduli space of surfaces of general type with $p_g=q=2$

We give first an easy construction of surfaces with $p_g=q=2, K^2=5$ and Albanese map of degree $3$, describing an irreducible connected component of the moduli space of surfaces of general type, which we show to be the only one of the Main Stream with these invariants and satisfying a mild condition. We call it the family of CHPP surfaces, since it contains the family constructed by Chen and Hacon, and coincides with the one considered by Penegini and Polizzi. We also give an easy construction of an irreducible connected component of the moduli space of surfaces of general type with $p_g=q=2, K^2=6$ and Albanese map of degree $4$, which we call the family of PP4 surfaces since it contains the family constructed by Penegini and Polizzi. Finally, we answer a question posed by Chen and Hacon, via three families of surfaces with $p_g=q$ whose Tschirnhaus module has a kernel realization with quotient a nontrivial homogeneous bundle. Two families have $p_g=q=3$, the third is a new family of surfaces with $p_g=q=2, K^2=6$ and Albanese map of degree $3$ (the existence of this family is based on arXiv:2212.14877, joint work of the second author with Edoardo Sernesi) .

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arXiv2023

Manifolds with trivial Chern classes I: Hyperelliptic Manifolds and a question by Severi

We give a negative answer to a question posed by Severi in 1951, whether the Abelian Varieties are the only projective manifolds with trivial Chern classes. By Yau' s celebrated result, compact Kähler manifolds with trivial Chern classes must be flat, that is, they belong to the class of Hyperelliptic Manifolds (quotients $T/G$ of a complex torus $T$ by the free action of a finite group $G$). We exhibit simple examples of projective Hyperelliptic Manifolds which are not Abelian varieties and whose Chern classes are zero not only in integral cohomology, but also in the Chow ring. We prove moreover that the Bagnera-de Franchis manifolds (quotients $T/G$ as above but where the group $ G$ is cyclic) have topologically trivial tangent bundle. Our results naturally lead to the question of classifying all compact Kähler manifolds with topologically trivial tangent bundle, and all the counterexamples to Severi's question.

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arXiv2022

The Hesse pencil and polarizations of type $(1,3)$ on Abelian surfaces

In this short note we prove two theorems, the first one is a sharpening of a result of Lange and Sernesi: the discriminant curve W of a general Abelian surface $A$ endowed with an irreducible polarization $D$ of type $(1,3)$ is an irreducible curve of degree $18$ whose singularities are exactly $36$ nodes and $72$ cusps. Moreover, we analyze the degeneration of the discriminant curve $W$ and its singularities as $A$ tends to the product of two equal elliptic curves. The second theorem, using the first one in order to prove a transversality assertion, shows that the general element of a family of surfaces constructed by Alessandro and Catanese is a smooth surface, thereby proving the existence of a new family of minimal surfaces of general type with $p_g=q=2, K^2=6$ and Albanese map of degree $3$.

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arXiv2022

Pluricanonical Maps and the Fujita Conjecture

We describe examples showing the sharpness of Fujita's conjecture on adjoint bundles also in the general type case, and use these examples to formulate related bold conjectures on pluricanonical maps of varieties of general type.

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arXiv2022

Singularities of normal quartic surfaces III (char=2, non-supersingular)

We show that the maximal number of singular points of a normal quartic surface $X \subset \mathbb{P}^3_K$ defined over an algebraically closed field $K$ of characteristic 2 is at most 12, if the minimal resolution of $X$ is not a supersingular K3 surface. We also provide a family of explicit examples, valid in any characteristic.

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arXiv2022

Singularities of normal quartic surfaces II (char=2)

We show, in this second part, that the maximal number of singular points of a quartic surface $X \subset \mathbb{P}^3_K$ defined over an algebraically closed field $K$ of characteristic 2 is at most 14, and that, if we have 14 singularities, these are nodes and moreover the minimal resolution of $X$ is a supersingular K3 surface. We produce an irreducible component, of dimension 24, of the variety of quartics with 14 nodes. We also exhibit easy examples of quartics with 7 $A_3$-singularities.

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arXiv2022

Singularities of normal quartic surfaces I (char=2)

We show, in this first part, that the maximal number of singular points of a normal quartic surface $X \subset \mathbb{P}^3_K$ defined over an algebraically closed field $K$ of characteristic $2$ is at most $16$. We produce examples with $14$, respectively $12$, singular points and show that, under several geometric assumptions ($\mathfrak S_4$-symmetry, or behaviour of the Gauss map, or structure of tangent cone at one of the singular points $P$, separability/inseparability of the projection with centre $P$), we can obtain smaller upper bounds for the number of singular points of $X$.

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arXiv2021

The number of singular points of quartic surfaces (char=2)

We show that the maximal number of singular points of a normal quartic surface $X \subset \mathbb{P}^3_K$ defined over an algebraically closed field $K$ of characteristic $2$ is at most $20$, and that if equality is attained, then the minimal resolution of $X$ is a supersingular K3 surface and the singular points are $20$ nodes. We produce examples with 14 nodes. In a sequel to this paper (in two parts, the second in collaboration with Matthias Schütt) we show that the optimal bound is indeed 14, and that if equality is attained, then the minimal resolution of $X$ is a supersingular K3 surface and the singular points are $14$ nodes. We also obtain some smaller upper bounds under several geometric assumptions holding at one of the singular points $P$ (structure of tangent cone, separability/inseparability of the projection with centre $P$).

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arXiv2021

Canonical Maps of general Hypersurfaces in Abelian Varieties

The main theorem of this paper is that, for a general pair $(A,X)$ of an (ample) Hypersurface $X$ in an Abelian Variety $A$, the canonical map $Φ_X$ of $X$ is birational onto its image if the polarization given by $X$ is not principal (i.e., its Pfaffian $d$ is not equal to $1$). We also show that, setting $g = dim (A)$, and letting $d$ be the Pfaffian of the polarization given by $X$, then if $X$ is smooth and $$Φ_X : X \rightarrow \mathbb{P}^{N:=g+d-2}$$ is an embedding, then necessarily we have the inequality $ d \geq g + 1$, equivalent to $N : = g+d-2 \geq 2 \ dim(X) + 1.$ We also formulate the following interesting conjecture, motivated by work of the second author: if $ d \geq g + 1,$ then, for a general pair $(A,X)$, $Φ_X$ is an embedding.

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arXiv2021

Kummer quartic surfaces, strict self-duality, and more

In this paper we first show that each Kummer quartic surface (a quartic surface $X$ with 16 singular points) is, in canonical coordinates, equal to its dual surface, and that the Gauss map induces a fixpoint free involution $γ$ on the minimal resolution $S$ of $X$. Then we study the corresponding Enriques surfaces $S/ γ$. We also describe in detail the remarkable properties of the most symmetric Kummer quartic, which we call the Cefalú quartic. We also investigate the Kummer quartic surfaces whose associated Abelian surface is isogenous to a product of elliptic curves through an isogeny with kernel $(\mathbb{Z}/2)^2$, and show the existence of polarized nodal K3 surfaces $X$ of any degree $d=2k$ with the maximal number of nodes, such that $X$ and its nodes are defined over $\mathbb{R}$. We take then as parameter space for Kummer quartics an open set in $\mathbb{P}^3$, parametrizing nondegenerate $(16_6, 16_6)$-configurations, and compare with other parameter spaces. We also extend to positive characteristic some results which were previously known over $\mathbb{C}$. We end with a section devoted to remarks on normal cubic surfaces, and providing some other examples of strictly selfdual hypersurfaces.

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arXiv2021

On topologically trivial automorphisms of compact Kähler manifolds and algebraic surfaces

In this paper, we investigate automorphisms of compact Kähler manifolds with different levels of topological triviality. In particular, we provide several examples of smooth complex projective surfaces X whose groups of $C^\infty$-isotopically trivial automorphisms, resp. cohomologically trivial automorphisms, have a number of connected components which can be arbitrarily large.

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arXiv2020

Teichmüller spaces of Generalized Hyperelliptic Manifolds

In this paper we achieve a description of the connected components of Teichmüller space corresponding to Generalized Hyperelliptic Manifolds $X$. These are the quotients $ X = T/G$ of a complex torus $T$ by the free action of a finite group $G$, and they are also the Kähler classifying spaces for a certain class of Euclidean cristallographic groups $Γ$, the ones which are torsion free and even.

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arXiv2020

$\mathfrak S_5$-equivariant syzygies for the Del Pezzo Surface of Degree 5

The Del Pezzo surface $Y$ of degree 5 is the blow up of the plane in 4 general points, embedded in $\mathbb{P}^5$ by the system of cubics passing through these points. It is the simplest example of the Buchsbaum-Eisenbud theorem on arithmetically-Gorenstein subvarieties of codimension 3 being Pfaffian. Its automorphism group is the symmetric group $\mathfrak S_5$. We give canonical explicit $\mathfrak S_5$-invariant Pfaffian equations through a $6 \times 6$ antisymmetric matrix. We give concrete geometric descriptions of the irreducible representations of $\mathfrak S_5$. Finally, we give $\mathfrak S_5$-invariant equations for the embedding of $Y$ inside $(\mathbb{P}^1)^5$, and show that they have the same Hilbert resolution as for the Del Pezzo of degree $4$.

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arXiv2020

The Moduli Space of smooth Ample Hypersurfaces in Abelian Varieties

We give a characterizaton of smooth ample Hypersurfaces in Abelian Varieties and also describe an irreducible connected component of their moduli space: it consists of the Hypersurfaces of a given polarization type, plus the iterated univariate coverings of normal type (of the same polarization type). The above manifolds yield also a connected component of the open set of Teichmüller space consisting of Kähler complex structures.

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arXiv2019

Rigid Group Actions on Complex Tori are Projective (after Ekedahl)

In this note we give a detailed proof of a result (sketched by Torsten Ekedahl in a discussion with the first author several years ago), describing complex tori admitting a rigid group action and showing explicitly their projectivity and their structure in terms of CM-fields. In the appendix, joint with Benoit Claudon, we show, using a method of Green-Voisin, that all group actions on complex tori deform to projective ones.

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arXiv2019

Fibred algebraic surfaces and commutators in the Symplectic group

We describe the minimal number of critical points and the minimal number $s$ of singular fibres for a non isotrivial fibration of a surface $S$ over a curve $B$ of genus $1$, constructing a fibration with $s=1$ and irreducible singular fibre with $4$ nodes. Then we consider the associated factorizations in the mapping class group and in the symplectic group. We describe explicitly which products of transvections on homologically independent and disjoint circles are a commutator in the Symplectic group $Sp (2g, \mathbb{Z})$.

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arXiv2019

The double point formula with isolated singularities and canonical embeddings

Motivated by the embedding problem of canonical models in small codimension, we extend Severi's double point formula to the case of surfaces with rational double points, and we give more general double point formulae for varieties with isolated singularities. A concrete application is for surfaces with geometric genus $p_g=5$: the canonical model is embedded in $\mathbb{P}^4$ if and only if we have a complete intersection of type $(2,4)$ or $(3,3)$.

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