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Fabrizio Catanese

Publications and source records attributed to Fabrizio Catanese.

At least 91 records · Page 5Linked to original sources
arXiv2010

Burniat surfaces III: deformations of automorphisms and extended Burniat surfaces

We continue our investigation of the connected components of the moduli space of surfaces of general type containing the Burniat surfaces, correcting a mistake in part II. We define the family of extended Burniat surfaces with K_S^2 = 4, resp. 3, and prove that they are a deformation of the family of nodal Burniat surfaces with K_S^2 = 4, resp. 3. We show that the extended Burniat surfaces together with the nodal Burniat surfaces with K_S^2=4 form a connected component of the moduli space. We prove that the extended Burniat surfaces together with the nodal Burniat surfaces with K_S^2=3 form an irreducible open set in the moduli space. Finally we point out an interesting pathology of the moduli space of surfaces of general type given together with a group of automorphisms G. In fact, we show that for the minimal model S of a nodal Burniat surface (G = (\ZZ/2 \ZZ)^2) we have Def(S,G) \neq Def(S), whereas for the canonical model X it holds Def(X,G) = Def(X). All deformations of S have a G-action, but there are different deformation types for the pairs (S,G) of the minimal models S together with the G-action, while the pairs (X,G) have a unique deformation type.

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arXiv2010

Irreducibility of the space of cyclic covers of algebraic curves of fixed numerical type and the irreducible components of $Sing (\bar{\mathfrak M_g})$

We prove irreducibility for the space of cyclic covers of fixed numerical type between smooth projective curves, and also for the space of cyclic covers of prime order and of fixed numerical-combinatorial type between moduli-stable projective curves. As an application we describe the irreducible components of the singular locus of the compactified moduli space of curves $\bar{\mahfrak M_g}$, extending the work of Cornalba, who described the irreducible components of the singular locus of the moduli space of curves $\mahfrak M_g}$.

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arXiv2010

Surfaces of general type with geometric genus zero: a survey

In the last years there have been several new constructions of surfaces of general type with $p_g=0$, and important progress on their classification. The present paper presents the status of the art on surfaces of general type with $p_g=0$, and gives an updated list of the existing surfaces, in the case where $K^2= 1,...,7$. It also focuses on certain important aspects of this classification.

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arXiv2010

Moduli spaces of surfaces and monodromy invariants

This paper is a survey of the authors' recent results on "abc-surfaces" and the monodromy of their natural Lefschetz fibrations and projections to P^1 x P^1, see (arXiv:0910.2142). The results being surveyed explore various fundamental questions about complex deformation vs. symplectomorphism vs. diffeomorphism from the perspective of one explicit family of examples; the paper also contains some remarks which were not in earlier papers.

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arXiv2009

On volume preserving complex structures on real tori

A basic problem in the classification theory of compact complex manifolds is to give simple characterizations of complex tori. It is well known that a compact Kähler manifold $X$ homotopically equivalent to a a complex torus is biholomorphic to a complex torus. The question whether a compact complex manifold $X$ diffeomorphic to a complex torus is biholomorphic to a complex torus has a negative answer due to a construction by Blanchard and Sommese. Their examples have however negative Kodaira dimension, thus it makes sense to ask the question whether a compact complex manifold $X$ with trivial canonical bundle which is homotopically equivalent to a complex torus is biholomorphic to a complex torus. In this paper we show that the answer is positive for complex threefolds satisfying some additional condition, such as the existence of a non constant meromorphic function.

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arXiv2009

Burniat surfaces II: secondary Burniat surfaces form three connected components of the moduli space

We prove in one go that each of the 4 families of Burniat surfaces with K^2 = 6,5,4, is a connected component of the moduli space of surfaces of general type. We prove also the rationality of each component. In the nodal case (one of the two families for K^2_S = 4) a very surprising and new phenomenon occurs. Both the moduli space for the minimal models and the Gieseker moduli space for canonical models are everywhere non reduced. But the nilpotence order is higher for the first.

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arXiv2009

Quotients of products of curves, new surfaces with $p_g=0$ and their fundamental groups

The first main purpose of this paper is to contribute to the existing knowledge about the complex projective surfaces $S$ of general type with $p_g(S) = 0$ and their moduli spaces, constructing 19 new families of such surfaces with hitherto unknown fundamental groups. We also provide a table containing all the known such surfaces with K^2 <=7. Our second main purpose is to describe in greater generality the fundamental groups of smooth projective varieties which occur as the minimal resolutions of the quotient of a product of curves by the action of a finite group. We classify, in the two dimensional case, all the surfaces with q=p_g = 0 obtained as the minimal resolution of such a quotient, having rational double points as singularities. We show that all these surfaces give evidence to the Bloch conjecture.

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arXiv2009

Moduli spaces and braid monodromy types of bidouble covers of the quadric

Bidouble covers $π: S \mapsto Q$ of the quadric Q are parametrized by connected families depending on four positive integers a,b,c,d. In the special case where b=d we call them abc-surfaces. Such a Galois covering $π$ admits a small perturbation yielding a general 4-tuple covering of Q with branch curve $\De$, and a natural Lefschetz fibration obtained from a small perturbation of the composition of $ π$ with the first projection. We prove a more general result implying that the braid monodromy factorization corresponding to $\De$ determines the three integers a,b,c in the case of abc-surfaces. We introduce a new method in order to distinguish factorizations which are not stably equivalent. This result is in sharp contrast with a previous result of the first and third author, showing that the mapping class group factorizations corresponding to the respective natural Lefschetz pencils are equivalent for abc-surfaces with the same values of a+c, b. This result hints at the possibility that abc-surfaces with fixed values of a+c, b, although diffeomorphic but not deformation equivalent, might be not canonically symplectomorphic.

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arXiv2009

Burniat surfaces I: fundamental groups and moduli of primary Burniat surfaces

This is the first in a series of articles on the moduli spaces of Burniat surfaces. We prove that Burniat surfaces are Inoue surfaces and we calculate their fundamental groups. As a main result of this paper we prove that every smooth projective surface which is homotopically equivalent to a primary Burniat surface (i.e., a Burniat surface with K^2 = 6) is indeed a primary Burniat surface.

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arXiv2008

On varieties whose universal cover is a product of curves

We investigate a necessary condition for a compact complex manifold X of dimension n in order that its universal cover be the Cartesian product $C^n$ of a curve $C = \PP^1 or \HH$: the existence of a semispecial tensor $ω$. A semispecial tensor is a non zero section $ 0 \neq ω\in H^0(X, S^nΩ^1_X (-K_X) \otimes η) $), where $η$ is an invertible sheaf of 2-torsion (i.e., $η^2\cong \hol_X$). We show that this condition works out nicely, as a sufficient condition, when coupled with some other simple hypothesis, in the case of dimension $n= 2$ or $ n= 3$; but it is not sufficient alone, even in dimension 2. In the case of Kähler surfaces we use the above results in order to give a characterization of the surfaces whose universal cover is a product of two curves, distinguishing the 6 possible cases.

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arXiv2008

Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, II

Let W -> X be a real smooth projective 3-fold fibred by rational curves. J. Kollár proved that, if W(R) is orientable, then a connected component N of W(R) is essentially either a Seifert fibred manifold or a connected sum of lens spaces. Our Main Theorem, answering in the affirmative three questions of Kollár, gives sharp estimates on the number and the multiplicities of the Seifert fibres and on the number and the torsions of the lens spaces when X is a geometrically rational surface. When N is Seifert fibred over a base orbifold F, our result generalizes Comessatti's theorem on smooth real rational surfaces: F cannot be simultaneously orientable and of hyperbolic type. We show as a surprise that, unlike in Comessatti's theorem, there are examples where F is non orientable, of hyperbolic type, and X is minimal. The technique we use is to construct Seifert fibrations as projectivized tangent bundles of Du Val surfaces.

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arXiv2008

The rationality of certain moduli spaces of curves of genus 3

We show, for each algebraically closed field, the rationality of the following two moduli spaces: M(3,3) parametrizing pairs (C, η) where C has genus 3 and ηis a 3-torsion divisor class, respectively of M(3,<3>) parametrizing pairs (C, <η>) as above and where <η> is the cyclic subgroup of order 3 in Pic_0(C) generated by η.

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arXiv2008

A characterization of surfaces whose universal cover is the bidisk

We show that the universal cover of a compact complex surface $X$ is the bidisk $\HH \times \HH$, or $X$ is biholomorphic to $\PP^1 \times \PP^1$, if and only if $K_X^2 > 0$ and there exists an invertible sheaf $η$ such that $η^2\cong \hol_X$ and $H^0(X, S^2Ω^1_X (-K_X) \otimes η) \neq 0$. The two cases are distinguished by the second plurigenus, $P_2(X)\geq 2$ in the former case, $P_2(X)= 0$ in the latter. We also discuss related questions.

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arXiv2007

Real singular Del Pezzo surfaces and threefolds fibred by rational curves, I

Let W -> X be a real smooth projective threefold fibred by rational curves. Kollár proved that if W(R) is orientable a connected component N of W(R) is essentially either a Seifert fibred manifold or a connected sum of lens spaces. Let k : = k(N) be the integer defined as follows: If g : N -> F is a Seifert fibration, one defines k : = k(N) as the number of multiple fibres of g, while, if N is a connected sum of lens spaces, k is defined as the number of lens spaces different from P^3(R). Our Main Theorem says: If X is a geometrically rational surface, then k <= 4. Moreover we show that if F is diffeomorphic to S^1xS^1, then W(R) is connected and k = 0. These results answer in the affirmative two questions of Kollár who proved in 1999 that k <= 6 and suggested that 4 would be the sharp bound. We derive the Theorem from a careful study of real singular Del Pezzo surfaces with only Du Val singularities.

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arXiv2007

The absolute Galois group acts faithfully on the connected components of the moduli space of surfaces of general type

We show that the Galois group $Gal(\bar{\Q} /\Q)$ operates faithfully on the set of connected components of the moduli spaces of surfaces of general type, and also that for each element $σ\in Gal(\bar{\Q} /\Q)$ different from the identity and from complex conjugation, there is a surface of general type such that $X$ and the Galois conjugate variety $X^σ$ have nonisomorphic fundamental groups. The result was announced by the second author at the Alghero Conference 'Topology of algebraic varieties' in september 2006. Before the present paper was actually written, we received a very interesting preprint by Robert Easton and Ravi Vakil (\cite{e-v}), where it is proven, with a completely different type of examples, that the Galois group $Gal(\bar{\Q} /\Q)$ operates faithfully on the set of irreducible components of the moduli spaces of surfaces of general type. We also give other simpler examples of surfaces with nonisomorphic fundamental groups which are Galois conjugate, hence have isomorphic algebraic fundamental groups.

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arXiv2007

A remarkable moduli space of rank 6 vector bundles related to cubic surfaces

We study the moduli space $\fM^s(6;3,6,4)$ of simple rank 6 vector bundles $\E$ on $\PP^3$ with Chern polynomial $1+3t+6t^2+4t^3$ and properties of these bundles, especially we prove some partial results concerning their stability. We first recall how these bundles are related to the construction of sextic nodal surfaces in $\PP^3$ having an even set of 56 nodes (cf. \cite{CaTo}). We prove that there is an open set, corresponding to the simple bundles with minimal cohomology, which is irreducible of dimension 19 and bimeromorphic to an open set $\fA^0$ of the G.I.T. quotient space of the projective space $\fB:=\{B\in \PP(U^\vee\otimes W\otimes V^\vee)\}$ of triple tensors of type $(3,3,4)$ by the natural action of $SL(W)\times SL(U)$. We give several constructions for these bundles, which relate them to cubic surfaces in 3-space $\PP^3$ and to cubic surfaces in the dual space $(\PP^3)^{\vee}$. One of these constructions, suggested by Igor Dolgachev, generalizes to other types of tensors. Moreover, we relate the socalled {\em cross-product involution} for $(3,3,4)$-tensors, introduced in \cite{CaTo}, with the Schur quadric associated to a cubic surface in $\PP^3$ and study further properties of this involution.

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arXiv2007

Canonical symplectic structures and deformations of algebraic surfaces

We show that a minimal surface of general type has a canonical symplectic structure (unique up to symplectomorphism) which is invariant for smooth deformation. We show that the symplectomorphism type is also invariant for deformations which allow certain normal singularities, provided one remains in the same smoothing component. We use this technique to show that the Manetti surfaces yield examples of surfaces of general type which are not deformation equivalent but are canonically symplectomorphic.

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