arXiv subjects
Publications and source records attributed to Filippo F. Favale.
Let X be a smooth complex irreducible projective variety of dimension $n \geq 2$ and $H$ be an ample line bundle on $X$. In this paper, we construct families of $μ_H$-stable vector bundles on $X$ having fixed determinant and rank $r$, which are generated by $r+1$ global sections, parametrized by Grassmanian varieties. This gives into the corresponding moduli spaces special subvarieties birational to Grassmannian.
We study the problem of the irreducibility of the Hessian variety $\mathcal{H}_f$ associated with a smooth cubic hypersurface $V(f)\subset \mathbb{P}^n$. We prove that when $n\leq5$, $\mathcal{H}_f$ is normal and irreducible if and only if $f$ is not of Thom-Sebastiani type, i.e., roughly, one can not separate its variables. This also generalizes a result of Beniamino Segre dealing with the case of cubic surfaces. The geometric approach is based on the study of the singular locus of the Hessian variety and on infinitesimal computations arising from a particular description of these singularities.
In this paper we deal with Brill-Noether theory for higher-rank sheaves on a polarized nodal reducible curve $(C,\underline{w})$ following the ideas of [arXiv:alg-geom/9511003v1]. We study the Brill-Noether loci of $\underline{w}$-stable depth one sheaves on $C$ having rank $r$ on all irreducible components and having small slope. In analogy with what happens in the smooth case, we prove that these loci are closely related to BGN extensions. Moreover, we produce irreducible components of the expected dimension for these Brill-Noether loci.
In this paper, we exploit some geometric-differential techniques to prove the strong Lefschetz property in degree $1$ for a complete intersection standard Artinian Gorenstein algebra of codimension $6$ presented by quadrics. We prove also some strong Lefschetz properties for the same kind of Artinian algebras in higher codimensions. Moreover, we analyze some loci that come naturally into the picture of "special" Artinian algebras: for them, we give some geometric descriptions and show a connection between the non emptiness of the so-called non-Lefschetz locus in degree $1$ and the "lifting" of a weak Lefschetz property to an algebra from one of its quotients.
Gordan and Noether proved in their fundamental theorem that an hypersurface $X=V(F)\subseteq \mathbb{P}^n$ with $n\leq 3$ is a cone if and only if $F$ has vanishing hessian (i.e. the determinant of the Hessian matrix). They also showed that the statement is false if $n\geq 4$, by giving some counterexamples. Since their proof, several others have been proposed in the literature. In this paper we give a new one by using a different perspective which involves the study of standard Artinian Gorenstein $\mathbb{K}$-algebras and the Lefschetz properties. As a further application of our setting, we prove that a standard Artinian Gorenstein algebra $R=\mathbb{K}[x_0,\dots,x_4]/J$ with $J$ generated by a regular sequence of quadrics has the strong Lefschetz property. In particular, this holds for Jacobian rings associated to smooth cubic threefolds.
Let $(C,\underline{w})$ be a polarized nodal reducible curve. In this paper we consider coherent systems of type $(r,d,k)$ on $C$ with $k < r$. We prove that the moduli spaces of $(\underline{w},α)$-stable coherent systems stabilize for large $α$ and we generalize several results known for the irreducible case when we chose a good polarization. Then, we study in details the components of moduli spaces containing coherent systems arising from locally free sheaves.
Let $C$ be a curve with two smooth components and a single node. Let $\mathcal{U}_C(r,w,χ)$ be the moduli space of $w$-semistable classes of depth one sheaves on $C$ having rank $r$ on both components and Euler characteristic $χ$. In this paper, under suitable assumptions, we produce a projective bundle over the product of the moduli spaces of semistable vector bundles of rank $r$ on each components and we show that it is birational to an irreducible component of $\mathcal{U}_C(r,w,χ)$. Then we prove the rationality of the closed subset containing vector bundles with given fixed determinant.
Let $C$ be a polarized nodal curve of compact type. In this paper we study coherent systems $(E,V)$ on $C$ given by a depth one sheaf $E$ having rank $r$ on each irreducible component of $C$ and a subspace $V \subset H^0(E)$ of dimension $k$. Moduli spaces of stable coherent systems have been introduced by King and Newstead and depend on a real parameter $α$. We show that when $k \geq r$, these moduli spaces coincide for $α$ big enough. Then we deal with the case $k=r+1$: when the degrees of the restrictions of $E$ are big enough we are able to describe an irreducible component of this moduli space by using the dual span construction.
We give a method for the computation of the plurigenera of a product-quotient manifold. We give two different types of applications to it: to the construction of Calabi-Yau threefolds and to the determination of the minimal model of a product-quotient surface of general type.
In this paper we investigate generalized theta divisors $Θ_r$ in the moduli spaces $\mathcal{U}_C(r,r)$ of semistable vector bundles on a curve $C$ of genus $2$. We provide a desingularization $Φ$ of $Θ_r$ in terms of a projective bundle $π:\mathbb{P}(\mathcal{V})\to\mathcal{U}_C(r-1,r)$ which parametrizes extensions of stable vector bundles on the base by $\mathcal{O}_C$. Then, we study the composition of $Φ$ with the well known theta map $θ$. We prove that, when it is restricted to the general fiber of $π$, we obtain a linear embedding.
Motivated by a conjecture of Xiao, we study families of coverings of elliptic curves and their corresponding Prym map $Φ$. More precisely, we describe the codifferential of the period map $P$ associated to $Φ$ in terms of the residue of meromorphic $1$-forms and then we use it to give a characterization for the coverings for which the dimension of $\ker(dP)$ is the least possibile. This is useful in order to exclude the existence of non isotrivial fibrations with maximal relative irregularity and thus also in order to give counterexamples to the Xiao's conjecture mentioned above. The first counterexample to the original conjecture, due to Pirola, is then analysed in our framework.
By a theorem of Reider, a twisted bicanonical system, that means a linear system of divisors numerically equivalent to a bicanonical divisor, on a minimal surface of general type, is base point free if $K^2_S \geq 5$. Twisted bicanonical systems with base points are known in literature only for $K^2=1,2$. We prove in this paper that all surfaces in a family of surfaces with $K^2=3$ constructed in a previous paper with G. Bini and J. Neves have a twisted bicanonical system (different from the bicanonical system) with two base points. We show that the map induced by the above twisted bicanonical system is birational, and describe in detail the closure of its image and its singular locus. Inspired by this description, we reduced the problem of constructing a minimal surface of general type with $K^2=3$ whose bicanonical system has base points, under some reasonable assumptions, to the problem of constructing a curve in $\mathbb{P}^3$ with certain properties.
We give an explicit example of log Calabi-Yau pairs that are log canonical and have a linearly decreasing Euler characteristic. This is constructed in terms of a degree two covering of a sequence of blow ups of three dimensional projective bundles over the Segre-Hirzebruch surfaces ${\mathbb F}_n$ for every positive integer $n$ big enough.
In this paper we are interested in quotients of Calabi-Yau threefolds with isolated singularities. In particular, we analyze the case when $X/G$ has terminal singularities. We prove that, if $G$ is cyclic of prime order and $X/G$ has terminal singularities, then $G$ has order $2,3$ or $5$.
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.
In this paper, we investigate quotients of Calabi-Yau manifolds Y embedded in Fano varieties X which are products of two del Pezzo surfaces - with respect to groups G that act freely on Y. In particular, we revisit some known examples and we obtain some new Calabi-Yau varieties with small Hodge numbers. The groups G are subgroups of the automorphism group of X, which is described in terms of the automorphism group of the two del Pezzo surfaces.