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Maria Lucia Fania

Publications and source records attributed to Maria Lucia Fania.

15 recordsLinked to original sources
arXiv2026

On Ulrich bundles on some decomposable threefold scrolls over $\mathbb F_a$

The study of Ulrich bundles provides profound insights into the underlying geometry and derived categories of projective varieties supporting them, yet their existence and modular properties remain sometimes largely obscure. In this paper we investigate moduli spaces of Ulrich bundles, the Ulrich complexity and the (Ulrich) representation type of decomposable threefold scrolls $X$ over Hirzebruch surfaces $\mathbb F_a$, for any integer $a \geqslant 0$. We completely classify Ulrich line bundles on them, in particular establishing that the Ulrich complexity of $X$ is consistently 1. By exploiting the multiple projective bundle structure of $X$, we uncover a novel geometric involution, which is distinct from the standard Ulrich involution and which dictates the behavior of higher-rank extensions and significantly streamlines their modular study. Thus, beyond line bundles, we first construct rank-two Ulrich bundles for several Chern classes, explicitly identifying those that cannot be obtained as suitable (twisted) pullbacks from the base surfaces. We also provide a comprehensive description of their associated moduli spaces, determining their dimension, their generic smoothness and the description of their birational structure. Ultimately, for noteworthy parameter cases, we prove that X is Ulrich wild by establishing the existence of generically smooth modular components of slope-stable Ulrich bundles of any rank $r \geqslant 1$ revealing the unbounded complexity of Ulrich modules supported on these threefold scrolls.

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arXiv2026

Ulrich wildness of some decomposable threefold scrolls over $\mathbb F_a$

The paper deals with Ulrich wildness of decomposable threefold scrolls $X$ over Hirzebruch surfaces $\mathbb{F}_a$, for any $a \geqslant 0$. Our Main Theorem enstablishes that for $a=0$, the moduli space of rank-$r$ Ulrich bundles, for any $r \geqslant 2$ and of given Chern classes, contains a generically smooth, unirational component $\mathcal{M}(r)$ of computed dimension whose general point corresponds to a slope-stable Ulrich bundle; in particular $X$ turns out to be Ulrich wild. When $a \geqslant 1$ and in presence of modular obstructions, $X$ is nevertheless shown to be Ulrich wild too.

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arXiv2024

On some "sporadic" moduli spaces of Ulrich bundles on some 3-fold scrolls over $\mathbb{F}_0$

We investigate on the existence of some "sporadic", rank-$r \geqslant 1$ Ulrich vector bundles on suitable $3$-fold scrolls $X$ over the Hirzebruch surface $\mathbb{F}_0$, which arise as tautological embeddings of projectivization of very-ample vector bundles on $\mathbb{F}_0$ that are uniform in the sense of Brosius and Aprodu--Brinzanescu. Such Ulrich bundles arise as deformations of ``iterative" extensions by means of "sporadic" Ulrich line bundles. We moreover explicitely describe irreducible components of the corresponding "sporadic" moduli spaces of rank $r \geqslant 1$ vector bundles which are Ulrich with respect to the tautological polarization on $X$. In some cases such irreducible components turn out to be a singleton, in some other cases such components are generically smooth, whose positive dimension has been computed and whose general point turns out to be a slope-stable vector bundle.

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arXiv2024

A note on some moduli spaces of Ulrich Bundles

We prove that the modular component $\mathcal M(r)$, constructed in the Main Theorem of a former paper of us (published in Adv. Math on 2024), paramatrizing (isomorphism classes of) Ulrich vector bundles of rank $r$ and given Chern classes, on suitable $3$-fold scrolls $X_e$ over Hirzebruch surfaces $\mathbb{F}_{e\geq 0}$, which arise as tautological embeddings of projectivization of very-ample vector bundles on $\mathbb{F}_e$, is generically smooth and unirational. A stronger result holds for the suitable associated moduli space $\mathcal M_{\mathbb F_e}(r)$ of vector bundles of rank $r$ and given Chern classes on $\mathbb{F}_e$, Ulrich w.r.t. the very ample polarization $c_1({\mathcal E}_e) = \mathcal O_{\mathbb F_e}(3, b_e),$ which turns out to be generically smooth, irreducible and unirational.

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arXiv2023

Ulrich Bundles on some threefold scrolls over $\mathbb{F}_e$

We investigate the existence of Ulrich vector bundles on suitable $3$-fold scrolls $X_e$ over Hirzebruch surfaces $\mathbb{F}_e$, for any integer $e \geqslant 0$, which arise as tautological embeddings of projectivization of very-ample vector bundles on $\mathbb{F}_e$ that are uniform in the sense of Brosius and Aprodu--Brinzanescu. We explicitely describe components of moduli spaces of rank $r \geqslant 1$ vector bundles which are Ulrich with respect to the tautological polarization on $X_e$ and whose general point is a slope-stable, indecomposable vector bundle. We moreover determine the dimension of such components, proving also that they are generically smooth. As a direct consequence of these facts, we also compute the Ulrich complexity of any such $X_e$ and give an effective proof of the fact that these $X_e$'s turn out to be geometrically Ulrich wild. At last, the machinery developed for $3$--fold scrolls $X_e$ allows us to deduce Ulrichness results on rank $r \geqslant 1$ vector bundles on $\mathbb{F}_e$, for any $e \geqslant 0$, with respect to a naturally associated (very ample) polarization.

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arXiv2020

Quadric surfaces in the Pfaffian hypersurface in $\mathbb{P}^{14}$

We study smooth quadric surfaces in the Pfaffian hypersurface in $\mathbb{P}^{14}$ parameterising $6 \times 6$ skew-symmetric matrices of rank at most 4, not intersecting the Grassmannian $\mathbb{G}(1,5)$. Such surfaces correspond to quadratic systems of skew-symmetric matrices of size 6 and constant rank 4, and give rise to a globally generated vector bundle $E$ on the quadric. We analyse these bundles and their geometry, relating them to linear congruences of lines in $\mathbb{P}^5$.

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arXiv2018

Ulrich bundles on three dimensional scrolls

In this paper we construct Ulrich bundles of low rank on three-dimensional scrolls (with respect to the tautological line bundle). We pay special attention to the four types of threefold scrolls in $\mathbb{P}^5$ which were classified in [Ott92].

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arXiv2015

Hilbert schemes of some threefold scrolls over F_e

Hilbert schemes of suitable smooth, projective 3-fold scrolls over the Hirzebruch surface F_e, with e > 1, are studied. An irreducible component of the Hilbert scheme parametrizing such varieties is shown to be generically smooth of the expected dimension and the general point of such a component is described. This article generalizes the study of Hilbert schemes done in arXiv:1110.5464 for e=1.

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arXiv2015

On families of rank-2 uniform bundles on Hirzebruch surfaces and Hilbert schemes of their scrolls

Several families of rank-two vector bundles on Hirzebruch surfaces are shown to consist of all very ample, uniform bundles. Under suitable numerical assumptions, the projectivization of these bundles, embedded by their tautological line bundles as linear scrolls, are shown to correspond to smooth points of components of their Hilbert scheme, the latter having the expected dimension. If e=0,1 the scrolls fill up the entire component of the Hilbert scheme, while for e=2 the scrolls exhaust a subvariety of codimension 1.

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arXiv2012

Hilbert scheme of some threefold scrolls over the Hirzebruch surface F_1

Hilbert schemes of suitable smooth, projective manifolds of low degree which are 3-fold scrolls over the Hirzebruch surface F_1 are studied. An irreducible component of the Hilbert scheme parametrizing such varieties is shown to be generically smooth of the expected dimension and the general point of such a component is described.

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arXiv2010

Vector spaces of skew-symmetric matrices of constant rank

We study the orbits of vector spaces of skew-symmetric matrices of constant rank 2r and type (N+1)x(N+1) under the natural action of SL(N+1), over an algebraically closed field of characteristic zero. We give a complete description of the orbits for vector spaces of dimension 2, relating them to some 1-generic matrices of linear forms. We also show that, for each rank two vector bundle on P^2 defining a triple Veronese embedding of P^2 in G(1,7), there exists a vector space of 8 x 8 skew-symmetric matrices of constant rank 6 whose kernel bundle is the dual of the given rank two vector bundle.

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arXiv2009

Skew-symmetric matrices and Palatini scrolls

We prove that, for m greater than 3 and k greater than m-2, the Grassmannian of m-dimensional subspaces of the space of skew-symmetric forms over a vector space of dimension 2k is birational to the Hilbert scheme of Palatini scrolls in P^(2k-1). For m=3 and k greater than 3, this Grassmannian is proved to be birational to the set of pairs (E,Y), where Y is a smooth plane curve of degree k and E is a stable rank-2 bundle on Y whose determinant is O(k-1).

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arXiv2004

The dimension of the Hilbert scheme of special threefolds

The Hilbert scheme of projective 3-folds of codimension 3 or more that are linear scrolls over the projective plane or over a smooth quadric surface or that are quadric or cubic fibrations over the projective line is studied. All known such threefolds of degree from 7 to 11 are shown to correspond to smooth points of an irreducible component of their Hilbert scheme, whose dimension is computed. A relationship with the locus of good determinantal subschemes is investigated

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arXiv2002

On the Hilbert scheme of Palatini threefolds

We study the Hilbert scheme of Palatini threefolds X in P^5. We prove that such a scheme has an irreducible component containing X which is birational to the Grassmannian G(3,14) and we determine the exceptional locus of the birational map.

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