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L. Costa

Publications and source records attributed to L. Costa.

At least 19 recordsLinked to original sources
arXiv2025

Some moduli spaces of $\alpha$-stable coherent systems on algebraic surfaces

Let $X$ be a smooth, irreducible, projective algebraic surface, and let $\alpha \in \mathbb{Q}[m]_{>0}$ be a polynomial. In this paper, we determine topological and geometric properties of the moduli space of $\alpha$-stable coherent systems of type $(n; c_{1}, c_{2}, k)$ with $k < n$ on $X$, for sufficiently large values of $\alpha$. We prove that, for $\alpha$ sufficiently large, the moduli space admits a description as a Grassmann bundle over a moduli space of $H$-stable torsion free sheaves. As a consequence, we obtain results on irreducibility, dimension. Our approach relies on establishing a correspondence between $\alpha$-stable coherent systems and extensions of $H$-stable torsion free sheaves by trivial bundles.

math.AG↗
arXiv2024

Cooperation and Federation in Distributed Radar Point Cloud Processing

The paper considers the problem of human-scale RF sensing utilizing a network of resource-constrained MIMO radars with low range-azimuth resolution. The radars operate in the mmWave band and obtain time-varying 3D point cloud (PC) information that is sensitive to body movements. They also observe the same scene from different views and cooperate while sensing the environment using a sidelink communication channel. Conventional cooperation setups allow the radars to mutually exchange raw PC information to improve ego sensing. The paper proposes a federation mechanism where the radars exchange the parameters of a Bayesian posterior measure of the observed PCs, rather than raw data. The radars act as distributed parameter servers to reconstruct a global posterior (i.e., federated posterior) using Bayesian tools. The paper quantifies and compares the benefits of radar federation with respect to cooperation mechanisms. Both approaches are validated by experiments with a real-time demonstration platform. Federation makes minimal use of the sidelink communication channel (20 {\div} 25 times lower bandwidth use) and is less sensitive to unresolved targets. On the other hand, cooperation reduces the mean absolute target estimation error of about 20%.

cs.LG↗
arXiv2024

Brill-Noether Theory of stable vector bundles on ruled surfaces

Let X be a ruled surface over a nonsingular curve C of genus $g\geq0$. Let $M_H:=M_{X,H}(2;c_1,c_2)$ be the moduli space of H-stable rank 2 vector bundles E on X with fixed Chern classes $c_i:=c_i(E)$ for $i=1,2$. The main goal of this paper is to contribute to a better understanding of the geometry of the moduli space $M_H$ in terms of its Brill-Noether locus $W_H^k(2;c_1,c_2)$, whose points correspond to stable vector bundles in $M_H$ having at least k independent sections. We deal with the non-emptiness of this Brill-Noether locus, getting in most of the cases sharp bounds for the values of k such that $W_H^k(2;c_1,c_2)$ is non-empty.

math.AG↗
arXiv2019

Theta divisors and Ulrich bundles on Geometrically ruled surfaces

We consider the following question: for which invariants $g$ and $e$ is there a geometrically ruled surface $S \rightarrow C$ over a curve $C$ of genus $g$ with invariant $e$ such that $S$ is the support of an Ulrich line bundle with respect to a very ample line bundle? A surprising relation between the existence of certain proper Theta divisors on some moduli spaces of vector bundles on $C$ with the existence of Ulrich line bundles on $S$ will be the key to completely solve the above question. The relation is realized by translating the vanishing conditions characterizing Ulrich line bundles to specific geometric conditions on the symmetric powers of the defining vector bundle of a given ruled surface. This general principle leads to some finer existence results of Ulrich line bundles in particular cases. Another focus is on the rank two case where, with very few exceptions, we show the existence of large families of special Ulrich bundles on arbitrary polarized ruled surfaces.

math.AG↗
arXiv2015

Homogeneous Ulrich bundles on Flag manifolds

Let $V$ be a $K$-vector space of dimension $n+1$. In this paper, we focus our attention into the existence of irreducible homogeneous Ulrich bundles on flag manifolds $\FF(p, q,n)$ which parameterizes all chains of linear subspaces $L_{p} \subset L_{q} \subset \PP(V)$ of dimension $p< q$, respectively. We determine all irreducible homogeneous Ulrich bundles on $\FF(0,n-1,n)$ and we prove that there are exactly $2^{n-1}$. Similarly, we prove that $\FF(0,n-2,n)$ and $\FF(1,n-1,n)$ are also the support of irreducible homogeneous Ulrich bundles. On the other hand, we prove that $\FF(0,1,n)$ do not support any irreducible homogeneous Ulrich bundle. We end posing a conjecture concerning the existence of irreducible homogeneous Ulrich bundles on $\FF(p,q,n)$ in terms of $p$ and $q$.

math.AG↗
arXiv2014

Rational families of instanton bundles on $P^{2n+1}$

This paper is devoted to the theory of symplectic instanton bundles on an odd dimensional projective space ${\mathbb P}^{2n+1}$ with $n\ge 2$. We study the 't Hooft instanton bundles introduced by Ottaviani and a new family of instanton bundles which generalizes one introduced on ${\mathbb P}^3$ independently by Rao and Skiti. The main result is the determination of the birational types of the moduli spaces of 't Hooft and of Rao-Skiti instanton bundles, respectively. Assuming a conjecture of Ottaviani, we show that the moduli space of all symplectic instanton bundles on ${\mathbb P}^{2n+1}$ with $n\ge 2$ is reducible.

math.AG↗
arXiv2012

AGATA - Advanced Gamma Tracking Array

The Advanced GAmma Tracking Array (AGATA) is a European project to develop and operate the next generation gamma-ray spectrometer. AGATA is based on the technique of gamma-ray energy tracking in electrically segmented high-purity germanium crystals. This technique requires the accurate determination of the energy, time and position of every interaction as a gamma ray deposits its energy within the detector volume. Reconstruction of the full interaction path results in a detector with very high efficiency and excellent spectral response. The realization of gamma-ray tracking and AGATA is a result of many technical advances. These include the development of encapsulated highly-segmented germanium detectors assembled in a triple cluster detector cryostat, an electronics system with fast digital sampling and a data acquisition system to process the data at a high rate. The full characterization of the crystals was measured and compared with detector-response simulations. This enabled pulse-shape analysis algorithms, to extract energy, time and position, to be employed. In addition, tracking algorithms for event reconstruction were developed. The first phase of AGATA is now complete and operational in its first physics campaign. In the future AGATA will be moved between laboratories in Europe and operated in a series of campaigns to take advantage of the different beams and facilities available to maximize its science output. The paper reviews all the achievements made in the AGATA project including all the necessary infrastructure to operate and support the spectrometer.

physics.ins-det↗
arXiv2011

Derived Category of Fibrations

Let X ->Y be a Zariski locally trivial fibration of smooth complex projective varieties, with fiber F. We give a structure theorem for the derived category of X provided both F and Z have a full strongly exceptional collection of line bundles.

math.AG↗
arXiv2010

Derived category of toric fibrations

The derived category of bounded complexes of coherent sheaves is one of the most important algebraic invariants of a smooth projective variety. An important approach to understand derived categories is to construct full strongly exceptional sequences. The problem of characterizing smooth projective varieties which have a full strongly exceptional collection and investigate whether there is one consisting of line bundles is a classical and important question in Algebraic Geometry. Not all smooth projective varieties have a full strongly exceptional collection of coherent sheaves. In this paper we give a structure theorem for the derived category of a toric fiber bundle X over Z with fiber F provided that F and Z have both a full strongly exceptional collection of line bundles.

math.AG↗
arXiv2010

Frobenius splitting and Derived category of toric varieties

The splitting of the Frobenius direct image of line bundles on toric varieties is used to explicitly construct an orthogonal basis of line bundles in the derived category D^b(X) where X is a Fano toric variety with (almost) maximal Picard number.

math.AG↗
arXiv2009

Stability and Unobstructedness of Syzygy Bundles

It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle $E_{d_1,..., d_n}$ on $\PP^N$ defined as the kernel of a general epimorphism $\xymatrix{ϕ:\cO(-d_1)\oplus...\oplus\cO(-d_n)\ar[r] &\cO}$ is (semi)stable. In this note, we restrict our attention to the case of syzygy bundles $E_{d,n}$ on $\PP^N$ associated to $n$ generic forms $f_1,...,f_n\in K[X_0,X_1,..., X_N]$ of the same degree $d$. Our first goal is to prove that $E_{d,n}$ is stable if $N+1\le n\le\tbinom{d+2}{2}+N-2$. This bound improves, in general, the bound $n\le d(N+1)$ given by G. Hein in \cite{B}, Appendix A. In the last part of the paper, we study moduli spaces of stable rank $n-1$ vector bundles on $\PP^N$ containing syzygy bundles. We prove that if $N+1\le n\le\tbinom{d+2}{2}+N-2$ and $N\ne 3$, then the syzygy bundle $E_{d,n}$ is unobstructed and it belongs to a generically smooth irreducible component of dimension $n\tbinom{d+N}{N}-n^2$, if $N \geq 4$, and $n\tbinom{d+2}{2}+n\tbinom{d-1}{2}-n^2$, if N=2.

math.AG↗
arXiv2008

Ultracold Heteronuclear Fermi-Fermi Molecules

We report on the first creation of ultracold bosonic heteronuclear molecules of two fermionic species, 6Li and 40K, by a magnetic field sweep across an interspecies s-wave Feshbach resonance. This allows us to associate up to 4x10^4 molecules with high efficiencies of up to 50%. Using direct imaging of the molecules, we measure increased lifetimes of the molecules close to resonance of more than 100 ms in the molecule-atom mixture stored in a harmonic trap.

cond-mat.other↗
arXiv2008

Brill-Noether theory for moduli spaces of sheaves on algebraic varieties

Let $X$ be a smooth projective variety of dimension $n$ and let $H$ be an ample line bundle on $X$. Let $M_{X,H}(r;c_1, ..., c_{s})$ be the moduli space of $H$-stable vector bundles $E$ on $X$ of rank $r$ and Chern classes $c_i(E)=c_i$ for $i=1, ..., s:=min\{r,n\}$. We define the Brill-Noether filtration on $M_{X,H}(r;c_1, ..., c_{s})$ as $W_{H}^{k}(r;c_1,..., c_{s})= \{E \in M_{X,H}(r;c_1, ..., c_{s}) | h^0(X,E) \geq k \}$ and we realize $W_{H}^{k}(r;c_1,..., c_{s})$ as the $k$th determinantal variety of a morphism of vector bundles on $M_{X,H}(r;c_1, ..., c_{s})$, provided $H^i(E)=0$ for $i \geq 2$ and $E \in M_{X,H}(r;c_1, ..., c_{s})$. We also compute the expected dimension of $W_{H}^{k}(r;c_1,..., c_{s})$. Very surprisingly we will see that the Brill-Noether stratification allow us to compare moduli spaces of vector bundles on Hirzebruch surfaces stables with respect to different polarizations. We will also study the Brill-Noether loci of the moduli space of instanton bundles and we will see that they have the expected dimension.

math.AG↗
arXiv2006

Cohomological characterization of vector bundles on multiprojective spaces

We show that Horrock's criterion for the splitting of vector bundles on $\PP^n$ can be extended to vector bundles on multiprojective spaces and to smooth projective varieties with the weak CM property (see Definition 3.11). As a main tool we use the theory of $n$-blocks and Beilinson's type spectral sequences. Cohomological characterizations of vector bundles are also showed.

math.AG↗
arXiv2006

m-blocks collections and Castelnuovo-Mumford regularity in multiprojective spaces

The main goal of the paper is to generalize Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on $n$-dimensional smooth projective varieties $X$ with an $n$-block collection $\cB $ which generates the bounded derived category $\cD ^b({\cO}_X$-$mod)$. To this end, we use the theory of $n$-blocks and Beilinson type spectral sequence to define the notion of regularity of a coherent sheaf $F$ on $X$ with respect to the $n$-block collection $\cB $. We show that the basic formal properties of the Castelnuovo-Mumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we compare our definition of regularity with previous ones. In particular, we show that in case of coherent sheaves on $\PP^n$ and for the $n$-block collection $\cB =(\cO_{\PP^n},\cO_{\PP^n} (1), ..., \cO_{\PP^n}(n))$ on $\PP^n$ Castelnuovo-Mumford regularity and our new definition of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a multiprojective space $\PP^{n_1}\times ... \times \PP^{n_r}$ with respect to a suitable $n_1+... +n_r$-block collection and we compare it with the multigraded variant of the Castelnuovo-Mumford regularity given by Hoffman and Wang.

math.AG↗