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Francesco Malaspina

Publications and source records attributed to Francesco Malaspina.

At least 19 recordsLinked to original sources
arXiv2026

Ulrich Bundles on decomposable threefold scrolls over $\mathbb F_a$

Ulrich bundles provide 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 study the geometry and the moduli spaces of Ulrich bundles on broad families of decomposable threefold scrolls $X$ over Hirzebruch surfaces $\mathbb F_a$, proving that their Ulrich complexity is 1. Exploiting the double-scroll structure enjoyed by $X$, we also introduce a (geometric) involution acting on the set of classified Ulrich line bundles which, together with the natural one, shapes the study of higher-rank extensions and significantly streamlines their modular study. We moreover prove that some higher-rank extensions yield indecomposable Ulrich bundles whose existence is intrinsically $3$-dimensional, i.e. going beyond natural pullbacks from the base surfaces. In rank two, for any choice of the parameters involved, we provide a comprehensive description of associated modular irreducible components, determining their dimensions, their generic smoothness and the description of their birational structure. Ultimately, for noteworthy parameter cases, we focus on the Ulrich representation type of $X$ proving that it is Ulrich wild by the existence of generically smooth modular components of slope-stable Ulrich bundles of arbitrary rank $r$ and of dimension growing quadratically with $r$, which reveals the unbounded complexity of Ulrich modules supported on these threefolds.

math.AG↗
arXiv2026

Stratification of moduli spaces of instantons on the Segre product of three lines via 't Hooft bundles

Let $X$ be the Segre product of three projective lines. For a fixed effective divisor $D$ on $X$, we introduce the notions of $D$-'t Hooft, $(D_i,D_j)$-special and $D$-sectional special bundle. The varieties parameterizing these bundles yield a natural stratification of the moduli space of stable instanton bundles with fixed Chern classes. After characterizing the curves associated with these bundles via Serre correspondence, we describe the corresponding Hilbert schemes. Using this description, we analyze the moduli spaces of $h_i$-'t Hooft bundles and the smaller strata of $(h_i,h_j)$-special and $(h_i)$-sectional special bundles. Finally, we provide a detailed study of the low-charge cases.

math.AG↗
arXiv2026

Ulrich bundles on smooth toric threefolds with Picard number $2$

In this paper, we study Ulrich bundles on smooth toric threefolds with Picard number$~2$, namely $\mathbb P(\mathcal O_{\mathbb P^{2}}(a_0) \oplus \mathcal O_{\mathbb P^{2}}(a_1))$. We construct resolutions and monads for Ulrich bundles of arbitrary rank, and provide explicit examples together with a complete classification of those arising as pullbacks from $\mathbb{P}^2$. As a consequence, we also show that these varieties are Ulrich wild.

math.AG↗
arXiv2026

H-Instanton Bundles on Three-Dimensional Smooth Toric Varieties with Picard Number Two

We study $H$-instanton bundles on the infinite family of smooth three-dimensional varieties $X_e=\mathbb{P}(\mathcal{O}_{\mathbb{P}^2} \oplus \mathcal{O}_{\mathbb{P}^2}(e))$, for $e \geq 0$. We provide two distinct monadic descriptions of $H$-instanton bundles on $X_e$, generalizing the classical monads on $\mathbb P^3$. We then characterize $H$-instanton bundles with second Chern class supported in a single degree, and investigate their existence and moduli spaces. Finally, for $e\leq 3$, we prove the existence of $H$-instanton bundles for all admissible second Chern classes. These results extend previous constructions on specific cases and contribute to the study of instanton bundles on threefolds with higher Picard number.

math.AG↗
arXiv2025

A low-loss, 24-mode laser-written universal photonic processor in a glass-based platform

We report the fabrication of the first 24-mode universal photonic processor (UPP) realized through femtosecond laser writing (FLW), marking the most complex UPP demonstrated to date. Optimized for quantum dot emission at 925 nm, the device exhibits total insertion losses averaging only 4.35 dB, enabling its direct application in advanced multi-photon quantum experiments. Leveraging the versatility of FLW, we introduce suspended waveguides and precisely engineered 2D and 3D microstructures, significantly enhancing thermal isolation and minimizing power dissipation. As a result, our processor operates efficiently at less than 10 W, requiring only a simple thermo-electric cooler for stable thermal management. The device exhibits exceptional performance after calibration, implementing Haar-random unitary transformations with an amplitude fidelity of 99.7 %. This work establishes FLW-based integrated photonics as a scalable and robust platform for advancing quantum computing, communication, and sensing technologies.

quant-ph↗
arXiv2025

't Hooft bundles on the complete flag threefold and moduli spaces of instantons

In this work we study the moduli spaces of instanton bundles on the flag twistor space $F:=F(0,1,2)$. We stratify them in terms of the minimal twist supporting global sections and we introduce the notion of (special) 't Hooft bundle on $F$. In particular we prove that there exist $μ$-stable 't Hooft bundles for each admissible charge $k$. We completely describe the geometric structure of the moduli space of (special) 't Hooft bundles for arbitrary charge $k$. Along the way to reach these goals, we describe the possible structures of multiple curves supported on some rational curves in $F$ as well as the family of del Pezzo surfaces realized as hyperplane sections of $F$. Finally we investigate the splitting behaviour of 't Hooft bundles when restricted to conics.

math.AG↗
arXiv2024

Selective linewidth control in a micro-resonator with a resonant interferometric coupler

Optical microresonators are characterized by a comb of resonances that preserve similar characteristics over a broad spectral interval. However, for many applications it is beneficial to selectively control of the quality factor (Q) of one or only some resonances. In this work we propose and experimentally validate the use of a resonant interferometric coupler to selectively change the Q-factor of a target resonance in an integrated silicon nitride microresonator. We show that its Q-factor can be continuously tuned from 65000 to 3 milions, leaving the untargeted resonances uperturbed. Our design can be scaled to independently control several resonances.

physics.optics↗
arXiv2023

Ulrich bundles on the degree six Segre fourfold

We study the resolution of an Ulrich bundle of arbitrary rank on the Segre fourfold $\PP^2\times\PP^2$. We characterize the Ulrich bundles $\Vv$ of arbitrary rank on $\PP^2\times\PP^2$ with $h^1(\Vv\otimesΩ\boxtimesΩ)=0$ or with $h^2(\Vv\otimesΩ(-1)\boxtimesΩ(-1))=0$ or obtained as pullback from $\PP^2$ and we construct more complicated examples.

math.AC↗
arXiv2023

Ulrich bundles of arbitrary rank on Segre-Veronese varieties

We generalize the results by Eisenbud and Schreyer about Ulrich bundles over Veronese varieties to Segre-Veronese varieties. We discuss the range where we have natural cohomology and we construct multigraded resolutions and monads for Ulrich bundles of any rank. Moreover we give cohomological characterizations for significant families of bundles.

math.AG↗
arXiv2021

Non-Ulrich representation type

We show that a smooth projective non-degenerate arithmetically Cohen-Macaulay subvariety X of P^N infinite Cohen-Macaulay type becomes of finite Cohen-Macaulay type by removing Ulrich bundles if and only if N = 5 and X is a quartic scroll or the Segre product of a line and a plane. In turn, we give a complete and explicit classification of ACM bundles over these varieties.

math.AG↗
arXiv2020

H-instanton bundles on three-dimensional polarized projective varieties

We propose a notion of instanton bundle (called $H$-instanton bundle) on any projective variety of dimension three polarized by a very ample divisor $H$, that naturally generalizes the ones on $\mathbb{P}^3$ and on the flag threefold $F(0,1,2)$. We discuss the cases of Veronese and Fano threefolds. Then we deal with $H$-instanton bundles $\mathcal{E}$ on three-dimensional rational normal scrolls $S(a_0,a_1,a_2)$. We give a monadic description of $H$-instanton bundles and we prove the existence of $μ$-stable $H$-instanton bundles on $S(a_0,a_1,a_2)$ for any admissible charge $k=c_2(\mathcal{E})H$. Then we deal in more detail with $S(a,a,b)$ and $S(a_0,a_1,a_2)$ with $a_0+a_1>a_2$ and even degree. Finally we describe a nice component of the moduli space of $μ$-stable bundles whose points represent $H$-instantons.

math.AG↗
arXiv2019

Instanton bundles on the flag variety F(0,1,2)

Instanton bundles on $\mathbb{P}^3$ have been at the core of the research in Algebraic Geometry during the last thirty years. Motivated by the recent extension of their definition to other Fano threefolds of Picard number one, we develop the theory of instanton bundles on the complete flag variety $F:=F(0,1,2)$ of point-lines on $\mathbb{P}^2$. After giving for them two different monadic presentations, we use it to show that the moduli space $MI_F(k)$ of instanton bundles of charge $k$ is a geometric GIT quotient and the open subspace $MI^s_F(k)\subset MI_F(k)$ of stable instanton bundles has a generically smooth component of dim $8k-3$. Finally we study their locus of jumping conics.

math.AG↗
arXiv2019

Instanton bundles on the Segre threefold with Picard number three

We study instanton bundles $E$ on $\mathbb{P}^1\times \mathbb{P}^1 \times \mathbb{P}^1$. We construct two different monads which are the analog of the monads for instanton bundles on $\mathbb P^3$ and on the flag threefold $F(0,1,2)$. We characterize the Gieseker semistable cases and we prove the existence of $μ$-stable instanton bundles generically trivial on the lines for any possible $c_2(E)$. We also study the locus of jumping lines.

math.AG↗
arXiv2019

Instanton bundles on the blow up of the projective $3$-space at a point

We propose a general definition of mathematical instanton bundle with given charge on any Fano threefold extending the classical definitions on $\mathbb P^3$ and on Fano threefold with cyclic Picard group. Then we deal with the case of the blow up of $\mathbb P^3$ at a point, giving an explicit construction of instanton bundles satisfying some important extra properties: moreover, we also show that they correspond to smooth points of a component of the moduli space.

math.AG↗
arXiv2018

ACM sheaves on the double plane

The goal of this paper is to start a study of aCM and Ulrich sheaves on non-integral projective varieties. We show that any aCM vector bundle of rank two on the double plane is a direct sum of line bundles. As a by-product, any aCM vector bundle of rank two on a sufficiently high dimensional quadric hypersurface also splits. We consider aCM and Ulrich vector bundles on a multiple hyperplanes and prove the existence of such bundles that do not split, if the multiple hyperplane is linearly embedded into a sufficiently high dimensional projective space. Then we restrict our attention to the double plane and give a classification of aCM sheaves of rank at most $3/2$ on the double plane and describe the family of isomorphism classes of them.

math.AG↗