arXiv subjects

Vincenzo Galgano

Publications and source records attributed to Vincenzo Galgano.

11 recordsLinked to original sources

arXiv2026

Schubert compactifications of matrix pencils

The Schubert compactification of a linear space of matrices $L \subseteq \textrm{Mat}_{n\times m}$ is a natural compactification of $L$ in the Grassmannian $\textrm{Gr}(n, n+m)$. This construction generalises classical Schubert varieties and matroid Schubert varieties, and it turns invariants of linear matrix spaces into projective and intersection-theoretic invariants of the associated variety. We develop this geometry completely in the case of matrix pencils, that is $\dim L=2$. The resulting Schubert surfaces are singular analogues of rational surfaces obtained as blow-ups of $\mathbb{P}^2$. We show that they are normal with quotient singularities, compute their degree, cohomology ring, and describe the distinguished rational curves and their intersection pairing in terms of the Kronecker invariants of the matrix pencil. Remarkably, this dictionary can be reversed: the invariants of the pencil are read off from intersection numbers and other invariants of the variety. In particular, we prove that the Schubert surface determines the regular component of the pencil, up to the natural action of $\mathrm{GL}_2 \times \mathrm{GL}_n \times\mathrm{GL}_m$. Finally, we prove that the ideal of Schubert surfaces of matrix pencils in the Plücker embedding is generated in degree $2$ and satisfies Green's property $N_p$ in a certain range.

math.AG↗

arXiv2026

Tensor analysis for lipid transport

High-dimensional biological datasets with molecular, spatial, and temporal dimensions are increasingly common. However, their analysis requires approaches that can integrate multiple data axes, accommodate noisy and missing measurements, and capture both dynamic interactions and localization changes. To address this, we provide an end-to-end tensor analysis pipeline that handles sparsity by employing tensor decomposition methods (HOSVD and CP) that are augmented with a binary mask for missing data and a framework for measurement error. We showcase this on a three-dimensional mammalian lipid transport dataset depending on lipid identities, organelle localizations, and time-series abundances. Our approach successfully identifies specific lipid-organelle pairs undergoing rapid temporal evolution, uncovers modules of lipids that co-vary across organelles and time, and extracts latent factors representing global redistribution trajectories. Direct comparison with a previous kinetic ODE model confirms that the tensor decompositions faithfully reproduce key lipid flux features.

q-bio.QM↗

arXiv2026

Discrete signature tensors for persistence landscapes

Signature tensors of paths are a versatile tool for mathematical data analysis. Recently, they have been applied in the context of vectorisation of persistent homology: after a choice of embedding of barcodes into a space of paths on a vector space, one applies the path signature map, resulting in tensors amenable to statistical and machine-learning methods. Among the different path embeddings, the persistence landscape embedding (PLE) is injective and stable, but PLE is composed with the signature map loses injectivity. Therefore, we address this by proposing a discrete alternative. Persistence landscapes are determined by the time-series of their critical points, of which we compute the discrete signature. We call this composition the {\em discrete landscape feature map} (DLFM), and give results on its injectivity, stability and computability. When studying the injectivity, we complete the proof of a general result due to Diehl, Ebrahimi-Fard and Tapia in the higher-dimensional setting. We showcase the DLFM on a knotted protein dataset, capturing sequence similarity and knot depth with statistical significance. We include an appendix with a preliminary study of Chen signatures of persistence landscapes from the point of view of algebraic geometry.

math.AT↗

arXiv2026

Some Classical Invariants, from Harmonic Quadruples to Triangle Groups

These notes are an expanded version of the lectures held in Tromso, in May 2025 at the "Lie-Stormer Summer School : Invariant Theory from classics to modern developments", in the framework of TiME events. We emphasize the analogy between binary quartics and ternary cubics (and subsequently modular forms) based on their harmonic and equianharmonic invariants. Triangle groups are presented in both the elliptic and the hyperbolic setting with their associated tilings. The topics include the discussion of a short Hilbert paper on polynomials which are powers, that was proposed to the participants. The appendix contains some exercises, with sketches of solutions, and a section devoted to Pfaffians edited by Vincenzo Galgano.

math.AG↗

arXiv2026

On the Coupled Cluster Doubles Truncation Variety of Four Electrons

We extend recent algebro-geometric results for coupled cluster theory of quantum many-body systems to the truncation varieties arising from the doubles approximation (CCD), focusing on the first genuinely nonlinear doubles regime of four electrons. Since this doubles truncation variety does not coincide with previously studied varieties, we initiate a systematic investigation of its basic algebro-geometric invariants. Combining theoretical and numerical results, we show that for $4$ electrons on $n\leq 12$ orbitals, the CCD truncation variety is a complete intersection of degree $2^{\binom{n-4}{4}}$. Using representation-theoretic arguments, we uncover a Pfaffian structure governing the quadratic relations that define the truncation variety for any $n$, and show that an exact tensor product factorization holds in a distinguished limit of disconnected doubles. We connect these structural results to the computation of the beryllium insertion into molecular hydrogen ({Be$\cdots$H$_2$ $\to$ H--Be--H}), a small but challenging bond formation process where multiconfigurational effects become pronounced.

math.AG↗

arXiv2025

The GameTheory package for Macaulay2

We describe the GameTheory package version 1.0 for computing equilibria in game theory available since version 1.25.05 of Macaulay2. We briefly explain the four equilibrium notions, Nash, correlated, dependency, and conditional independence, and demonstrate their implementation in the package with examples.

math.AG↗

arXiv2025

Secant varieties of generalised Grassmannians

Secant varieties of a homogeneously embedded generalised Grassmannian $G/P$ inherit the natural group action, and one can reduce the study of their local geometric properties to $G$-orbit representatives. The case of secant varieties of lines is particularly elegant as their $G$-orbits are induced by $P$-orbits in both $G/P$ and $\mathfrak{g}/\mathfrak{p}$. Parabolic orbits are a classical problem in Representation Theory, well understood when $G/P$ is cominuscule. Exploiting them, we provide a complete and uniform description of both the identifiable and singular loci of the secant variety of lines to any cominuscule variety. We also introduce a finer version of the $2$-nd Terracini locus, called $2$-nd strong-Terracini locus, and we determine it for cominuscule varieties. Finally, we analyse the non-cominuscule case of isotropic Grassmannians for comparison, and we highlight a few differences.

math.AG↗

arXiv2023

Identifiability and singular locus of secant varieties to Grassmannians

Secant varieties are among the main protagonists in tensor decomposition, whose study involves both pure and applied mathematical areas. Grassmannians are the building blocks for skewsymmetric tensors. Although they are ubiquitous in the literature, the geometry of their secant varieties is not completely understood. In this work we determine the singular locus of the secant variety of lines to a Grassmannian Gr(k,V) using its structure as SL(V)-variety. We solve the problems of identifiability and tangential-identifiability of points in the secant variety: as a consequence, we also determine the second Terracini locus to a Grassmannian.

math.AG↗

arXiv2023

Identifiability and singular locus of secant varieties to spinor varieties

In this work we analyze the $Spin(V)$-structure of the secant variety of lines $σ_{2}(\mathbb{S})$ to a Spinor variety $\mathbb{S}$ minimally embedded in its spin representation. In particular, we determine the poset of the $Spin(V)$-orbits and their dimensions. We use it for solving the problems of identifiability and tangential-identifiability in $σ_2(\mathbb S)$, and for determining the second Terracini locus of $\mathbb{S}$. Finally, we show that the singular locus $Sing(σ_{2}(\mathbb{S}))$ contains the two $Spin(V)$-orbits of lowest dimensions and it lies in the tangential variety $τ(\mathbb{S})$: we also conjecture what it set-theoretically is.

math.AG↗

arXiv2023

Equivariant Euler characteristics on permutohedral varieties

By the work of J.Huh, one can interpret binomial coefficients as a solution to an intersection problem on a permutohedral variety $X_E$. Applying Hirzebruch-Riemann-Roch, this intersection problem is equivalent to computing Euler characteristic of a specific element of $K$-theory of $X_E$. This element has a natural lifting to equivariant $K$-theory and thus the Euler characteristic may be upgraded to a Laurent polynomial. We provide and implement three different approaches, in particular a recursive one, to computing these polynomials.

math.AG↗

arXiv2022

Graph States and the Variety of Principal Minors

In Quantum Information theory, graph states are quantum states defined by graphs. In this work we exhibit a correspondence between graph states and the variety of binary symmetric principal minors, in particular their corresponding orbits under the action of $SL(2,\mathbb F_2)^{\times n}\rtimes \mathfrak S_n$. We start by approaching the topic more widely, that is by studying the orbits of maximal abelian subgroups of the $n$-fold Pauli group under the action of $\mathcal C_n^{\text{loc}}\rtimes \mathfrak S_n$, where $\mathcal C_n^{\text{loc}}$ is the $n$-fold local Clifford group: we show that this action corresponds to the natural action of $SL(2,\mathbb F_2)^{\times n}\rtimes \mathfrak S_n$ on the variety $\mathcal Z_n\subset \mathbb P(\mathbb F_2^{2^n})$ of principal minors of binary symmetric $n\times n$ matrices. The crucial step in this correspondence is in translating the action of $SL(2,\mathbb F_2)^{\times n}$ into an action of the local symplectic group $Sp_{2n}^{\text{loc}}(\mathbb F_2)$ on the Lagrangian Grassmannian $LG_{\mathbb F_2}(n,2n)$. We conclude by studying how the former action restricts onto stabilizer groups and stabilizer states, and finally what happens in the case of graph states.

quant-ph↗