Subject / 6 records

Algebra and algebraic geometry

Nilpotent derivations, Lie-theoretic constructions, resonance, and group extensions.

Classification: Second Brain

6 records

• 10.5281/zenodo.237601692026

  Finite incidence models for elliptic resonance varieties

  Martino Lupini · Mathematics

  Let E = A ⊕ B be a sum of line bundles of degrees 2 ≤ a ≤ b on a smooth complex elliptic curve. We determine all irreducible components of the locus of two-dimensional spaces of sections that generate a sheaf of rank one. Their number is a if A ≇ B, and max{1, a−2} if A ≅ B. Relative Grassmannians and projective bundles give smooth finite normalization models for these components. In saturation degree two, the normalization maps are closed embeddings.

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• 10.5281/zenodo.236826802026

  Jordan spectra of nilpotent derivations from scalar geometry

  Martino Lupini · Orderability, Groups

  We determine the complete Jordan spectrum of nilpotent derivations of a nonabelian class-two Lie algebra I over a characteristic-zero field. We assume that its scalar bracket space has a decomposable basis over the ground field and that its reduced geometric decomposable locus admits no nontrivial additive-group action. A plane-preserver algebra and a hyperplane arrangement determine the horizontal rank set R; an integer criterion then decides every full Jordan type. Every admitted type occurs over the ground field, and the spectrum remains unchanged under field extension. For u = dim(I/Z(I)), d = dim I′, f = dim(Z(I)/I′), and r = max R, the greatest partition is (2^r, 1^(u-2r)) + (f) + (1^d), with componentwise addition. The full spectrum recovers R and is a dominance order ideal exactly when R has no gaps. For fixed free central dimension f, finite scalar loci with identical support-sum rank functions can have different spectra; support dimension six is minimal. A family has Jordan strata and commuting varieties whose component dimensions differ without bound. Geometric consequences include a smooth rational greatest-type stratum, a crepant resolution, and a classification of nilpotent linear spaces of largest dimension. Hall polynomials give uniform Jordan-stratum point counts at almost all places for number-field models.

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• 10.5281/zenodo.234376442026

  Spectral criteria for extending partial left orders

  Martino Lupini · Orderability, Groups

  We answer affirmatively two extension questions posed by Bodart, Ciobanu, and Metcalfe in 2024: every partial left order extends to a total left order on the group of positive rational affine transformations, and on every group ℤ² ⋊T ℤ with T ∈ SL₂(ℤ) and tr(T) > 2. Both follow from a metabelian criterion: it suffices that the group be torsion-free, its derived subgroup have finite rational rank, and all conjugation eigenvalues on that subgroup be positive real. We derive more general extension results from structural theorems of Dong and Bodart–Dong. For groups with a finite-rank nilpotent normal subgroup and abelian quotient, extension of arbitrary partial left orders is equivalent to extension of total left orders from subgroups. This connects the cyclic case with Pierce’s classical theorem. A unified spectral criterion allows a locally nilpotent derived subgroup: under local finiteness and positive spectrum modulo its center, full extension is equivalent to the requirement that a central eigenvalue whose positive power is positive real is itself positive real. The proofs explain how local affine certificates yield orders on the prescribed ambient group.

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• 10.5281/zenodo.231060642026

  Constructive normal identities in metabelian groups

  Martino Lupini · Groups

  No abstract on this deposit.

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• 10.5281/zenodo.228428972026

  Twisted Iwahori gluing and central extensions of SL2(Z[1/2])

  Martino Lupini · Mathematics

  No abstract on this deposit.

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• 10.5281/zenodo.228341372026

  Congruence stability for an Iwahori gluing problem

  Martino Lupini · Mathematics

  No abstract on this deposit.

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