versal deformation
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2021 ◽  
Vol 28 (02) ◽  
pp. 269-280
Author(s):  
I.D.M. Gaviria ◽  
José A. Vélez-Marulanda

Let [Formula: see text] be a fixed algebraically closed field of arbitrary characteristic, let [Formula: see text] be a finite dimensional self-injective [Formula: see text]-algebra, and let [Formula: see text] be an indecomposable non-projective left [Formula: see text]-module with finite dimension over [Formula: see text]. We prove that if [Formula: see text] is the Auslander–Reiten translation of [Formula: see text], then the versal deformation rings [Formula: see text] and [Formula: see text] (in the sense of F.M. Bleher and the second author) are isomorphic. We use this to prove that if [Formula: see text] is further a cluster-tilted [Formula: see text]-algebra, then [Formula: see text] is universal and isomorphic to [Formula: see text].


Author(s):  
Ashis Mandal ◽  
Satyendra Kumar Mishra

We study the deformation of Courant pairs with a commutative algebra base. We consider the deformation cohomology bi-complex and describe a universal infinitesimal deformation. In a sequel, we formulate an extension of a given deformation of a Courant pair to another with extended base, which leads to describe the obstruction in extending a given deformation. We also discuss the construction of versal deformation of Courant pairs. As an application, we compute universal infinitesimal deformation of Poisson algebra structures on the three-dimensional complex Heisenberg Lie algebra. We compare the second deformation cohomology spaces of these Poisson algebra structures by considering them in the category of Leibniz pairs and Courant pairs, respectively.


2020 ◽  
Vol 19 (3) ◽  
Author(s):  
Martin Klimeš ◽  
Christiane Rousseau

AbstractIn this note we present variants of Kostov’s theorem on a versal deformation of a parabolic point of a complex analytic 1-dimensional vector field. First we provide a self-contained proof of Kostov’s theorem, together with a proof that this versal deformation is indeed universal. We then generalize to the real analytic and formal cases, where we show universality, and to the $${\mathcal {C}}^\infty $$ C ∞ case, where we show that only versality is possible.


2020 ◽  
Vol 20 (3) ◽  
pp. 319-330
Author(s):  
D. A. H. Ament ◽  
J. J. Nuño-Ballesteros ◽  
J. N. Tomazella

AbstractLet (X, 0) ⊂ (ℂn, 0) be an irreducible weighted homogeneous singularity curve and let f : (X, 0) → (ℂ2, 0) be a finite map germ, one-to-one and weighted homogeneous with the same weights of (X, 0). We show that 𝒜e-codim(X, f) = μI(f), where the 𝒜e-codimension 𝒜e-codim(X, f) is the minimum number of parameters in a versal deformation and μI(f) is the image Milnor number, i.e. the number of vanishing cycles in the image of a stabilization of f.


2020 ◽  
Vol 20 (2) ◽  
pp. 285-296
Author(s):  
Jean-Marc Drézet

AbstractSome coherent sheaves on projective varieties have a non-reduced versal deformation space; for example, this is the case for most unstable rank 2 vector bundles on ℙ2, see [18]. In particular, some moduli spaces of stable sheaves are non-reduced. We consider some sheaves on ribbons (double structures on smooth projective curves): let E be a quasi locally free sheaf of rigid type and let 𝓔 be a flat family of sheaves containing E. We find that 𝓔 is a reduced deformation of E when some canonical family associated to 𝓔 is also flat. We consider also a deformation of the ribbon to reduced projective curves with two components, and find that E can be deformed in two distinct ways to sheaves on the reduced curves. In particular some components M of the moduli spaces of stable sheaves deform to two components of the moduli spaces of sheaves on the reduced curves, and M appears as the “limit” of varieties with two components, whence the non-reduced structure of M.


2017 ◽  
Vol 92 (8) ◽  
pp. 1846-1857 ◽  
Author(s):  
Itziar Baragaña ◽  
Ferran Puerta

2015 ◽  
Vol 444 ◽  
pp. 81-123 ◽  
Author(s):  
Gebhard Böckle ◽  
Ann-Kristin Juschka

2014 ◽  
Vol 16 (1) ◽  
pp. 179-198 ◽  
Author(s):  
Alice Fialowski ◽  
Goutam Mukherjee ◽  
Anita Naolekar

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