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      Questions tagged [deformation-theory]

      for questions about deformation theory, including deformations of manifolds, schemes, Galois representations, and von Neumann algebras.

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      Free almost commutative vertex algebras

      Given a commutative $k$-algebra $A$, we can freely generate a commutative vertex algebra by formally adjoining a derivation. We obtain a functor $CAlg_{k} \rightarrow CVAlg_{k} $, which I'll denote $\...
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      1answer
      174 views

      Failure of $H^1(X, \mathcal{T}_X)$ to act freely on the isomorphism classes of liftings of a deformation

      It is well know that the isomorphism classes of first order deformations of a nonsingular variety $X$ are in $1$ to $1$ correspondence with $H^1(X,\mathcal{T}_X)$. It is also known that given any ...
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      32 views

      Field of definition from deformation rigidity

      It is known that a smooth complex quasi-projective variety which is deformation-rigid (e.g. any holomorphic deformation inside an ambient space is trivial) can be defined over a number field. Can one ...
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      1answer
      71 views

      Connected sum of algebraic curves, handlebody decomposition, and induction on genus

      Are there any nice methods of taking algebraic curves $C_1, C_2$ of genera $g_1,g_2$ and producing a curve $C_3 = C_1 \# C_2$ of genus $g_3 = g_1+g_2$? I'm imagining doing this over any field, but ...
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      1answer
      97 views

      Monomials in products in power series ring on several variables

      Let $A \colon= K[[X_1,\ldots,X_m,Y_1,\ldots,Y_n]]$ be a power series ring over a field $K$ in $m + n$ variables and ${\frak m}$ be the unique maximal ideal of $A$. For arbitrary two elements $\alpha ...
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      0answers
      429 views

      Theorem from Deformation Theory

      My question refers to some steps it the proof of Theorem 3.3 part (b) in Christensen's paper treating Deformation theory (see pages 9-11): https://mathematics.stanford.edu/wp.../A.-Christensen-Draft....
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      235 views

      Obstructions to locally trivial deformations

      Let $X$ be a complex projective variety. If $X$ is smooth, then the first-order infinitesimal deformations are given by $H^1(X, T_X)$ and the obstructions are in $H^2(X, T_X)$. Now assume that $X$ is ...
      3
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      0answers
      94 views

      non-associative deformation quantization [closed]

      Several physicists now consider non-Poisson bivectors but still apply Maxim’s morphism for deformation quantization. The result is a `non-associative star product’, So a deformation as a ________? ...
      5
      votes
      1answer
      133 views

      Deformations of Vertex Algebras

      As the title suggests, I'm interested in deformation theory of vertex algebras and their representations. In the paper https://arxiv.org/abs/1806.08754, the authors construct, for a vertex algebra $...
      2
      votes
      0answers
      93 views

      Families over Artin Rings and Deformations

      Let us work with a class of schemes over an algebraically closed field $k$ such that any two schemes in this class are isomorphic. An example of such a class would be genus zero nonsingular curves ...
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      151 views

      Fibers of blow up in families

      Let $T$ be a smooth curve over $\mathbb{C}$ and $p:\mathbb{P}^n \times T \to T$ the natural projection. Let $V \subset \mathbb{P}^n_T$ be a $T$-flat subscheme of codimension at least $2$ and $\pi: \...
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      0answers
      56 views

      The underlying curve of a family of genus zero $n$ punctured curves

      Let $X$ be a curve of genus zero over an algebraically closed field $k$ so that $X \cong \mathbb{P}_k^1$. Let $(C, s_1, \cdots, s_n)$ a $n$ punctured genus zero curve over $k$ where $s_i: k \to C$ are ...
      5
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      1answer
      181 views

      Coarse moduli space versus Kuranishi family

      We will work over complex number field $\mathbb{C}$. Let $\mathscr{M}_h$ be the moduli functor for canonically polarized manifolds with $h$ the Hilbert polynomial. Let us denote by $M_h$ the coarse ...
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      0answers
      75 views

      Log deformations in obstructed case

      I'm going to assume reader is aware of semi-stable log structures either in Kawamata-Namikawa version or later approaches. Anyway, let $X$ be a d-semistable variety. I want to know whether I can ...
      1
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      1answer
      170 views

      Example of a nonsmoothable scheme

      I try to understand Iarrobinos example of a nonsmoothable 0-dimensional scheme with the help of Artins notes on it: http://www.math.tifr.res.in/~publ/ln/tifr54.pdf (pages 4-6) But I have some ...

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