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deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions $P_\epsilon$, where $\epsilon$ is a small number, or vector of small quantities.

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exact sequence of deformations

Suppose that $C\cong P^1$ and $Def(f)$ denote the first order deformation of pointed stable map $(C,{p_i},f:C\longrightarrow X)$. I read that we have short exact sequence: $0\longrightarrow H^0(C,T_C)...
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Residues of $\frac{1}{\prod_{i=1}^n (x-P_i)^{e_i}}$

This is a problem occurring in my research about deformations of $\mathbb{Z}/p^n$-covers over a ring of power series. Given an algebraically closed field $k$ of characteristic $p>0$, suppose $1< ...
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Unobstructedness of nodal holomorphic curve in symplectic manifold

Suppose $(X,\omega)$ is a compact symplectic manifold and $J$ is an $\omega$-compatible almost complex structure on $X$ (the symplectic structure seems to be irrelevant for this question actually). ...
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Completion of a local noetherian $\Gamma$-algbera

I am in trouble with a result . Let $\Gamma$ is a complete local noetherian $k$-algbera with residue field $k$ and $(R,m)$ is a local noetherian $\Gamma$-algebra with residue field $k$. Also let $\...
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DGLA controlling deformation of holomorphic curves

Suppose $C$ is a compact Riemann surface and $X$ is a compact Kähler manifold. Suppose $f:C\to X$ is a stable holomorphic map. Then, the deformations of $f$ are controlled by the complex $L^\bullet = ...
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Hilbert scheme of Grassmannians

Let $X=\mathbb{G}(k,n)$ be the Grassmannian of $k$-planes in $n$-space. Let $Y\subseteq X$ a subvariety and let $H$ be the connected component of the Hilbert scheme of $X$ that contains $Y$. Is $H$ ...
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Deformation theory of holomorphic vector bundles in Donaldson-Kronheimer

There is the Proposition 6.4.3 in Donaldson-Kronheimer as follows: Proposition (6.4.3) (i) There is a holomorphic map $\psi$ from a neighborhood of $0$ in $H^1(\operatorname{End} \mathscr{E})...
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Hilbert polynomial of structure sheaf of hypersurfaces

Is there an example of a hypersurface $X$ of some projective space $\mathbb{P}^n$ such that there exists an invertible sheaf $\mathcal{L}$ on $X$, not isomorphic to the structure sheaf $\mathcal{O}_X$,...
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Higher Braces algebra and operads

1) In [HIGHER OPERATIONS ON HOCHSCHILD COMPLEX], Gerstenhaber and Voronov showed that the Hochschild complex $C_1(\mathcal A)$ of any associative algebra (or e_1 algebra) $\mathcal A$ is naturally ...
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Generalisation of the notion of operad

Let $\mathscr P$ be an operad in the category of vector spaces. An algebra (of the type encoded by $\mathscr P$) on the vector space $V$ is a morphism of operads $\mu:\mathscr P\to End_V$ with $End_V$ ...
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Degeneration of smooth curves and Picard-Lefschetz formula

Let $\pi:\mathcal{C} \to \Delta$ be a family of projective curves of genus $g \ge 2$ over the unit disc $\Delta$, smooth over the punctured disc $\Delta\backslash \{0\}$ and central fiber $\pi^{-1}(0)$...
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Why is every deformation of the universal enveloping algebra of a complex semisimple Lie algebra trivial?

I have read in these lecture notes that every deformation $U_h(\mathfrak{g})$ of $U(\mathfrak{g})$ is trivial, i.e. isomorphic to $U(\mathfrak{g})[[h]]$ as associative $\mathbb{C}[[h]]$-algebras. Why ...
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When is the moduli of generalized parabolic bundles with fixed determinant smooth?

Let $X$ be a smooth, projective curve of genus at least $2$, $x_1, x_2$ two distinct closed points, $d$ an odd integer and $\alpha$ a positive real number less than $1$. By a generalized parabolic ...
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158 views

Local to global deformation of invertible sheaves

Let $\pi:X \to S$ be a flat, projective morphism, $S$ irreducible. Suppose that for all $s \in S$, the fiber $X_s$ satisfies $h^2(\mathcal{O}_{X_s})=0$. This means in particular that given an ...
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Pull-back of line bundles and field extension

Let $X$ be a smooth, projective variety over a field $K$ of characteristic $0$ (not necessarily algebraically closed) and $L$ an invertible sheaf on $X_{\bar{K}}=X \times_K \mbox{Spec}(\bar{K})$, ...
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158 views

Bertini-type theorem for reducible schemes

Let $X \subset \mathbb{P}^n$ be a reducible, projective subscheme. Assume that $X$ is reduced (meaning that every local ring is reduced i.e., does not contain nilpotent element). Denote by $S_d$ the ...
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Absolute approximation of formal schemes

Let $\mathfrak{X}_j$ be an inverse system of qcqs $p$-adic formal scheme, flat over $\mathbf{Z}_p$, with affine transition maps, and assume $\mathcal{O}_{\mathfrak{X}_j}$ is a coherent sheaf of ...
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90 views

Representability of Flattening stratification functor

Let $f:X \to Y$ be a projective morphism between noetherian scheme. Is there any know condition under which there exists a functorial stratification of $Y$ i.e., there exists a filtration by locally ...
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1answer
117 views

Variation of global sections of line bundles

The underlying field is $\mathbb{C}$. Let $\pi:\mathcal{C} \to \mathbb{A}^n$ be a flat family of projective curves (not necessarily smooth) of genus $g \ge 2$. Assume $\mathcal{C}$ is regular. Let $\...
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Can operads (or category theoretic structures more generally) be compared?

I was reading John Baez’s paper on operads and phylogenetics trees where he formalizes a Jukes–Cantor model of phylogenetics. Because biological questions receive different answers depending on the ...
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Clifford algebras as deformations of exterior algebras

$\def\Cl{\mathcal C\ell} \def\CL{\boldsymbol{\mathscr{C\kern-.1eml}}(\mathbb R)}$ I'm not an expert in neither of the fields I'm touching, so don't be too rude with me :-) here's my question. A well ...
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1answer
338 views

Bertini's theorem over non-algebraically closed field

Let $K$ be a non-algebraically closed (infinite) field of characteristic $0$ and $X$ a smooth, projective $K$-variety. Does there exist an ample invertible sheaf $\mathcal{L}$ on $X$ such that a ...
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A general definition of an equisingular family of singular varieties?

This question is about the existence of a definition. I'm far from being an expert in the field in question I apologize in advance for any inaccuracies or stupid and wrong assumptions. Let $X$ be a ...
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170 views

Does the local Bertini theorem in mixed characteristic imply the global Bertini theorem

Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. Assume that the characteristic of $K$ is $0$ and of $k$ is $p>0$. Let $\pi:X \to \mbox{Spec}(R)$ be a flat, ...
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Deformation invariance of rational connectedness in positive/mixed characteristic

Let $f:X \to S$ be a smooth morphism and $S$ the spectrum of a discrete valuation ring. If the generic fiber of $f$ is rationally (chain) connected then is the special fiber of $f$ also rationally (...
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115 views

On regularity of flat families over a DVR

Let $k$ be an algebraically closed field of characteristic zero and $R$ a discrete valuation ring over $k$. Let $\pi:X \to \mathrm{Spec}(R)$ be a flat, projective morphism such that the generic fiber ...
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Deformations of Calabi-Yau manifolds

Let $X$ be a compact complex smooth manifold with holomorphically trivial canonical class. It is true that any (sufficiently small?) deformation of the complex structure of $X$ also has ...
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When is an unobstructed formal deformation convergent?

Let $X$ be a smooth finite type scheme over $\mathbb{C}$ and let $ \mathcal{X} \to Spec(\mathbb{C}[[x]])$ be a formal deformation of $X$ (flat family with special fiber $X$). Which of the following ...
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Building conilpotent coalgebras from co-square-zero-extensions

Let $\mathrm{K}$ be a field of char. 0. Given a chain complex $\mathrm{X} $ over $\mathrm{K}$ denote $\mathrm{E}(\mathrm{X})$ the co-square-zero-extension on $\mathrm{X}, $ i.e. the cocommutative ...
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Fedosov vs. Kontsevich deformation quantization : a beginner survey

I'm a condensed matter physicist who tries to understand the details of deformation quantization. In my self-made training, I've found two huge pieces of work, namely Fedosov, B. V. (1994). "A ...
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Lie algebras : Deformations and Rigidity

I am studying deformation (as it is introduced in https://arxiv.org/pdf/math/0611793.pdf or http://web.cs.elte.hu/~fialowsk/pubs-af/condefnew2.pdf) and rigidity of some infinite dimensional Lie ...
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Deformation of “Hecke modification”

Let $X$ be a smooth curve over $\mathbb{C}$. I wish to compute the deformation of the following data $(E,F,x)$. $E$ and $F$ are locally free sheaves over $X$ and $x$ is a point on $X$. They satisfy: ...
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523 views

Generators of an ideal of $K[[X_1,X_2,X_3]]$

Let us consider an irreducible polynomial $$f \colon= \alpha_e + \alpha_{e-1}X_1 + ... + \alpha_1X_1^{e-1} + X_1^e \in K[[X_2,X_3]][X_1]$$ and set $$\iota_1 \colon K[[X_2,X_3]] \hookrightarrow K[[X_1,...
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Compute action of the gauge group in deformation theory of an algebra

I am working on Balazs Szendroi's introduction to deformation theory, but I got stuck on the exercise on the bottom of page 6. Consider a vector space $A$ with a multiplication $m$ that makes it into ...
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Holomorphic family of Riemann surfaces

Let $2g+m\ge 3$. A holomorphic family of (non-singular, compact) Riemann surfaces of type $(g,m)$ is a triple $(X,Y,\pi,s_1,\ldots,s_m)$, where $X,Y$ are complex manifolds of (complex) dimension $n+1,...
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Exercise 1.1.(c) in Hartshorne's Deformation Theory

Exercise 1.1.(c) in Hartshorne's Deformation Theory: Over an algebraically closed field $k$, we define a curve in $\mathbb P^2_k$ to be the closed subscheme, defined by a homogeneous polynomial $f(...
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Behaviour of the number of generators of a certain ideals

Let us define $A_n, f_n, {\frak a}_n, k(n), \iota_n, {\frak b}_n$ and $l(n)$ by the followings$\colon$ $A_n \colon= K[[X_1,...,X_n]]$, i.e. a $n$-variable formal power series ring over a field $K$. ...
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1answer
344 views

Elementary (English) reference for the cotangent complex?

I'm trying to understand cotangent complexes and their role in deformation theory, and later the statement that they're somehow natural in a derived scheme/stack. I understand that the standard ...
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Degeneration of curves inside a family of surfaces

We are interested in understanding how a higher genus curve in a smooth surface of general type can degenerate into the non-normal locus of the limit of a degeneration of surfaces. More precisely: ...
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1answer
212 views

Deformation Theoretic Interpretation of $H^1(C,T_C(-2p))$

Suppose $C$ is a (non-singular) compact Riemann surface of genus $g$ and with $n$ (distinct) marked points $p_1,\ldots,p_n$. If we assume the stability condition ($2-2g-n<0$), then it is proved in "...
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A differential graded Lie algebra with the Hochschild differential

Let $(V,\cdot)$ be an associative algebra and $W$ be a vector space endowed with a bimodule structure $\triangleright:V\otimes W\to W$ and $\triangleleft:W\otimes V\to W$ such that the following ...
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1answer
138 views

Degeneration of coadjoint orbits

Let $X$ be a projective manifold and we have a degeneration of fibers such that they are biholomorphic to coadjoint orbits, i.e, $X\to \Delta$ , and fibers $X_t$ are biholomorphic to coadjoint orbits ...
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155 views

Identification of cohomology sheaf in the definition of the Kodaira-Spencer morphism for abelian schemes

Let $p:A \to S$ be a projective abelian scheme, where $S$ is some smooth scheme over a base field $k$. Then we have the Kodaira-Spencer morphism $$ \kappa : T_{S/k} \to R^1p_*T_{A/S} $$ where $T_{S/k}$...
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Finite dimensional approximation of Donaldson theory

In addition to the Seiberg-Witten invariant there has been further success with "finite dimensional approximations" of the Seiberg-Witten theory: Bauer-Furuta's stable (co)homotopy invariants, and ...
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Deformation-Obstruction Theory of YM Instantons

In Donaldson-Kronhiemer Section 4.2.5. (local models of the moduli space of YM instantons) they first get local models of the moduli space $M$ inside the space of all connections modulo gauge $\...
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Vanishing of space of first order infinitesimal deformations for irreducible algebraic stack

This question has a few bits, and apologies if some questions are phrased poorly since I am not knowledgeable on the language of stacks or deformation theory. Suppose $\mathscr{X}$ is an algebraic ...
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165 views

Can two singular points collapse to a new singular point?

It is about deformation theory on algebraic surfaces. If there are two singular points on an algebraic surfaces, is it possible that two singular points collapse to a new point as the surface deforms? ...
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155 views

Modern reference for Fischer-Grauert theorem and its generalization by Schuster

From this MSE question and its answer, and from this MO question I have learned of the following remarkable theorem of Wolfgang Fischer and Haus Grauert. Theorem. A proper holomorphic submersion with ...
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179 views

Clarification on relationship between Grothendieck-Messing and Honda systems

It's well-known, due to work of Fontaine (see e.g. this), that if $k$ is a perfect field of characteristic $p$, then one can classify $p$-divisible groups over $W(k)$ by Honda systems. Namely, these ...
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428 views

deformation theory in positive characteristic

The idea "Formal deformation theory in characteristic zero is controlled by a differential graded Lie algebra (dgla)" goes back to Goldman-Millson, Deligne, Drinfeld among others; see Lurie's ICM talk....