The study of how mathematical objects (complex manifolds, associative algebras, Lie algebras) can be deformed into similar mathematical objects, at least infinitesimally.

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54 views

why does infinitesimal lifting imply triviality of infinitesimal deformations?

I'm trying to learn some deformation theory, but I'm stuck on the proof of corollary 4.7 in https://math.berkeley.edu/~robin/math274root.pdf Let $X$ be an affine nonsingular scheme of finite type ...
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37 views

Computing the cotangent complex: what's the ring?

As far as I understand, deformation theory of schemes may be calculated via the cotangent complex. I have read that in general the cotangent complex may be difficult to compute. However, I have a ...
4
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89 views

Smooth points with obstructed deformations

Let $k$ be an algebraically closed field, e.g. $k=\mathbb C$. Let $Art_k$ be the category of local Artin $k$-algebras with residue field $k$. A deformation functor is a functor $D:Art_k\to Sets$ such ...
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36 views

Deformation theory in a paper of Bogomolov and Tschinkel

I am trying to read this paper http://www.math.nyu.edu/~tschinke/papers/yuri/00ajm/ajm.pdf, by Bogomolov and Tschinkel. I had 0 preparation in the theory of deformation of complex structures, but in ...
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35 views

Complete intersections with respect to different sections

Let us consider the Whitney sum $E$ of holomorphic line bundles $L_1,\dots, L_k$ on a smooth (projective) variety $X$. For a generic global section $s$ of $E$, the zero locus $Z_s := s^{-1}(0_E)$ ...
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45 views

Deformations of associative algebras and Hochschild cohomology.

I am studying the deformation theory of associative algebras (and Poisson algebras) and came across a question for which I cannot find an answer: Let $(A,\mu)$ be a commutative associative algebra ...
5
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1answer
98 views

Hartshorne's Algebraic geometry Chapter III ex. 9.10

I'm struggling with the exercise said in the title (Hartshorne, III, ex. 9.10). No problems in showing that $\mathbb{P}^1$ is rigid. In the second part, we want to show that $X_0$ being rigid does not ...
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41 views

Examples of complex manifolds satisfying the Kodaira-Spencer condition for unobstructed deformations

The theorem of existence proved by Kodaira and Spencer shows that a complex manifold $M$ for which the second cohomology group $H^2(M, \Theta T^{1,0} M)$ is trivial, $\Theta T^{1,0} M$ being the sheaf ...
7
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238 views

Studying Deformation Theory of Schemes

Versal Property Local Deformation Space Mini-versal deformation space I came across these words while studying these papers a) Desingularization of moduli varities for vector bundles on curves, ...
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18 views

Show that after the deformation the line element is given by the following equation

Assume throughout that $X_i,x_i$ (i=1,2,3) are respectively the material and spatial coordinates of a point referred to a common rectangular Cartesian coordinate system with origin $0$, and the the ...
2
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1answer
142 views

Question about homotopy equivalence

I have this proof but I don't understand why $i\circ j$ induces a homotopy equivalence, and how to see $j_*$ is injective at the level of homology? $X$ is a Banach space
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39 views

deformation space inside cohomology

For which smooth projective varieties $X$ is $H^1(X,T_X)$ (canonically ) contained in $H^\cdot(X,\mathbb C)$? If $K_X$ is trivial this is true. But are there other type of varieties?
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96 views

A reference on Kodaira-Spencer deformation theory

I am looking for an introduction to Kodaira-Spencer deformation theory. I have a background in Teichmüller theory, but I know almost nothing of (and am not so interested in) algebraic geometry. Do ...
4
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1answer
88 views

Lifting vector bundles along thickenings

Let $X_0 \to X$ be a nilpotent closed immersion of schemes. Is every vector bundle on $X_0$ the pullback of a vector bundle on $X$? The answer is yes when $X$ is affine. In general, there may be ...
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57 views

Infinitesimal deformation of coherent sheaves

Let $\mathcal F$ a coherent sheaf over an affine subset U, then we can consider it as an R module, if $U=Spec(R)$. Let $R$ an algebra over an algebraically closed field $\mathbb K$ and consider the ...
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76 views

Difference between isomorphisms, deformations, diffeomorphisms and homeomorphis,s

I am learning about complex algebraic varieties and have encountered several equivalence relations: isomorphism, birational, deformation equivalent, diffeomorphic and homeomorphic. I consider all of ...
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1answer
111 views

Difference between Kuranishi's existence theorem and Kodaira-Spencer's version.

Conserning infinitesimal deformation of a complex compact manifold $M$, Kuranishi showed in his generalized existence theorem that a local moduli space exists and is unique up to isomorphism. I want ...
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60 views

Double lines in $X$, implies $X \cong \mathbb{P}^{n}$

Let $X$ be a smooth, complex, projective variety. How to prove that if through two general points of $X$ there exists a double line, then $X \cong \mathbb{P}_{\mathbb{C}}^{n}$?
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32 views

unobstructed hypersurfaces in $\mathbb{P}^3$

Why is a smooth complex hypersurface in $\mathbb{P}^3$ of degree 6 or greater, unobstructed?
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160 views

Endomorphism of a local $k$-algebra inducing an automorphism modulo $m^2$ is an automorphism

The following is exercise 4.1 of Hartshorne's Deformation Theory, used in the proof given there of the sufficiency of the infinitesimal lifting criterion of smoothness: Let $(A,m)$ be a local ...
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1answer
134 views

Deformation Theory reference

I was recommended a book by Oort sometime ago to read with reference to deformation theory, in particular in positive characteristic with hyperelliptic curves. I couldn't find anything that seemed to ...
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69 views

Compute $h^1(X,T_X)$ for a plane curve $X\subset \mathbb P^2$

I would like to share a (partial) computation which I made, and might be completely wrong. My goal was to compute $h^1(X,T_X)$ for a uninodal curve $X\subset \mathbb P^2$ of degree $d$. I do not even ...
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1answer
52 views

Deformations of a cuspidal plane cubic

To practice with deformations, I am trying to compute the space of first order deformations of the cuspidal curve $X=\textrm{Spec }B$, where $B=P/I$, $P=k[x,y]$ and $I=(f)=(y^2-x^3)$. The conormal ...
5
votes
1answer
261 views

First order and infinitesimal deformations

Terminology. Let $X$ be an algebraic variety over some algebraically closed field $k$. By an infinitesimal deformation of $X$, I mean a flat surjective map $\mathfrak X\to S=\textrm{Spec }A$, where ...
4
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1answer
74 views

Exact sequences in the Mumford example

This question concerns the proof for the Mumford example of an Hilbert Scheme that has a non-reduced component. I am studying the proof given on R. Hartshone, "Deformation Theory", pp. 91-94. I do not ...
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42 views

Lifting extensions of sheaves

Consider a curve $X_0$ over an algebraically closed field $k$. Let $A$ be a local Artinian ring over $k$ and $X$ a scheme over $A$ with $X \otimes_A k = X_0$. For a sheaf $F$ on $X$ we denote by $F_0$ ...
4
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1answer
45 views

Versal extensions of algebras

I am reading the beginning of Sernesi's book "Deformations of algebraic schemes", and I am stuck on an example of a versal extension. Background: $A\to R$ is a fixed ring homomorphism. By an ...
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1answer
97 views

Deformations of complex structures and Deformations in quantum algebra

I am a bit confused about the meaning of the word "Deformation". For one, as in Wikipedia, it seems to refer to fixing a compact surface and varying the complex structure. For another, as in ...
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128 views

(Graded) deformations of algebras

I'm reading the article of Braverman and Gaitsgory, Poincaré-Birkhoff-Witt theorem for quadratic algebras of Koszul type, but I'm stuck in a point near the beginning. Let $A$ be a (positively) graded ...
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80 views

Infinitesimal deformation of projective schemes

Let $S$ be the coordinate ring $\mathbb{C}[X_0,...,X_n]$ in $n$-variables. Let $X=\mathrm{Proj}(S/I)$ be a projective scheme where $I$ is an ideal in $S$. Is there a $1-1$ correspondence of first ...
3
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1answer
140 views

Deformation of family of divisors

Given a family of complex compact surfaces $X \to B$ over a smooth curve $B$ and a one-dimensional family of effective Cartier divisors $\{D_t\}$ in the central fiber $X_0$. For simplicity, let's just ...
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250 views

Calculate the curvature of an arc (curved beam) with an initial displacement field

Given the arc showed below, I need to calculate the local radius of the deformed arc (dashed line), with an initial displacement field $u^{(0)}$, which is a function of $\theta$. The undeformed radius ...
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1answer
54 views

Space modelled on ring

I am learning some deformation theory. The dimension of the moduli space for some deformation problem is bound by the dimension of (usually) the first cohomology group of the object that is to be ...
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409 views

Classification of local Artin (commutative) rings which are finite over an algebraically closed field.

A result in deformation theory states that if every morphism $Y=\operatorname{Spec}(A)\rightarrow X$ where $A$ is a local Artin ring finite over $k$ can be extended to every $Y'\supset Y$ where $Y'$ ...
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1answer
82 views

Some exact sequences of cohomology on picard schemes

I'm looking for the answer of this question. $X$ is a variety over a field $k$, and $Art_k$ is the category of local artinian $k$-algebras whose residue field is $k$. I consider the formal completion ...
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57 views

How do I check whether an orbifold admits deformations?

Orbifolds $\mathbb{C}^2/\mathbb{Z}_n$, given by the action $(x, y) \mapsto (\zeta x, \zeta^{-1} y)$ with $\zeta$ a primitive $n^\text{th}$ root of unity, admit smooth deformations. This is because ...
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118 views

Genus of an embedded curve in projective smooth manifold deformed in its homology class

Let $C$ be a smooth embedded curve in an $n$-dimensional complex projective smooth manifold $X$ of class $[C]=\beta \in H_2(X,\mathbb{Z}).$ Can one make arbitrary arithmetic genus by deforming ...
5
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1answer
82 views

What is the universal deformation of $\widehat{\mathbb{G}}_a$ over $\mathbb{F}_p$?

Lubin and Tate show in their paper Formal moduli for one-parameter formal Lie groups that for any formal group over a field $k$ of characteristic $p>0$ with height $h<\infty$, the functor of ...
2
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1answer
63 views

Map from $\operatorname{Ext}^1(M,M)$ to $H^1(G, \operatorname{End}(M))$

The setting is as follows: $(R,m)$ is a local ring (assume noetherian, complete, if you need) and $\rho\colon G\to \operatorname{Aut}(M)$ is a group representation on the free, finite-rank ...
3
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1answer
160 views

Formal deformations of modules

This question is based on an exercise in Etingof's Introduction to Representation Theory. Let $A$ be an associative algebra (say over a field $k$) and $M$ an $A$-module. Write $\rho : A \to ...
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82 views

the existence of a closed subset of $\mathbb{A}^8\times \mathbb{A}^1$ which is flat over $\mathbb{A}^1$

I've been doing some reading on deformation theory and one way it is used is to study singularities on varieties while perturbing the varieties. I would actually like to use deformation theory to ...
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1answer
53 views

Reducing a flat morphism $\psi:X\rightarrow\mathbb{A}_{\mathbb{C}}^1\;$ to $\;\psi|_{Y}: X\cap Y\rightarrow \mathbb{A}_{\mathbb{C}}^1$

Suppose $\psi: X\rightarrow \mathbb{A}_{\mathbb{C}}^1$ is a flat morphism, where $X\subseteq \mathbb{A}_{\mathbb{C}}^n$ with $X$ not needing to be smooth, with $\psi^{-1}(0)$ being a complete ...
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1answer
77 views

explicitly constructing a certain flat family

Is it possible to construct a flat family $$ \phi:\mathbb{A}_{\mathbb{C}}^8=\operatorname{Spec} \mathbb{C}[x,y,z,w,a,b,c,d]\longrightarrow \operatorname{Spec} \mathbb{C}[t_1, t_2, t_3] ...
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1answer
83 views

If $\phi^{-1}(0)$ in $\phi:X\rightarrow Spec\; \mathbb{C}[t]$ is a complete intersection, then is $\phi$ flat?

This is a simple question so I am hoping the answer is quite simple as well. Suppose $\phi:X\rightarrow Spec\; \mathbb{C}[t]$ is a map such that the algebraic variety $\phi^{-1}(0)$ is a complete ...
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99 views

Deformations preserving dual graph

Let $C$ be a nodal curve over an algebraically closed field $k$. A deformation of $C$ over an Artinian ring $A$ over $k$ consists of a flat scheme $C'$ over $A$ and a closed immersion $i: C \to C'$ ...
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2answers
106 views

infinitesimal versus local deformation

When does one use infinitesimal deformation (over the ring of dual numbers $k[t]/(t^2)$) versus local deformation (over $k[t]$ or $k[t_1,\ldots, t_n]$)? It seems that one works over the ring of dual ...
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391 views

a flat deformation

The following is an example that I made up in order to understand a certain concept in one of Eisenbud's books. Consider $R = k[x_1,x_2,x_3,x_4]$ and let $I = \left< x_1 x_2+x_3 x_4 +x_2 + x_3, ...
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200 views

What does $H^0(Y',f^*N_Y)$ measure?

Let $X$ be a smooth variety and let $Y\subset X$ be a smooth subvariety. Let $f:Y'\to Y$ be a (say, finite surjective) morphism. When $f$ is the identity, the cohomology group $H^0(Y',f^*N_Y)$ ...
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1answer
144 views

Associativity of Moyal-like products

The Moyal product of two smooth functions $f,g$ on $\mathbb R^{2n}$ can be defined as $$ f\star g = \exp\left(-\omega^{ij} \frac{\partial}{\partial y^i} \frac{\partial}{\partial z^j}\right) f(y)g(z) ...
10
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2answers
425 views

Does every Poisson bracket on a commutative algebra come from a second-order deformation?

Let $A$ be a commutative algebra over a field $k$ (of characteristic not equal to $2$ to be safe). Recall that $f : A \otimes A \to A$ is a Hochschild $2$-cocycle if it satisfies $$f(ab, c) + f(a, b) ...