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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18
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318 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 $Y'$ ...
14
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0answers
180 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 $H^0(Y',f^*N_Y)$ ...
10
votes
2answers
381 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) ...
6
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1answer
294 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, ...
5
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1answer
136 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 ...
5
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0answers
67 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 ...
5
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1answer
71 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 ...
4
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1answer
38 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 ...
4
votes
1answer
63 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 ...
4
votes
1answer
37 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 ...
4
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1answer
40 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 ...
4
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39 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
votes
0answers
98 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 ...
4
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0answers
79 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 ...
3
votes
1answer
122 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 ...
3
votes
1answer
115 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 ...
3
votes
1answer
141 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 ...
3
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0answers
55 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 $X \cong \mathbb{P}_{\mathbb{C}}^{n}$?
3
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0answers
93 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 ...
2
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1answer
49 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 ...
2
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1answer
82 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 ...
2
votes
1answer
108 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) ...
2
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1answer
58 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 ...
2
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0answers
31 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 ...
2
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0answers
24 views

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

Why is a smooth complex hypersurface in $\mathbb{P}^3$ of degree 6 or greater, unobstructed?
2
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0answers
48 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 ...
1
vote
1answer
73 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 ...
1
vote
0answers
59 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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0answers
94 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 $i: C \to C'$ ...
0
votes
1answer
74 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] ...
0
votes
1answer
72 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 ...
0
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0answers
52 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 ...
0
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0answers
16 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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0answers
36 views

Question about deformation

Why if $H\setminus X$ has no crititical point then $X$ is a deformation retract of $H$? $H$ is a Hilbert space $X$ is a subspace of $H$ ($X=\varphi^b=\lbrace x\in H, \varphi(x)\leq b\rbrace$ , ...
0
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0answers
24 views

Question on the Deformation lemma

I have a functional $J$ defined on a hilbert space, with a finit number of a critical point $v_1,...,v_m$ let $b>\max\lbrace J(v_1),...,J(v_2)\rbrace$, and i want to prove that the set ...
0
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0answers
38 views

Deformation of sympletic manifolds and deformation of complex manifolds

I know that for a complex manifold you can determine the variation of the parameter by picking a $\theta (t) \in H^1(\mathscr{M}, \Theta)$ (where $\Theta$ is the sheaf of vector fields over ...
0
votes
1answer
71 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 ...
0
votes
0answers
136 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 ...
0
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0answers
41 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 ...
0
votes
1answer
68 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 ...