# Tagged Questions

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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### Affine smooth variety has only trivial first-order deformations

Does someone have a quick and direct argument that infinitesimal deformations of an affine smooth variety over a field $k$ are only the trivial infinitesimal deformations? (without previous ...
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### Behaviour of Chern class under deformation

The setting is the following: $f: X \rightarrow T$ is a smooth projective map of complex algebraic varieties, and $L$ is a line bundle on $X$. My question is the following: is $c_1(L_t)$ (in ...
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### Ribbons of abelian varieties

Let $X$ be a scheme over $\mathbb C$, such that $A=X_{\textrm{red}}$ is an abelian variety. Suppose the tangent spaces to $X$ are all of dimension $\dim X+1$. Then $X$ is non-reduced everywhere, but I ...
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### analytic deformation of a compact set in the complex plane

Let $K$ be an uncountable compact set in $\mathbb{C}$ such that zero is a limit point of $\partial K$, and such that $|k|\leq 1$ for all $k\in K$. I would like to find an analytic function ...
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### What is the difference between isometric deformation and conformal defomation?

What is the difference between an isometric deformation and a conformal deformation? In fact, even the definition of both deformations is still unclear clear for me. Is it possible to define an ...
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### How can we continuously deform a height 1 formal group law into a height 2 formal group law?

A Quick Review: The complex elliptic curve $\mathbb{C}/(\mathbb{Z} + \tau \mathbb{Z})$ may be rewritten using the exponential, $\text{exp(}{2 \pi i \tau}) =: q$ as $\mathbb{C}^\times/q^\mathbb{Z}$ . ...
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### Is a formal deformation of a Lie algebra an example of a formal group law?

I stumbled across the following definition of the formal deformation of a Lie algebra, and it looks like a group object in the category of formal schemes (not necessarily commutative or ...
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### Obstruction map for Quot schemes is surjective

I am reading "Lectures on vector bundles" by Le Potier and am confused about a statement in the proof of the existence theorem on page 144, after Lemma 8.6.6. Let $X$ be a projective curve (can ...
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### 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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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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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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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### 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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### A reference on Kodaira-Spencer deformation theory

I am looking for an introduction to Kodaira-Spencer deformation theory. I have a background in TeichmĂŒller theory, but I know almost nothing of (and am not so interested in) algebraic geometry. Do ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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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### 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 $F_0$ ...
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### 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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### 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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I'm reading the article of Braverman and Gaitsgory, PoincarĂ©-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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
### 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 ...