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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Arithmetically Cohen-Macaulay curve on a quadric

If $Y$ is a curve of bidegree $(a,b)$ on a smooth quadric surface $Q\subset \mathbb{P}^3$, how do we see that it is arithmetically Cohen-Macaulay (ACM, for short) iff $|a-b|\leq 1$? If (like me) ...
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39 views

Hartshorne Deformation theory Exercise7.1

This is a exercise in "Deformation Theory [Hartshorne]". Let C be a local Artin ring with residue field k. Let X be a scheme flat over C , and let$X_0=X\times _Ck$. If F is a coherent sheaf on X ...
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30 views

Existence of a certain nodal quartic curve

I am reading this (https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/Univcodim2invent.pdf) paper of Voisin, but I am having some trouble with the proof of Sublemma 2.8 (it might be something very ...
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25 views

Details about the definition of: “deformation of a family”

Let $f:X\to Y$ be a flat, surjective morphism of $k$-schemes with connected fibres i.e. $f$ is a family. Definition: Let $T$ be a $k$-scheme. A deformation of $f$ (over $T$) is a family $g:\mathfrak ...
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$\mathcal{T}^i(X/Y,\mathcal{F})$ forms a sheaf

In Hartshorne's Deformation Theory, given an $A$-algebra $B$ and a $B$-module $M$, he defines these functors $T^i$ for $i=0,1,2$ that outputs $B$-modules $T^i(B/A,M)$. In Exercise 3.5, he asks the ...
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55 views

Normal bundle to doubled line on cubic surface

Let $X\subset \mathbb{P}^3$ be a smooth cubic surface, and $L\subset X$ be a line. Let $Y\subset X$ be the subscheme cut out by the divisor $2L$. How do we compute ...
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51 views

$T^i$ functors in Hartshorne's Deformation Theory

In chapter 3 of Hartshorne's Deformation Theory, he defines functors $T^i$ for $i=0,1,2$ that take as input a ring homomorphism $A\rightarrow B$ and a $B$-module $M$ and outputs $T^i(B/A,M)$, a ...
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50 views

Flat schemes over artinian local ring with isomorphic special fibers

I'm sure this is standard, but I don't know where to find it. If $A$ is a local artinian $k$-algebra, $X_1,X_2$ are finite type schemes flat over $A$, and $f:X_1\rightarrow X_2$ is a morphism over ...
4
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60 views

Serre duality explicitly on curves

Consider a Riemann surface $X$, with genus and marking so that a suitable moduli space exists. It's a well-known fact that the tangent space to that moduli space at $X$ (in other words, the space of ...
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133 views

How is the smoothness of the space of deformations related to unobstructedness?

As a beginning differential geometer, I've been trying to learn about deformation theory. Other than Kodaira's book, I've found virtually no references from the point of view of differential ...
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86 views

Exercise from “Moduli of Curves”

I am struggling with Exercise 1.5 from the book "Moduli of Curves" by Harris and Morrison. Namely, I want to show that $\mathbb{P}^1_{\mathbb{C}}$ is a fine moduli space of lines through the origin in ...
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51 views

Normal Space of an orbit at a point

Let $X$ be a curve of genus $g$. If $d>n(2g-1)$,then for any vector bundle $E$ of rank $n$ and degree $d$ over $X$ has $H^1(X,E)=0$ and $E$ is generated by its global sections. For such a vector ...
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98 views

Deformation theory introduction without unnecessary machinery

I would like to find an introduction (book, article and/or lecture course) in deformation theory that does not use unnecessary machinery (for example, schemes instead of complex varietie, or ...
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63 views

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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43 views

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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45 views

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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1answer
25 views

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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58 views

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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72 views

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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85 views

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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39 views

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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1answer
92 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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105 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 ...
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112 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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47 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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36 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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98 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 ...
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1answer
117 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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49 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 ...
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551 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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146 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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41 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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181 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 ...
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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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83 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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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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177 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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66 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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34 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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172 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
180 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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70 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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64 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
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378 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 ...
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80 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 $F_0$ ...
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1answer
52 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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121 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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141 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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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 ...