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

# Questions tagged [deformation-theory]

203 questions

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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 $Y'$ is an infinitesimal thickening of $Y$, then $X$ is…

user16544

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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, Int. Symp on Algebraic Geometry by C. S. Seshadri…

Babai

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### 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
$$f(ab, c) + f(a, b) c = f(a, bc) + a f(b, c)$$
and recall that $\{ -,…

Qiaochu Yuan

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### 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 geometry. As such, my understanding is hazy.
I am…

Jesse Madnick

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### Glueing of algebraic surface along two intersecting curves

My question arises as part of understanding an analogue of the normalization of singular curves. Assume that $C$ is such a curve and that $p\in C$ is a singular point of $C$ (and the only one for simplicity). Then the normalization $\widetilde C\to…

user526015

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### 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 $I = \left< x_1 x_2+x_3 x_4 +x_2 + x_3, x_1 x_4+ x_2 x_3+ x_2 +x_3 \right>$ and $J = \left<…

math-visitor

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### Cotangent complex of dual numbers

Let $k$ be a ring, put $k[\epsilon] := k[t]/(t^2)$. What is the cotangent complex of $k[\epsilon] \to k$?
I know $\Omega^1_{k/k[\epsilon]]}$ is going to be zero. But I don't see any way around explicitly writing down a cofibrant replacement to get…

Maanroof

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### Where can I learn about differential graded algebras?

I want to learn more about differential graded algebras so that I can construct explicit examples of derived schemes over characteristic 0, compute smooth resolutions of morphisms of schemes, and compute examples of cotangent complexes of morphisms…

user1876508

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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 $A$ is a local Artin $k$-algebra, such that $X$ is…

Brenin

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### (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 associative algebra over a field $k$. They define…

Mauro Porta

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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 order infinitesimal deformations of $X$ in…

Chen

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### Smoothness of total spaces with nice singular fibers

Let $f:X \to Y$ be a (flat and projective) family of hypersurfaces over a smooth curve $Y$. If every fiber of $f$ is smooth, it is clear that the total space $X$ is also smooth. I would like to know if the smoothness still holds when we also allow…

Akatsuki

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### Why are infinitesimal deformation defined to be over Artin rings?

I am wondering what is the motivation for defining infinitesimal deformations to be over the spectrum of Artinian rings, i.e rings that have a finite number of prime maximal ideals.
I have been trying to understand the connection between…

Yuugi

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### What are $q$-deformations?

This question has already appeared in a lot of different ways and here is another one.
First of all, many people know the typical quantum group $U_q(\mathfrak{sl}_2)$ by generators and relations. This thing is often called a $q$-deformation of…

Mathematician 42

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### 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 simple but I don't exactly see it).
Namely, let…

pozio

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