# 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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### 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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### 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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### 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 $R/m^n$-...
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### 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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### 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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### 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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### 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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### 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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### unobstructed hypersurfaces in $\mathbb{P}^3$

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