# 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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### Fourier Transform Deformation [on hold]

Let f(x,y) denote an original image containing an object. After deformation the image is expressed as f(a⋅x+b⋅y, c⋅x+d⋅y). Find the Fourier transform of f(a⋅x+b⋅y,c⋅x+d⋅y) in terms of F(w1,w2).
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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 ...
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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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### 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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### 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....
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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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### 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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### 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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### 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 ...
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 ...