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If you want to look at Deformation Theory from a complex-analytic point of view ( i.e. in the spirit of Kodaira-Spencer "deformations of complex structures" ), you need to solve the Maurer-Cartan equ …

Probably Illusie wrote "Zariski's Main Theorem", but he intended the Theorem of Formal Functions (which is the key result needed in the modern proof of Zariski's Theorem). In fact, the Theorem of For …

Let's assume that we are working over $\mathbb{C}$. First of all, hypersurfaces in $\mathbb{P}^n$ are unobstructed, so their first-order deformations always correspond to small deformations (deformat …

There is actually the following general result. Let us consider a morphism of algebraic schemes $f \colon X \to Y$, where $X$ is reduced and projective and $Y$ smooth. Then the first order deformatio …

Yes, it is possible, as shown by the following simple example. Think of a double cover $S$ of $\mathbb{P}^2$ branched on two smooth conics intersecting transversally: it has four singularities of ty …

Question 1. The answer is yes, and the classical example is as follows. Is it possible to find a one-parameter family $\psi \colon \mathcal{X} \to \Delta$ such that $X_0$ is isomorphic to the cone ove …

You can look at Manetti's paper Deformation theory via differential graded Lie algebras, arXiv:math/0507284. As the title suggest, it follows the philosophy that every deformation problem is governed …

The answer is no. In fact, flat maps are not required to be smooth maps in general. For instance, a flat family of smooth curves degenerating to a nodal curve is clearly not locally trivial. Howeve …

The answer is the following and can be found in Hartshorne's book Deformation Theory, see in particular Exercise 10.5 page 83. We work over an algebraically closed field $k$. Then there is an exact …

The answer is no, since first-order deformations can be obstructed, so that they do not give necessarily global embedded deformations of the subscheme. For instance, Mumford gives an example of a sm …

These curves may actually exist, as the following example shows. Let $C$ be a smooth curve of genus $g$. Then the diagonal $\Delta \subset C \times C$ is isomorphic to $C$ and has self intersection $ …

The answer is no, in fact there exist examples of non-reduced projective curves which are non smoothable. Perhaps the easiest example is the double line, i.e. the scheme $X=2L$, where $L$ is a line o …

The way I see this is the following, which I learnt from Sernesi's book "Deformations of algebraic schemes". Assume that you have an infinitesimal deformation $\xi$ of a nonsingular scheme $X$ over $ …

Let us assume that we are working over $\mathbb{C}$. If $F$ is a stable sheaf then the answer to your first question is yes. In fact, in this case there exists a quasi-projective moduli space $M$ an …

Regarding your last question, the answer is yes, since there are terminal singularities that are not rigid. For instance, in the recent preprint by Taro Sano On deformations of Fano threefolds with te …