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One of my favourite examples is the following theorem, due to S. Mori: Theorem A. Let $X$ be a smooth complex projective variety such that $-K_X$ is ample. Then $X$ contains a rational curve. In f …

The answer is yes, in fact the following result holds. Theorem. Let $f \colon X \to T$ be a flat deformation of a Fano variety $X_0:=f^{-1}(0)$ having at most terminal, $\mathbb{Q}$-factorial singula …

As far as I know, the prototypes of obstruction theories in algebraic geometry originated from the more general Kodaira-Spencer theory of deformation of complex manifolds [see Kodaira-Spencer, On defo …

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 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 rigidity of quotient singularities in dimension greater or equal than $3$ was established by Schlessinger in his paper Rigidity of quotient singularities, Invent. Math. 14 (1971). Roughly speaking …

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 …

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 …

Du Val singularities are indeed smoothable and, in fact, more is true: they are of class $T$, namely, they are quotient singularities admitting a $1$-parameter $\mathbb{Q}$-Gorenstein smoothing. See D …

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 …

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 …

A good survey on this problem is the paper by Greuel, Lossen and Shustin Equisingular families of projective curves, see http://arxiv.org/pdf/math/0612310.pdf. In particular, at page 5 one can find t …

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 …

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 $ …