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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 …

The answer is yes: the germ of complex space $(M_h, \, [X])$ is analytically isomorphic to the quotient $S/\mathrm{Aut}(X)$. This is true not only for moduli spaces of hyperbolic curves, but also i …

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 …

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 …

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 …

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 …

A candidate for a counterexample is the Cartwright-Steger surface, see http://arxiv.org/abs/1412.4137. It is complex surface $X$ of general type with $$p_g(X)=q(X)=1, \quad K_X^2=9,$$ hence $K_X^2=9 …

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. 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 …

This is a $A_2$ surface singularity, in fact it is isomorphic to the quotient $\mathbb{A}^{2}/\mu_{3}$ where the action is given by $$ \begin{array}{ccc} \mu_{3}\times\mathbb{A}^{2} & \longrightarrow …

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 …

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 …

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 …

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 $ …

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 …