I had been trying to understand when a family of rationally connected varieties is singular or non-reduced. However, I could not find a good reference for this topic. Could someone …

We know in differential geometry, given a $C^k$ manifold for $k>1$, the tangent space at a point in this manifold is parametrized by curves passing through this point modulo certai …

I just start to study Hartshorne's 'Deformation theory'. I have a stupid question about $T^i$-functors. It may be an easy algebra. Exercise 1.3.3. Let $B=k[x,y]/(x^2,xy,y^2)$. S …

For any finite dimensional Lie algebra $\mathfrak{g}$, we know that the universal enveloping algebra $U(\mathfrak{g})$ is a deformation of the symmetric algebra $S(\mathfrak{g})$. …

Say I want to construct a flat family of affine schemes using deformation theory. To be more specific, suppose I have a local $k$-algebra $A$, and I want to find a flat family $\m …

Let $X, Y$ be irreducible projective varieties and $Y$ be smooth. Let $f:X \to Y$ be a flat projective morphism. Assume that a special fiber of $f$ is non-reduced i.e., there exist …

(Cross-post from math.stackexchange, where it has received no attention.) Orbifolds $\mathbb{C}^2/\mathbb{Z}_n$, given by the action $(x, y) \mapsto (\zeta x, \zeta^{-1} y)$ with …

Fix two positive integers $d, e$ and assume $d>e$. Is it true that a general degree $e$ curve which lies in a complete intersection of a degree $e$ and a smooth degree $d$ surface …

Let $\mathcal{X} \to B$ be a family of smooth projective varieties over a field $K$ (possibly finite). Assume that each fiber $\mathcal{X}_b$ of the family has a $K$-rational point …

Let $X, Y$ be projective varieties in $\mathbb{P}^n$ for $n>10$. Assume that dimensions of $X,Y$ are greater than $n/2$. My first question is as follows: Is there a …

As is well-known, for the compact semi-simple Lie groups $G$, there exist non-commutative Hopf algebra deformations ${\cal O}_q[G]$ of their coordinate algebras ${\cal O}[G]$, the …

The following baby version of virtual fundamental cycle is well known: Let $M\subset V$ be the zero locus of a section $s$ of a vector bandle $E \to V$, in general $s$ is not tran …

Let $X$ be a smooth projective variety over a finite field $\mathbb{F}_p$. What are the conditions for the existence of a projective variety $X'$ over $\mathbb{Q}_p$ such that $X$ …

Let $X=Y\times Z$ be a product of complex manifolds $Y,Z$. Is it true that there exists no rigid curve on $X$? Here I mean by a rigid curve a curve which is not a member of any fam …

Suppose we have a family of objects $\Xi \to S$ over a base smooth projective scheme $S$. Take a closed point $p\in S$ and consider the tangent space to $S$ at $p$. Can one constru …