GIT-quotient of affine variety

Let $G$ be a reductive group, $X$ a $G$-variety, $L$ a line $G$-bundle on $X$. Then the GIT-quotient is defined as
$$X //_L G:=\operatorname{Proj} \left( \bigoplus_{n \geqslant 0} H^0(X, L^{\otimes n})^G \right).$$
If $X$ is affine and $L=\mathcal O_X(\chi)$ for a character $\chi: G \to \mathbb G_m$, then the GIT-quotient may be considered as

• first taking an affine quotient $\operatorname{Spec} (\mathbb k[X]^{G_\chi})$ by $G_\chi:=\operatorname{Ker}(\chi: G \to \mathbb G_m)$,
• then taking a quotient by $\mathbb G_m=G / G_\chi$ via replacing $\operatorname{Spec}$ to $\operatorname{Proj}$ of non-negative gradings.

For the details on the construction see
Mukai An introduction to invariants and moduli, remark 6.14.

The first step is fairly stupid, but a proof that GIT-quotient is very general is given in
Mumford Geometric invariant theory, converse 1.13 of chapter 1.4.

From MO:253379.

Tangent space to moduli space of sheaves

Let $X$ be a projective variety. Suppose that the moduli space $\mathcal M$ of nice sheaves with fixed rank and Hilbert polynomial exists. Then

$$T_{[E]}\mathcal M=\mathrm{Ext}^1(E, E).$$

For a partial proof see
Hartshorne Lectures on Deformation Theory, theorem 2.6

From MO:253578.

Castelnuovo-Mumford regularity

Let $X$ be a projective variety. Then a coherent sheaf $F$ is called $m$-regular by Castelnuovo-Mumford if
$$H^i(F(m-i))=0.$$
It looks strange that one does not simply require
$$H^i(F(m))=0.$$
But then $m$-regularity would not imply $(m+1)$-regularity. For example, $\Omega^1_{\mathcal P^n}$ with $n>1$ is $(-1)$-regular, but not $0$-regular.

For properties of Castelnuovo-Mumford regularity, see
Mumford Lectures on Curves on an Algebraic Surface, lecture 14.

From MSE:2008517.