11

Somehow that question slipped my radar, sorry!
The truth is that shamefully I'm not able to say much, as I don't have a strong knowledge of resolution of singularities. But at least so far, the flow of information has been from characteristic $p$ to characteristic $0$, or more specifically to mixed characteristic. So maybe I wouldn't be surprised if once ...

7

I personally like the notes by Eugene Xia: Abelian and Non-Abelian Cohomology to build intuition.
But for a definitive source, I would read Simpson: Moduli of representations of the fundamental group of a smooth projective variety I and Moduli of representations of the fundamental group of a smooth projective variety II
True, the above results of Simpson ...

7

The $\tilde{\rho}$-invariants contain the $\tilde{\rho}_G$-invariants, at least.
The map $\Sigma_g \to \Sigma_b$ defines a pullback map $ H^1( \Sigma_b, \mathbb Q) \to H^1(\Sigma_g, \mathbb Q) $ (i.e. cohomology is a contravariant functor).
The image of this pullback map is the $G$-invariants in $H^1(\Sigma_g, \mathbb Q)$. The image of this pullback map ...

6

I prefer primes like 1000003 and 1000000007 because it is easy to recognize small integers and rational numbers with small denominators in the output. For instance, modulo 1000003 we have
1/3=666669,
-1/3=333334,
1/5=600002.
I know that's not what you are asking from the mathematical point of view, but from practical point of view it is very helpful.
EDIT: ...

4

Condition (8) implies that the map is unramified, and since it is also flat by (6) (which by the way implies $S$ is CM by Miracle Flatness), it is etale, and so $S$ is regular. Note you don't need (5) here.
Without condition (8) you can take $R=\mathbf{C}[[x,y]]$ and $S=R[w]/(w^2 - xy)$.

4

I never liked the definition of formal completion (as defined in EGA or Hartshorne), so I'll use the following definition from Brian's formal GAGA paper.
Definition: Let $X$ be a locally Noetherian scheme, and $I \subseteq \mathcal{O}_X$ a coherent ideal. The formal completion of $X$ along $V(I)$, is the ringed topos $(X_{\text{Zar}}, \varprojlim \mathcal{O}...

4

The universal property of the total space $$\mathbf{V}(E^\vee) = \operatorname{Spec}_M \operatorname{Sym} E^\vee $$ of a vector bundle (locally free sheaf) $E$ on some scheme $M$ is: giving a map $T\to \mathbf{V}(E^\vee)$ corresponds to giving a map $f\colon T\to M$ and a section $\omega$ of $f^* E$.
We apply this to $T=\operatorname{Spec} k[\varepsilon]$, $...

4

This is true and follows from:
Claim: Let $x$ be a $n\times n$ matrix with $\mathbb{C}$-coefficients. Then the centralizer $C(x)$ of $x$ in $GL_n(\mathbb{C})$ fits into a short exact sequence $1\rightarrow U\rightarrow C(x) \rightarrow \prod_{n_i} GL_{n_i}\rightarrow 1$, where $U$ is a unipotent group and $n_i$ a sequence of integers.
Since an extension of ...

3

No.
The pre-closure of the join of two irreducible projective varieties is NOT necessarily quasi-projective.
Let $X$ be a smooth plane conic and let $Y$ be a single point of $X$. The pre-closure of the join of $X$ and $Y$ is the union of all the lines through points $x \in X$, $y \in Y$, $x \neq y$. Of course there is only one point in $Y$. So we get the ...

3

Yes. Any elliptic curve which is not base-changed from $\mathbb Q$ will do the trick.
If there were such a nonconstant map, there would be a surjection from the Jacobian of the modular curve to the elliptic curve. There are many ways to rule this out.
One way is to look at $\ell$-adic Tate module of the elliptic curves. It's easy to check that the action of $...

2

$\DeclareMathOperator\Fr{Fr}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Diff{Diff}\DeclareMathOperator\Sym{Sym}$Another, more geometric, approach to this question is as follows: let $G$ be a Lie group acting on $M$, and let $P$ be a principal $H$-bundle over $M$. (This addresses your question (2), but it includes (1) if you take $P = \Fr(V)$ to be the ...

2

For polynomials with rational coefficients, the cylindrical decomposition in real algebric geometry gives a way to find whether a polynomial (or system of polynomials) has finitely many roots in $\mathbb{R}^n$ or not, and if it has, it can compute the coordinates of each root, in the form of the root of a single-variable polynomial (and specify which root by ...

1

Let me point out that if $G$ is disconnected, then the action might not extend. The simplest example I can think of would be a real line bundle over a torus with $w_1$ equal to one of the cohomology generators. Then a ${\mathbb Z}_2$ action that switches the two generators of cohomology will not extend, because the bundle is not preserved. One can do the ...

1

Your question can be rephrased as follows: I know something about split square-zero extensions. How can I (categorically) conclude something about all square-zero extensions? The key observation is that square-zero extensions are the torsors for the split square-zero extensions. I first learned about this general idea from section 7.4.1 of Lurie's Higher ...

1

Daniel Halpern-Leistner is teaching a foundational course on moduli theory based on stacks at Cornell. Very nice addition to the literature in my opinion.

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