ag.algebraic geometry – Pro-representability and the obstruction to deformations of “stable curve of genus one + a section”

I have 2 questions about the theorem III.1.2 of Deligne-Rapoport’s “Les shemas de modules de courbes elliptiques”.


Let $k$ be a field, $Lambda$ a complete noetherian local ring with the residue field $k$, $C_0$ an elliptic curve or the Neron $N$-gon ($N$ relatively prime to the characteristic of $k$) with the standard generalized elliptic curve structure over $k$.
Then the deformation functor $D$ of $C_0$ (as a generalized elliptic curve, i.e., considering its morphism $m : C_0^text{sm} times C^0 to C^0$) on $mathscr{C}_Lambda$ is pro-representable.

First, the authors show that $C_0$ has no infinitisimal automorphisms over $k(epsilon)/epsilon^2$.
From this they conclude this highlighted statement.
I don’t understand why we get the pro-representability from this.

It seems that we can apply the arguments of (3.7) of Schelessinger’s “Functors of artin rings” even in this situation.
If so, then by this and by (3.8) of this Schlessinger’s paper we have the pro-representability.

But I don’t understand why we can apply the argument of (3.7), even if we add a morphism $C_0^text{sm} times C^0 to C^0$.


Let $k$ be a field, $(C_0, e)$ the Neron 1-gon with the “identity section” over $k$ or an elliptic curve with its identity.
Let $A, A’$ be artin local rings with the residue fields $k$ and $A’ to A$ a small extension.
(i.e., a surjection whose kernel is annihilated by the maximal ideal of $A’$ and is 1-dimension over $k$.)
Consider $(C, e)$, where $C$ is proper flat scheme over $A$, $e in C^text{sm}(A)$, and $C times_A k cong C_0$, compatible with $e$.
Then the obstruction to deformation of this $(C, e)$ to $A’$ is in $operatorname{Ext}^2(Omega_{C_0 / k}(e), mathscr{O}_{C_0})$,
and its first order deformation is parametrized by $operatorname{Ext}^1(Omega_{C_0 / k}(e), mathscr{O}_{C_0})$.

Since $C_0/k$ is locally complete intersection, I know that the obstruction to deformation of $C$ (just as a scheme) to $A’$ is in $operatorname{Ext}^2(Omega_{C_0 / k}, mathscr{O}_{C_0})$, and its first order deformation is parametrized by $operatorname{Ext}^1(Omega_{C_0 / k}, mathscr{O}_{C_0})$.
(By the section 10 of Hartshorne’s “Deformation theory”.)
(This, for a stable curve of genus $ge 2$, is also described in Deligne-Mumford.)

Why do we need $Omega_{C_0 / k}(e)$ instead of $Omega_{C_0 / k}$, if we add a section?

And I think that the obstruction is in $operatorname{Ext}^2(Omega_{C_0 / k}, mathscr{O}_{C_0})$ even in this case.
Indeed, since $e$ is in $C^text{sm}(A)$, if there exists a deformation $C’$ of $C$ to $A’$, then by the formally smoothness, $e in C’^text{sm}(A)$ lifts to $e’ in C’^text{sm}(A’)$.
So the existence of the deformation of $C$ automatically induces the existence of the deformation of the section.

This argument is described in the proof of ( of Kai-Wen Lan’s “Arithmetic Compactifications of PEL-Type Shimura Varieties”, to show the formally smoothness of the local moduli of abelian varieties.

This post is related to this, but it has no answer.

Obstruction to the existence of an invariant symplectic connection

Let $M$ be a symplectic manifold with a symplectic action of a Lie algebra $mathfrak{g}$. I am interested whether there exists a $mathfrak{g}$-invariant symplectic connection on $M$. Where does the obstruction live and how to construct it?

dg.differential geometry – Obstruction to the quantification problem for non-Abelian groups

Consider the Lie group $ S ^ 1 $. Recall that the associated Lie algebra is $ mathbb {R} $.

Let $ M $ to be a collector. Consider the second group of de-Rham cohomology $ H ^ 2 (M, mathbb {R}) $.

Let $ Omega in H ^ 2 (M, mathbb {R}) $ be a class of integral cohomology; that is to say, it is the image of an element of $ H ^ 2 (M, mathbb {Z}) $ under the natural map $ H ^ 2 (M, mathbb {Z}) rightarrow H ^ 2 (M, mathbb {R}) $. Then we know that we can build a principal $ S ^ 1 $-package $ P rightarrow M $ on the collector $ M $, a connection $ 1 $-form $ omega $ on the collector $ P $ so that the associated form of curvature (which is a $ 2 $-form on $ P $, but can only be projected on a $ 2 $-form on $ M $) is precisely the $ 2 $-form $ Omega $ with which we started.

Question: How far can we go to relax the condition that the structure group is abelian?

Question: Either $ G $ to be a Lie group and $ mathfrak {g} $ to be his Lie algebra. Let $ M $ to be a collector and $ Omega $ be a $ mathfrak {g} $-valued $ 2 $-form on $ M $. Under what conditions can a principal be found $ G $ group on collector $ M $and a connection on $ P (M, G) $ whose curvature is $ Omega $?

topology at.algebraic – Definition of the 1st degree obstruction class

Recently, I took an obstruction lesson illustrated by Milnor.
He defined $ mathfrak {o} _i $by an element $ H ^ i (M; pi_ {i-1} (V_ {n-i + 1} (F)) $, which is cohomology with local coefficients.

But the 0th homotopy group has no group structure, and the definition of $ mathfrak {o} _1 $ does not work. So is there a 1st degree obstruction class and if it exists, how do you define it?

at.algebraic topology – obstruction cocycle for non-simple spaces using local coefficients

This question is similar to here, but I was hoping for a concrete theorem statement surrounding the obstruction cocycle for non-single spaces.

I hope for a theorem like this:

Let $ A subset X $ such as $ pi_1 (A) = pi_1 (X) $ and $ f: A to Y $ to be a function. leasing $ pi_n (Y) $ be a $ mathbb Z ( pi_1 (A)) $ module, where $ pi_1 (A) $ acts via $ f _ * $ and the usual action of $ pi_1 (Y) $. Assume that $ H ^ * (X, A, pi_n (Y)) = 0 $ for everyone $ n in mathbb N $. There is then an extension of $ f $ to all $ X $. More generally if $ pi_1 (X) = pi_1 (A) / N $ and $ N subset ker f _ * $, then if $ H ^ * (X, A, pi_n (A) / N) = 0 $, there is an extension.

The slightly awkward generalization is due to my desire to prove universal property for construction Quillen & # 39; s more presented in these notes, prop 1.1.2.

A naive assumption on my part would be to apply the usual obstruction theorem to universal coverage of $ Y $, then if we use local coefficients, we can make sure that $ f: A à tilde {Y} $ is a lift of $ f: A to Y $, but I'm not completely sure.

Should the Democrats in the House be tried for abuse of power and obstruction in Congress?

Yes, it lasts from the first day. They did not have time to discover that there is nothing. They tried with Muller report, NO collusion. Judged with a prostitute – it went well. – I've tried with all the lawyers, I only have them with tax prospects. – Now a phone call that the transcript is open to everyone.

All the while, they are focusing on Trump's removal of his duties.

It does not matter that millions of people voted for Trump.

Do you think that Lewandowski, in his idiotic testimony, contributed more to implicating Trump in the obstruction of justice that he did to defend him?

The pathetic sycophant with the imbecile face acted as a witness for the crowd!

What a silly little man, acting like he did, he perfectly explained that Trump is guilty!

Oh, but of course, he's running for the Senate race in New Hampshire and hopes Trump will take him there. The silly is just another silly sap that will sing like a canary in years !!!

Aggressive geometry – Image of the obstruction map for a relative quotient schema

Let $ f: X to S $ to be a projective morphism between projective schemes of finite type, and $ mathcal {O} _ {X} (1) $ a $ f $-pack of lines. Given a $ S $– consistent flat $ mathcal {O} _ {X} $-module $ mathcal {H} $ and a polynomial $ P $we can form the relative rating scheme $ pi: Q = text {Quot} _ {X / S} ( mathcal {H}, P) to S $ setting quotients $ mathcal {H} to mathcal {F} $ with the Hilbert polynomial $ P $, up to equivalence. According to Huybrechts and Lehn Proposition 2.2.7 (, we obtain the following result:

For everyone $ s in S $ and $ q_ {0} in pi ^ {- 1} (s) $ corresponding to the quotient $ mathcal {H} _ {s} to F $ with the kernel K $ like sheaves $ X_ {s} = f ^ {- 1} (s) $we have the exact sequence
$$ 0 to text {Hom} _ {X_ {s}} (K, F) to T_ {q_ {0}} Q xrightarrow () {d pi} T {{s} S xrightarrow ( ) { mathfrak {o}} text {Ext} ^ {1} _ {X_ {s}} (K, F) $$
or $ mathfrak {o} $ is the obstruction card.

My question is: what are the other entries in this exact sequence? Basically, I try to calculate the cokernel of the obstruction map $ mathfrak {o} $but I have trouble understanding how this is defined abstractly in Huybrechts and Lehn. Is there a geometrically intuitive way to understand what the image or core of an obstruction card code? References that could help with the calculation?

[ Politics ] Open question: NO COLLUSION! NOT OBSTRUCTION! EXONERATION COMPLETE !!! Is your cup of tears left overflowing?

[Politics] Open question: NO COLLUSION! NOT OBSTRUCTION! EXONERATION COMPLETE !!! Is your cup of tears left overflowing?

Why do Republicans think the obstruction of justice is acceptable?

Whatever Democrats accuse Republicans is almost always something the Democrats do themselves.

The Democrats created the Ku Klux Klan after the civil war. The Democrats brought us to Vietnam. Lying Ted Kennedy was a democrat. Democrats have illegally spied on the Trump campaign BEFORE the 2016 presidential election. The Democrats want the tide of illegal immigrants to continue entering the country. Call of name. Want to politics. Racial politics.

Yet Democrats call themselves compassion party and many are still fearful.