Families over Artin Rings and Deformations

Let us work with a class of schemes over an algebraically closed field $ k$ such that any two schemes in this class are isomorphic.

An example of such a class would be genus zero nonsingular curves over $ k$ . Let $ X$ act as a representative of this class.

Is the set of isomorphism classes of such families over an Artin ring $ A$ equivalent to the set of isomorphism classes deformations of $ X$ over $ A$ ?

The only problem I am encountering with verifying such a statement is we allow families $ \mathcal{X}$ over $ B$ to have varying fibers while a deformation fixes the fiber.

** Please assume any nice property you may wish. I did not want to specify the class to be nonsingular genus zero curves because I do not want $ H^1(X, \mathcal{T}X) =(0)$ ., i.e I am interested in scheme which have nontrivial deformations.**

Fibers of blow up in families

Let $ T$ be a smooth curve over $ \mathbb{C}$ and $ p:\mathbb{P}^n \times T \to T$ the natural projection. Let $ V \subset \mathbb{P}^n_T$ be a $ T$ -flat subscheme of codimension at least $ 2$ and $ \pi: \mathrm{Bl}_V \mathbb{P}^n_T \to X$ be the blow-up of $ \mathbb{P}^n \times T$ along $ V$ . Under what conditions on $ V$ can we say that for all $ t \in T$ , the fiber over $ t$ of the morphism $ \pi$ , is the blow-up of $ \mathbb{P}^n$ along $ V_t$ ? Any idea or referece will be most welcome.

An exact complex of tensor families

Given a field $ \mathbb{K} = \mathbb{R}$ or $ \mathbb{C}$ , some natural numbers $ q,n,m$ with $ q \leq n$ and $ m < n$ and a polynomial algebra $ S := \mathbb{K}[\eta_1, \dots, \eta_n]$ . We can consider $ \otimes_\mathbb{K}^q S$ an $ S$ -module via $ $ \eta_i \cdot p_1 \otimes \dots \otimes p_q := \sum_{j=1}^q p_1 \otimes \dots \otimes \eta_i \cdot p_j \otimes \dots \otimes p_q \quad \forall p_1, \dots, p_q \in S$ $

Fuks claims that the following statement on pages 85 to 86 of his book “Cohomology of infinite-dimensional Lie algebras” is obvious:

If there is a family of tensors $ \{c_{i_1 \dots i_m} \in \otimes_{\mathbb{K}}^q S \mid i_1, \dots, i_m = 1, \dots, n\}$ which is skew-symmetric with respect to its indices, with the property that

$ $ \sum_{r=1}^{m + 1} (-1)^r \eta_{i_r} \cdot c_{i_1 \dots \hat{i_r} \dots i_{m+1}} = 0 \quad \forall i_1, \dots, i_{m+1},$ $

then it follows that there exists some family of tensors $ \{b_{j_1 \dots j_{m-1}} \in \otimes_{\mathbb{K}}^q S \mid j_1, \dots, j_{m-1} = 1, \dots, n\}$ , also skew-symmetric with respect to its indices, with the property that

$ $ \sum_{r=1}^{m} (-1)^r \eta_{i_r} \cdot b_{i_1 \dots \hat{i_r} \dots i_{m}} = c_{i_1 \dots i_m} \quad \forall i_1, \dots, i_{m}.$ $

If one tries to write it out in coefficients, one can sort of see why this might be true, and clearly this looks like it arises from exactness of some “complex of tensor families”, but I’m unsure how to write down any of these approaches in a way that one could call “obvious”. I am wondering if I am missing a point of view from which this indeed is an obvious statement. Thanks in advance!

(The context of this statement is the triviality of the Lie algebra cohomology of $ n$ -dimensional formal vector fields in order $ q = 1, \dots, n$ . While I did find another source that proves this, the approach in Fuks’ book still confuses me, and I’d be happy to understand it!)

Integral lifts of families of varieties over a finite field

Let $ X\rightarrow \mathrm{Spec}\:F_q[[t]]$ be a flat morphism with smooth proper geometrically connected fibers. Suppose the central fiber lifts to a scheme $ X’_0$ smooth proper over $ W(F_q)$ . Is our family the base change of a flat morphism with smooth proper fibers $ X’\rightarrow \mathrm{Spec}\:W(F_q)[[t]]$ whose central fiber is $ X’_0$ ? is our family the base change of any flat morphism with smooth proper fibers $ X’\rightarrow \mathrm{Spec}\:W(F_q)[[t]]$ ?

If the answer is “no”, what additional conditions are required? Is it enough to demand that the central and the generic fiber lift to smooth proper schemes in possibly unrelated ways?

System.Data.SqlClient.SqlError: The media set has 2 media families but only 1 are provided. All members must be provided. (Microsoft.SqlServer.Smo)

I want to Backup/Restore our managed metadata managed service’s database from live to test which are both sharepoint 2013 farms. So i did these steps:-

1- Inside the live database i select the related managed metadata database >> right click on it >> Tasks >> Back Up… >> and i got a file named ttd.bak of size 1.9 MB, as follow:- enter image description here

2- then inside our test database , i created a new database with the same name as the live >> right click on the “Databases” >> click on “Restore Database” >> i select the new database and the ttd.back, but i got this error:-

TITLE: Microsoft SQL Server Management Studio ------------------------------  Restore failed for Server 'WIN-SPDEV'.  (Microsoft.SqlServer.SmoExtended)  For help, click: http://go.microsoft.com/fwlink?ProdName=Microsoft+SQL+Server&ProdVer=10.50.4000.0+((KJ_PCU_Main).120628-0827+)&EvtSrc=Microsoft.SqlServer.Management.Smo.ExceptionTemplates.FailedOperationExceptionText&EvtID=Restore+Server&LinkId=20476  ------------------------------ ADDITIONAL INFORMATION:  System.Data.SqlClient.SqlError: The media set has 2 media families but only 1 are provided. All members must be provided. (Microsoft.SqlServer.Smo)  For help, click: http://go.microsoft.com/fwlink?ProdName=Microsoft+SQL+Server&ProdVer=10.50.4000.0+((KJ_PCU_Main).120628-0827+)&LinkId=20476  ------------------------------ BUTTONS:  OK ------------------------------ 

as follow:- enter image description here

so can anyone advice on this? now our live environment (which i backup-ed the DB from) has the SP database on separate VM, while our test server (where i am trying to restore the DB) has the database server inside the same VM as sharepoint.. not sure if this is related to my problem?

Characterization of summable families

Could somenone help me with to prove the following result?

If $ C=\{\lambda\in{\Lambda}:x_\lambda\neq{0}\}$ is countable and for every bijective mapping $ \tau:\mathbb{N}\longrightarrow{C}$ $ \sum_{n\geq{1}}x_{\tau(n)}$ is a convergent serie, then $ \{x_\lambda:\lambda\in{\Lambda}\}$ is a summable family.

I have proved the contrary implication, but I dont’t know how to prove this one.

Thanks.

Connected components in flat families

Let $ f:X\rightarrow S$ be a proper flat family of schemes with non-empty geometrically reduced fibers and $ S$ Noetherian integral scheme. Is it true that $ X$ has finitely many subschemes such that the restriction of $ f$ to each of them is a proper and flat family of schemes over $ S$ with non-empty geometrically connected and geometrically reduced fibers?

Behavior of L-function in families of elliptic curves

I’m curious about how the L-functions of elliptic curves behave as the elliptic curves vary in families. In other words, if we regard the L-function $ L(E_{/K},s)$ of an elliptic curve $ E$ over a number field $ K$ as a function on the moduli space of elliptic curves over $ K$ , $ \mathcal{M}_{E/K}$ , in the first variable, what can be said about its properties, both for fixed values of the second variable $ s$ , and as a two-variable function on $ \mathcal{M}_{E/K} \times \mathbb{C}$ ?

The question has a natural extension to other moduli spaces of arithmetic varieties.

References to where it is discussed in detail would be appreciated.

Google Families Notifications

I am trying to turn off email notifications so that i dont get an email each time my son installs something. I have gone to all the steps guided in the support link:

https://support.google.com/families/answer/7184159?hl=en

Manage notifications When you manage your child’s Google Account, you can decide if you want to get notifications through email and the Family Link app.

Note: Certain notifications can’t be turned off.

Open the Family Link app Family Link. In the top left, tap Menu Menu and then Notifications settings. Tap the type of notification you would like to change. For each child, turn notifications on or off. You can also change your notification settings from the web:

Visit families.google.com. Click Menu Menu and then Notifications settings. For each child, turn notifications on or off.

The issue is when i go to the “Manage Notifications” there is no place to change these settings. only the below view: (only the “Learn more” is hyperlinked. Same in the actual app itself.)

enter image description here

Any help would be great thanks.

Eigenvalues of products of exponential families

I have a question about a close cousin of the multiplicative eigenvalue problem.

Let $ U$ be a special unitary matrix with diagonalization $ D = \operatorname{diag}(e^{2 \pi i a_1}, \ldots, e^{2 \pi i a_n})$ . The $ a_j$ may be normalized so as to satisfy $ a_1 \le a_2 \le \cdots \le a_n \le a_1 + 1$ and $ a_1 + \cdots + a_n = 0$ . These extra conditions have the advantage of producing a canonical sequence of logarithms: we may define a function $ \operatorname{LogSpec}$ by $ $ \operatorname{LogSpec} U = (a_1, \ldots, a_n).$ $

It also has the disadvantage of being not smooth. Given a point in $ \mathbb R^n$ satisfying only the equality $ a_1 + \cdots + a_n = k$ for $ k \in \mathbb{Z}$ , this point can be moved into by the region satisfying the family of inequalities (without modifying its image through $ t \mapsto e^{2 \pi i t}$ ) by repeated reflection. Let’s call this assignment $ R$ , as in $ $ R\colon\thinspace \left\{ a_* \in \mathbb R^n \mid a_1 + \cdots + a_n = 0\right\} \to \left\{a_* \in \mathbb R^n \middle| \begin{array}{c} a_1 + \cdots + a_n = 0, \ a_j \le a_{j+1}, \; a_n \le a_1 + 1 \end{array} \right\}.$ $ By consequence, curves like $ $ \gamma(t) = \operatorname{LogSpec} \exp\left(\begin{array}{cccc} it & 0 \ 0 & -it \end{array}\right),$ $ which are smooth in $ SU(2)$ before postcomposition with $ \operatorname{LogSpec}$ , become a kind of sawtooth function. For ease of reference below, I’ll call the image of a convex set through $ R$ folded-convex.

Question: I would like to know a reference for (or, indeed, a proof of) the following result:

$ \DeclareMathOperator{\LogSpec}{LogSpec}$ ??Theorem??: Let $ \xi_1, \ldots, \xi_m$ be a sequence of $ n \times n$ anti-Hermitian matrices, each exponentiating to a closed subgroup of $ U(n)$ . The assignment $ $ (t_1, \ldots, t_m) \mapsto \LogSpec \left( \prod_{j=1}^m \exp(\xi_j t_j) \right)$ $ sends convex sets in $ \mathbb R^m$ to folded-convex sets in $ \mathbb{R}^n$ .

In the classical version of the multiplicative eigenvalue problem, the set $ $ L_{m,n} = \left\{(\LogSpec U_j)_{j=1}^m \in \mathbb{R}^{n \cdot m} \middle| \begin{array}{c} \text{$ U_j$ unitary}, \ U_1 \cdots U_m = 1 \end{array}\right\} \subseteq \mathbb{R}^{n \cdot m}$ $ is shown to be convex by a clever application of symplectic reduction. The method of proof in Meinrenken and Woodward’s A symplectic proof of Verlinde factorization involves giving an explicit model for the moduli of flat connections on the trivial $ U(n)$ –bundle over a punctured Riemann sphere, then using its symplectic structure and a symplectic convexity theorem (suitably augmented to cope with loop groups) to deduce the convexity of $ L_{m,n}$ .

Their methods are especially well-suited to dealing with formulas like $ $ 1 = \operatorname{Ad}_{c_1}(t_1) \cdots \operatorname{Ad}_{c_m}(t_m),$ $ where $ t_j \in \mathfrak t_j \subseteq \mathfrak{su}(n)$ are anti-Hermitian diagonal and $ c_j \in SU(n)$ are special unitary. I’m new to this material and to symplectic geometry broadly, and so I’ve been unable to tweak these methods into saying something about this more restricted problem, where there are far fewer adjoint actions in play. Despite that, this seems like the kind of problem that would have attracted classical attention, and so I’m hopeful that there exists a resource that works this out. I’m also happy to hear about adjacent results—maybe I can make do with one of them.

Caveat lector: the theorem seems true in numerical experiment, but without a proof, there may well be edge cases unaccounted for. I’d be very, very happy to hear about those.