ag.algebraic geometry – Which infinite-dimensional Lie algebras have realizations as algebras of global sections of vector bundles with special structure?

I would not be surprised by downvotes since this question is at the same time very naïve, very vague and might be asking about things well known for decades. Specifically vagueness comes from the word “special” in the title. As for that third weakness, I have a weak defence against it by choosing the reference request tag.

The starting point is the trivial observation that for a finite-dimensional semisimple Lie algebra $mathfrak g$, the algebra $mathfrak g(t,t^{-1})$ of Laurent polynomials with coefficients in $mathfrak g$ can be viewed as the algebra of polynomial global sections of the trivial vector bundle with fibre $mathfrak g$ over the punctured plane $mathbb C^times$.

My first question is whether it is possible to somehow modify this bundle in such a way that the global sections will turn out to be the affine Lie algebra $hat{mathfrak g}$ (the universal central extension of $mathfrak g(t,t^{-1})$); similarly for twisted versions of $hat{mathfrak g}$.

Second question – is it known what does one obtain with nontrivial vector bundles, and with some projective curve in place of $mathbb C^times$? I have no idea whether one indeed obtains a Lie algebra at all or something different. Does the algebraic group structure of $mathbb C^times$ have any significance in this context? Accordingly, is it important whether I take an elliptic curve or curves of all genera give more or less similar results?

Finally, I remember that although sections of the tangent bundle form a Lie algebra, it is not a Lie algebra bundle in the sense that fibres do not have any Lie algebra structure. Still, if I am not mistaken, the Virasoro algebra is the universal central extension of the algebra of global sections of a tangent bundle, right? Again, is there some modified form of the tangent bundle such that the Virasoro algebra would be global sections of this modified bundle? And again, is the group structure significant here? What are algebras of sections of tangent bundles over projective curves, in particular, over elliptic curves? Do they have nontrivial central extensions?

schemes – Ideal sheaf is quasi-coherent if and only if its generated by local sections.

My confusion is lies in Schemes Lemma 10.1 of the Stacks project.

First, Modules Definition 8.1 states that a sheaf $mathcal{F}$ of $mathcal{O}_X$-modules is locally generated by sections if for all $xin X$, there is a neighborhood $U$ of $x$ and a surjection $mathcal{O}_U^{(I)}tomathcal{F}|_U$ where $mathcal{O}_U^{(I)} = bigoplus_{iin I}mathcal{O}_U$.

Now Schemes Lemma 10.1 states “Let $(X,mathcal{O}_X)$ be a scheme, $i:Zto X$ be a closed immersion of locally ringed spaces:

(1) The locally ringed space $Z$ is a scheme.
(2) The kernel $mathcal{I}$ of the map $mathcal{O}_Xto i_*mathcal{O}_Z$ is a quasi-coherent sheaf of ideals.
(3) for every affine open $U = operatorname{Spec}(R)$ of $X$, the morphism $i^{-1}(U)to U$ can be identified with $operatorname{Spec}(R/I)tooperatorname{Spec}(R)$ for some ideal $I$ of $R$, and
(4) we have $mathcal{I}|_U = widetilde{I}$.

In particular, any sheaf of ideals locally generated by sections is a quasi-coherent sheaf of ideals (and vice versa), and any closed subspace of $X$ is a scheme.”

Question: It is this remark “In particular…” at the end which I do not understand at all. I don’t even see where a module that is locally generated by sections appears in the statement, how am I supposed to conclude that a module locally generated by sections is quasi-coherent?

I am quite new to scheme theory, so I would very much appreciate a reasonably detailed response.

Sections of fibrations of Kodaira dimension zero

Let $X$ be a projective variety with a morphism $f:Xrightarrow mathbb{P}^1$, and let $F$ be a general fiber of $f$.

Assume that $F$ in turn has a fibration $g_{F}:Frightarrow S$ with rational fibers, where $S$ is a (singular) K3 surface.

Are there results yielding, under suitable hypotheses, the existence of a section $s:mathbb{P}^1rightarrow X$ of $f$?

ag.algebraic geometry – Rational sections of tropical conics

Let us consider the family of Fermat conics in $(mathbb{C}^*)^2subsetmathbb{C}^2$.
$$picolon V(ax^2+by^2-1)subset(mathbb{C}^*)^2_{a,b}times(mathbb{C}^*)^2_{x,y}to(mathbb{C}^*)^2_{a,b}$$
We know that $pi$ does not admit rational sections: the generic conic is non-split.

Taking tropicalization functor, we get the morpphism $$mathrm{Trop}(pi)colon mathrm{Trop}(ax^2+by^2-1)subsetmathbb{R}^4tomathbb{R}^2$$

Does $mathrm{Trop}(pi)$ admit tropical rational sections? (Sections over $mathbb{R^2}backslash W$ for some proper tropical subvariety $Wsubsetmathbb{R}^2$?)

Ella – Multipurpose Shopify Sections Theme

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hard drive – how can I repair my partition that has been splited into two sections and isn’t available anymore?

I had a 540GB NTFS partition named “D”. After one restart (I might have done something wrong) my ubuntu partition turn into a unallocated partition. Then I use a live boot Ubuntu and used “testdisk” to recover that partition and that works fine. but after that operation, my D drive wasn’t accessible any more. After I checked the partition manager it seemed that my drive D is split into one 140GB NTFS and 400GB unallocated and the unallocated part become a part of an extended partition (that I have no clue what it is)(there will be a picture of partition manager). This drive only contained data and not something like software or OS. Is there any way I can recover my data ?

Windows Disk Manager Image

javascript – Show contents in a book chapter and sections in a highly normalized repository

I want to present books that has chapters and sections, and contents in each book, chapter and section.

This is my schema:

type BookId = string
type ChapterId = string
type SectionId = string
type ContentId = string


export type SourceId =
| ChapterId
| SectionId
| BookId


type Book = {
  id: BookId,
  name: string,
}

type Chapter = {
  id: ChapterId,
  bookId: BookId,
  name: string
}
type Section = {
  id: SectionId,
  chapterId: ChapterId,
  name: string
}

type Content = {
  id: ContentId,
  sourceId: SourceId,
  name: string,
  content: string
}

This is how I query all the books, chapters and sections:

 list() {
    return this.bookRepo.all().then(books =>
      Promise.all(books.map(book =>
        Promise.all((
          this.bookRepo.contents(book.id),
          this.bookRepo.chapters(book.id)
        )).then(((contents, chapters)) =>
          Promise.all(chapters.map(chapter =>
            Promise.all((
              this.bookRepo.contents(chapter.id),
              this.bookRepo.sections(chapter.id)
            )).then(((contents, sections)) =>
              Promise.all(sections.map(section =>
                this.bookRepo.contents(section.id).then(contents => (
                  section, contents
                ))))
                .then(sections => (chapter, contents, sections)))))
            .then(chapters => (book, contents, chapters))))));
  }

I just render as json at the moment, but the query code seems hard to maintain.

Divide a list into n sections with the sum of each section having the least possible range

Consider such a list :

  • list: "12 43 92 85 10 29 48 50 82 75 20 49 18 57 82 99 20 39 48 57 19 28 57 93 22"

Get an algorithm to get :

  • subgroups (separated buy "|") : 12 43 92 85 10 29 48 50 82 | 75 20 49 18 57 82 99 | 20 39 48 57 19 28 57 93 22

The algorithm has to minimize the range between the average of each subgroup.

at.algebraic topology – Does homeomorphism between cones imply homeomorphism between sections

For any topological space $A$, the cone $C(A)$ is defined to be $A times (0,infty)$ with $A times 0$ identified to a point (cone point).

Let $X$ and $Y$ be two compact Hausdorff spaces such that there is a homeomorphism between $C(X)$ and $C(Y)$ which preserves the cone points. Can we prove that $X$ and $Y$ are homeomorphic?

conic sections – Area of triangle within an ellipse and hyperbola that have the same foci

Question: An ellipse and a hyperbola have the same foci, $F_1$ and $F_2$. These curves cross at 4
points – let $P$ be one of the points. These curves also intersect the line $overleftrightarrow{AB}$ at 4 points labelled $Q$, $R$, $S$ and $T$ in that order. If $RS$ = 20, $ST$ = 14 and $∆PF_1$ $F_2$ is isosceles,
compute the area of $∆PF_1$ $F_2$ .

I have no idea how to go about solving this. I know that the easiest way of getting the area is probably by SSS and Heron’s Formula, and I did get that the major axis is $14 + 20 + 14=48$ so the $PF_1+PF_1=48$ and probably that $PF_1$ is congruent to $F_1F_2$