Questions tagged [algebraic-vector-bundles]

Use this tag for questions related to algebraic vector bundles, which are morphism of varieties $E \to X$ that locally (in the Zariski topology) have the structure of a projection of a direct product $k^n × X$ onto $X$ such that the gluing preserves the linear structure of the vector space.

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54 views

Does $K_0$ of a regular (affine) scheme always surjects onto $K_0$ of any open subscheme?

Let $X$ be a Noetherian separated regular scheme and let $U$ be an open subscheme. Then does there always exist an exact sequence $K_0(X)\to K_0(U)\to 0$ ? If not, then is it at least true when $X=\...
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41 views

Induced map on the $K_0$ of punctured spectrum of completion

Let $(R,\mathfrak m)$ be a local Gorenstein ring (may also assume excellent) of dimension at least $2$, and let $(\hat R,\hat {\mathfrak m})$ be the $\mathfrak m$-adic completion. Let $U:=\text{Spec}(...
2
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1answer
81 views

When two Algebraic vector bundles on a Noetherian quasi-affine scheme are equal in $K_0$ of the scheme

Let $X$ be a (connected) Noetherian scheme and $K_0(X)$ denote the Grothendieck group of the category of Algebraic vector bundles (coherent sheaves that are locally free and of constant rank ( as $X$ ...
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0answers
26 views

Given a locally free sheaf, compute its associated vector bundle

I have this situation, $r$ and $s$ are positive integers: we define $\mathcal{E}$ to be the vector bundle on $\mathbb{P}^s$ whose associated locally free sheaf is $$ \mathcal{O}_{\mathbb{P}^s} \oplus ...
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38 views

Bass-Quillen conjecture for non-affine case

Bass-Quillen conjecture expects that any vector bundle on $U\times \mathbb{A}^1$, to be extended from $U$. Here $U$ is a regular affine scheme. Being affine is an essential part of the conjecture, you ...
2
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106 views

vector bundles $\mathcal F, \mathcal G$ such that $\mathcal Hom (\mathcal F, \mathcal G) $ is a direct sum of copies of $\mathcal O_X$

Let $(R, \mathfrak m)$ be a Noetherian local ring of depth at least $2$. Let $X=$Spec$(R)\setminus \{\mathfrak m\}$ be the punctured spectrum and $\mathcal O_X$ be the structure sheaf on $X$ induced ...
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1answer
33 views

the moduli spaces that contain direct sums of line bundles

Let $U_X(n,d)$ be the moduli space of semistable vector bundle of rank $n$ and degree $d$ over a smooth projective curves over the complex numbers. How do I know if $U_X(n,d)$ contains direct sums of ...
2
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39 views

Full and faithful functor from quasi-coherent $\mathcal{O}_S$-modules to $S$-vector bundles

Let $S$ be a scheme. I want to show that the composition $\text{Spec}_S(-)\circ \text{Sym}(-)$ is a full and faithful contravariant functor from the category of quasi-coherent $\mathcal{O}_S$-...
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2answers
39 views

Where do I find this article?

I have been searching for this paper for days, unfortunately I cannot find it. Does any of you know how to find it? Hirschowitz A. Problèmes de Brill–Noether en rang supérieur, C. R. Acad. Sci. ...
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36 views

“Transition morphism” of fiber bundle

I want categorical approach to the notion "transition functions of fiber bundle". I have found the page https://ncatlab.org/nlab/show/fiber+bundle#definitions in nLab. According to this page, Fiber ...
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1answer
54 views

Cohen-Macaulay sheaves in Picard group of Cohen-Macaulay schemes

Let $X$ be a Noetherian, integral, separated, CM (Cohen-Macaulay) scheme. Is it true that the set $\{ [L] \in Pic (X) : L$ is CM $ \}$ is finite ? If this is not true in general, then what if we ...
3
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1answer
94 views

On the ring structure of $K_0$ of the punctured spectrum of a regular local ring

Let $(R, \mathcal m, k)$ be a regular local ring of dimension at least $3$. Let $K_0(X)$ be the Grothendieck group of algebraic vector bundles over the punctured spectrum $X =Spec R \setminus \{\...
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65 views

Line bundles correspoding to a hyperplane

Assume we have a smooth projective variety $X$ over a field and a hyperplane section $H$ on it. For each Weil divisor on $X$ you can construct a line bundle on $X$. For $H$ this line bundle which is ...
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70 views

What is transition function of scheme theoretic vector bundle ?? [duplicate]

Let X be a scheme X and $\mathscr{F}$ be a locally free sheaf of rank $n$ over $X$. I want to consider a vector bundle over an algebraic variety $X$ , that is , the relative spec over $X$ of ...
3
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319 views

Pushforward of a line bundle along a finite morphism of curves

Let $f:X\rightarrow Y$ be a finite morphism (a branched covering) of degree $n$ of smooth complex algebraic curves. It is a known result that for any line bundle $L$ on $X$, the pushforward $f_* L$ ...
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96 views

chern class of bundle in Blochs “Cycles on arithmetic schemes”

In Blochs "Cycles on arithmetic schemes and Euler characteristics of curves" he defines the bivariant chern class for bundles on a scheme. Now I wanted to calculate a bivariant chern class with $n=1$ ...
3
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212 views

Grothendieck group of a smooth complex projective curve

I'm reading the section of Le Potier's 'Lectures on Vector Bundles' where he proves that the Grothendieck group $K(X)$ of a smooth complex projective curve $X$ (which is the free abelian group on ...
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97 views

Degree of sub sheaf of locally free sheaf is bounded

Let $X$ be smooth projective curve over a field $k$. Let $\mathcal F$ be a finite rank locally free sheaf on $X$. Now, consider the set $ D = \{\operatorname {deg}(\mathcal G) : \mathcal G$ is a ...
3
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0answers
198 views

How can I find a locally free resolution of $\mathcal{O}_X^\vee \to \mathcal{L}_\bullet$?

Given a smooth complex projective variety $X$, I want to try and learn how to find a locally free resolution $$ 0 \to \mathcal{O}_X^\vee \to \mathcal{L}_0 \to \mathcal{L}_1 \to \cdots $$ The starting ...
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1answer
55 views

How can I find the rank of this algebraic vector bundle?

Let $P(n,d)$ be the vector space of degree $d$ polynomials in $\mathbb{C}[x_0,\ldots,x_n]$. If I consider the variety $$ Z = \{(x,f) \in \mathbb{P}^n\times P(n,d) : f(x) = 0 \} $$ then the projection $...