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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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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58 views

A question about Gieseker compactification

Let $\ell_{\infty} \subset \Bbb P^2$ be a fixed line, and $G = GL_r$. Let $\mathcal U^a_G$ be the Gieseker partial compactification of the moduli space $Bun^a_G(\Bbb A^2)$. The space $Bun^a_G(\Bbb A^2)...
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What is transition function of scheme theoretic vector bundle ??

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 ...
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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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88 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 $n=1$ ...
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131 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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64 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 ...
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141 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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50 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 $...