Questions tagged [line-bundles]

Question 1

Let $M,N$ be both $A$-modules where $A$ is a commutative ring and let $m\subset A$ be a maximal ideal, is it true that $s\in mM$ and $t\in mN$ implies $s\otimes t\in m(M\otimes N)$? This question is ...

Question 2

Let us consider the complex projective space $\mathbb{CP}^n$ as a complex manifold. The holomorphic line bundle $\mathcal{O}(1)$ over $\mathbb{CP}^n$ may be defined as the dual bundle of the ...

Question 3

I have few questions about explicit proof that the relative spectrum associated to invertible sheaf $\mathcal{O}_{\Bbb P^2_{\Bbb C}}(-1)$ coinsides with incident variety $I \subseteq \mathbb{P}^2 \...

Question 4

In Corollary B.0.44 of Complex Geometry, Huybrechts shows via a Čech cohomology argument that if $X$ is a smooth manifold, then $\operatorname{Vect}_1(X) \cong H^1(X, (C^\infty)^\times)$, where $\...

Question 5

This is from the paper "Quantum geometry, stability, and modularity" by Alexandrov et al. [arXiv:2301.08066v2]. Let $X$ be a variety (it is a Calabi Yau threefold in the paper, but I don't ...

Question 6

I am interested in understanding the Picard group of orthogonal Grassmanians in the following two cases. If someone could give me the proofs or a hint or a good reference it would make me happy. Case ...

Question 7

I was reviewing my notes when I suddenly had a bit of confusion regarding the identification of the hyperplane bundle with $\mathcal{O}(1)$. Let $\mathbb{P}^n = \operatorname{Proj} \mathbb{C}[Z_0, \...

Question 8

I am studying Complex Manifolds by Morrow & Kodaira, and when I came across the proof that the first Chern class of a holomorphic line bundle is represented in the de Rham cohomology group $H^2_{\...

Question 9

Let $ X $ be a projective scheme over a noetherian ring $ A $, $ \mathcal{F} $ a coherent sheaf and $ \mathcal{L} $ an ample line bundle, suppose $ H^1(X, \mathcal{F} \otimes \mathcal{L}^{\otimes m}) =...

Question 10

I am trying to understand the proof from Griffiths and Harris on page 146 that \begin{align} K_{\mathbb{P}^n} = [-(n+1)H]\ \ (= \mathcal{O}_{\mathbb{P}^{n}}(-n-1)) \end{align}, where $K_{\mathbb{P}^...

Question 11

I want to understand how the correspondence between divisors and holomorphic line bundles on a compact Riemann surface $S$ works. Griffiths and Harris describe this correspondence in detail in their ...

Question 12

Let $M$ be a K3 surface, defined as a complex manifold of dimension 2 with $b_1=0$ and $c_1(M,\mathbb{Z})=0$. It follows that $M$ has trivial canonical bundle. Let $L$ be an ample bundle on $M$. We ...

Question 13

Let $X$ be a smooth projective irreducible curve and let $\mathcal{L}$ be a line bundle on $X$. We denote by $D=\sum_{i=1}^n m_iP_i$ the divisor on $X$ corresponding to $\mathcal{L}$, i.e. such that $\...

Question 14

I'm considering a birational morphism $f:X \to Y$ between smooth projective varieties, and a divisor $D$ on $X$. I'm trying to study $f_*(\mathcal O_X(D))$. Its structure depends on $D$ and on how $f$ ...

Question 15

I am currently doing an exercise that asks me to show that there is a bijection between the isomorphism classes of real line bundles over a finite CW complex $X$ and $H^1 (X; \mathbb Z / 2 \mathbb Z)$....

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Questions tagged [line-bundles]

Is it true that $s\in mM$ and $t\in mN$ implies $s\otimes t\in m(M\otimes N)$?

Global holomorphic sections of $\mathcal{O}(1)$

Explicit Isomorphism from $\operatorname{Spec}(\operatorname{Sym}^_{\mathcal{O}_X}(\mathcal{O}_{\Bbb P^2}(-1)^))$ to an Incidence Variety

Two different parametrisations of real line bundles. Is $H^1(X, (C^{\infty})^\times) \cong H^1(X, \mathbb Z_2)$?

What is the definition of the determinant and the dual of a complex in the derived category of coherent sheaves?

What is the Picard group of orthogonal grassmanians?

The identification of the hyperplane bundle with $\mathcal{O}(1)$

de Rham Representation of First Chern class

Globally Generated Property of $\mathcal{F} \otimes \mathcal{L}^{m+1}$ Given $H^1(X, \mathcal{F} \otimes \mathcal{L}^{m}) = 0$

The canonical bundle of $\mathbb{P}^n$ is $[-(n+1)H]$ via Griffiths and Harris

Correspondence between divisors and line bundles

Ample line bundle on a K3 surface implies $L^{\otimes 2}$ is base-point free

Degree of a line bundle is equal to the degree of the corresponding divisor

Pushforward of Line Bundles under Birational Morphisms

Bijection between isomorphism classes of real line bundles over a finite CW complex and its first mod 2 cohomology group.

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