The invertible-sheaves tag has no usage guidance.

**4**

votes

**0**answers

118 views

### Obstruction to lifting of global sections of invertible sheaves

Are there known examples of smooth projective hypersurface in $\mathbb{P}^3$, say $X$ and an invertible sheaf $L$ on $X$ with $H^0(X,L)>0$ satisfying the following property: There exists an ...

**2**

votes

**0**answers

126 views

### Universal property of limits of invertible sheaves

Let $R$ be a discrete valuation ring, $m$ the maximal ideal and $f:X \to \mathrm{Spec}(R)$ be a flat, proper morphism of relative dimension $1$. Assume further that $X$ is regular. For any $n>0$, ...

**4**

votes

**1**answer

1k views

### What can be said about a pullback of a very ample line bundle w.r.t birational maps?

Let $X$ be a smooth projective variety and $\phi: X \to \mathbb P^n$ be a map. If $\phi$ is an embedding then $E=\phi^*(O(1))$ is very ample. But can one say something if $\phi$
is birational (but not ...

**10**

votes

**3**answers

1k views

### Justification of the term “invertible sheaf”

Let $X$ be a locally ringed space (or a scheme) and $M,N$ two $\mathcal{O}_X$-modules such that $M \otimes N \cong \mathcal{O}_X$. Does it follow that $M$ is invertible in the usual sense, namely that ...