# Questions tagged [invertible-sheaves]

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### Examples of degree zero, rank one reflexive sheaves without r-th roots

Let $X$ be a normal, projective surface (or more generally a variety) over $\mathbb{C}$ (i.e., $X$ is irreducible). Fix a polarisation on $X$. I am looking for examples of rank one, degree zero (...
222 views

### Flatness of direct image sheaf over local artinian ring

Let $\pi:X \to \mbox{Spec}(\mathbb{C}[t]/(t^2))$ be a smooth, projective morphism and $L$ be an invertible sheaf on $X$. Denote by $L_0$ the restriction of $L$ to the closed fiber, say $X_0$ of $\pi$. ...
166 views

### Reference request: Vanishing of first cohomology term in Riemann-Roch theorem for singular projective curves over a field

$\newcommand{\F}{\mathcal{F}}$ $\newcommand{\ox}{\mathcal{O}_X}$ Let $f:X \to \operatorname{Spec}(k)$ be a projective scheme of dimension one over a field $k$. The Riemann-Roch equation for such ...
245 views

### Local to global deformation of invertible sheaves

Let $\pi:X \to S$ be a flat, projective morphism, $S$ irreducible. Suppose that for all $s \in S$, the fiber $X_s$ satisfies $h^2(\mathcal{O}_{X_s})=0$. This means in particular that given an ...
139 views

### Roots of the Hilbert polynomial of an invertible sheaf

Let $X$ be a smooth, projective variety over an algebraically closed field of characteristic zero. Fix a polarisation on $X$. Let $\mathcal{L}$ be an invertible sheaf on $X$ with Hilbert polynomial, ...
101 views

### Pull-back of line bundles and field extension

Let $X$ be a smooth, projective variety over a field $K$ of characteristic $0$ (not necessarily algebraically closed) and $L$ an invertible sheaf on $X_{\bar{K}}=X \times_K \mbox{Spec}(\bar{K})$, ...
265 views

### Examples of smooth projective varieties with “nice” Picard group

I am looking for examples of smooth projective varieties $(X,H)$ with $H$ a polarization on $X$, $\dim \mbox{Pic}^0(X)=0$, $\mbox{Pic}(X) \not= \mathbb{Z}$ satisfying the property: for any two line ...