Questions tagged [invertible-sheaves]
The invertible-sheaves tag has no usage guidance.
18
questions
2
votes
1
answer
627
views
How far is ample from globally-generated
Most examples of ample line bundles that are not globally generated have less number of global sections than the dimension of the variety. Assuming ampleness, is the existence of "enough" ...
3
votes
1
answer
145
views
Examples of jumping base locus of complete linear systems
I am looking for examples of invertible sheaves in smooth, projective families such that the associated base locus (i.e., the intersection of all the effective divisors in the complete linear system) ...
3
votes
1
answer
258
views
When is a sheaf $\mathcal{L}_1 \subset \mathcal{F} \subset \mathcal{L}_2$ sandwiched between two line bundles also a line bundle?
This question is in the interest of answering one part of this question, but I think it is distinct enough to warrant a separate question.
Let $X$ be a regular 2-dimensional Noetherian scheme, for ...
4
votes
1
answer
286
views
Is the pull-back of canonical sheaf invertible (modulo torsion)?
Let $X$ be a $\mathbb{Q}$-Gorenstein (isolated) singularity of dimension at least $2$ and $f:Y \to X$ be a resolution of singularities. In this case the canonical sheaf $K_X$ is not necessarily ...
0
votes
0
answers
160
views
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 (...
1
vote
2
answers
277
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 $\pi$. ...
3
votes
0
answers
204
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 ...
3
votes
1
answer
294
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 ...
2
votes
1
answer
167
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, ...
1
vote
0
answers
124
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 $X_{\bar{K}}=X \times_K \mbox{Spec}(\bar{K})$, ...
4
votes
1
answer
355
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 ...
1
vote
1
answer
267
views
Variation of global sections of line bundles
The underlying field is $\mathbb{C}$.
Let $\pi:\mathcal{C} \to \mathbb{A}^n$ be a flat family of projective curves (not necessarily smooth) of genus $g \ge 2$. Assume $\mathcal{C}$ is regular. Let $\...
0
votes
0
answers
131
views
Vector bundles on families of rational curves
Let $\pi:\mathcal{X} \to S$ be a flat, projective family of rational curves ($S$ is noetherian) over an algebraically closed field $k$. Assume $S$ is irreducible. Let $E$ be a locally-free sheaf on $\...
3
votes
2
answers
404
views
Is the zero locus of a global section flat?
Let $f:X \to Y$ be a surjective, smooth projective morphism of noetherian schemes. Let $\mathcal{L}$ be an inverible sheaf on $X$ satisfying $f_*\mathcal{L}$ is locally free and $s \in H^0(\mathcal{L})...
4
votes
0
answers
218
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
159
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 $n>0$, ...
5
votes
1
answer
3k
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 ...
12
votes
3
answers
3k
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 ...