7
votes
Accepted
Local to global deformation of invertible sheaves
The answer is no, even for locally constant families. Let $F$ be a smooth projective variety with an automorphism $\sigma$, and let $L$ be a line bundle on $F$ such that $\sigma^* L$ is not isomorphic ...
- 16.1k
6
votes
Examples of smooth projective varieties with "nice" Picard group
Here is a simple example. Let $X = P^1 \times Q^3$ and $H = O(1,1)$. The Hilbert polynomial of the line bundle $O(a,b)$ is
$$
P(t) = (t+a+1)(t+b+1)(t+b+2)(t+b+3/2)/3.
$$
It has three integral roots $-...
- 39.3k
6
votes
Accepted
How far is ample from globally-generated
The answer is no for curves, and then a product construction show that the answer is no in every dimension.
Take a hyperelliptic curve $X$ of genus $g \geq 3$ and let $P$ be a 2-torsion line bundle on ...
- 66.3k
5
votes
Accepted
Examples of jumping base locus of complete linear systems
Take $X$ to be $\mathbb P^2$ blown up at three $R$-points which are colinear on the special fiber and not on the generic, and take $L$ to be $\mathcal O(2)$ minus the three exceptional divisors. The ...
- 148k
5
votes
Flatness of direct image sheaf over local artinian ring
$\newcommand{\C}{\mathbb{C}}\newcommand{\D}{\C[t]/t^2}\newcommand{\im}{\mathrm{im}\,}$A module $M$ over $\D$ is flat if and only if the inclusion $(t)\subset \D$ remains injective after tensoring with ...
- 7,377
3
votes
Accepted
Flatness of direct image sheaf over local artinian ring
This follows easily from the theory of modules over $R = \mathbb C[t]/(t^2)$. Indeed, we have a short exact sequence
$$0 \to H^0(\mathscr L_0) \to H^0(\mathscr L) \to H^0(\mathscr L_0) \to 0,$$
...
- 28.7k
2
votes
Accepted
Is the pull-back of canonical sheaf invertible (modulo torsion)?
I think that $f^*K_X$ is not invertible in general. For instance, take as $X$ a quotient surface singularity of type $\frac{1}{4}(1, \, 1)$. Then straightforward computations give $$K_Y=f^*K_X - \frac{...
- 66.3k
2
votes
Roots of the Hilbert polynomial of an invertible sheaf
For the sake of simplicity, let us assume that $\mathcal{L}$ is a polarization, and that we are computing the Hilbert polynomial of $\mathcal{L}$ with respect to itself, i.e. $P(t)=\chi(X, \, \mathcal{...
- 66.3k
2
votes
Accepted
Variation of global sections of line bundles
For a general smooth base $S$ it is not true that $T$ is regular. Assume even that $\mathcal{C} = C \times S$ for a fixed smooth curve $C$. The line bundle $\mathcal{L}$ then defines a morphism $f \...
- 39.3k
1
vote
When is a sheaf $\mathcal{L}_1 \subset \mathcal{F} \subset \mathcal{L}_2$ sandwiched between two line bundles also a line bundle?
No. Locally, $\mathcal L_2$ is isomorphic to the coordinate ring $R$ and $\mathcal F$ is an ideal $I \subseteq R$ which contains a principal ideal. So, e.g $R = k[x,y]$ and $I = (x,y)$ containing $(...
- 1,347
Only top scored, non community-wiki answers of a minimum length are eligible