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3 votes
1 answer
160 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) ...
user43198's user avatar
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1 vote
2 answers
283 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 $\pi$. ...
Chen's user avatar
  • 1,593
3 votes
1 answer
306 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 ...
user43198's user avatar
  • 1,981
1 vote
0 answers
126 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 $X_{\bar{K}}=X \times_K \mbox{Spec}(\bar{K})$, ...
Jana's user avatar
  • 2,022
1 vote
1 answer
315 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 $\...
user43198's user avatar
  • 1,981
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 $\...
Jana's user avatar
  • 2,022
3 votes
2 answers
410 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})...
user43198's user avatar
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4 votes
0 answers
223 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 ...
Ron's user avatar
  • 2,126