### Free pencils on divisors.

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Given a symplectic fibration $E\to M$, with $M$ compact and symplectic and the fibres complex projective, we produce symplectic submanifolds of $E$ analytic in the vertical direction, and apply this to complex vector bundles on symplectic manifolds.

Motivated by the notion of Seshadri-ampleness introduced in [11], we conjecture that the genus and the degree of a smooth set-theoretic intersection $C\subset {\mathbb{P}}^{3}$ should satisfy a certain inequality. The conjecture is verified for various classes of set-theoretic complete intersections.

A notion of positivity, called Seshadri ampleness, is introduced for a smooth curve $C$ in a polarized smooth projective $3$-fold $\left(X,A\right)$, whose motivation stems from some recent results concerning the gonality of space curves and the behaviour of stable bundles on ${\mathbb{P}}^{3}$ under restriction to $C$. This condition is stronger than the normality of the normal bundle and more general than $C$ being defined by a regular section of an ample rank-$2$ vector bundle. We then explore some of the properties of Seshadri-ample curves....

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