Curves with decomposable normal vector bundles and automorphism groups

If a smooth projective threefold $X$ satisfies a certain Property A (see below for definition), then any automorphism of $X$ has zero entropy. Let $Y$ be a smooth projective threefold satisfying Property A. Let $π:X\rightarrow Y$ be a blowup at either a point or at a smooth curve $C\subset Y$ with the following two properties: i) $c_1(Y).C$ is an odd number, and ii) the normal vector bundle $N_{C/Y}$ is decomposable. Then we show that $X$ also satisfies Property A. As a further application of Property A we prove the following result. Let $X_1$ be the blowup of $X_0=\mathbb{P}^3$ at a finite number of points, and let $X=X_2$ be the blowup of $X_1$ at a finite number of pairwise disjoint smooth curves (here the images of these curves in $X_0$ may intersect). Then any automorphism of $X$ has the same first and second dynamical degrees. Under some further conditions, then any automorphism of $X$ has zero entropy. The result is also valid for threefolds $X_0$ satisfying a certain condition on the second Chern class. Some explicit examples are given.

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