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On families of rank-$$2$$ uniform bundles on Hirzebruch surfaces and Hilbert schemes of their scrolls. (English) Zbl 1341.14012
It seems natural to relate the study of (moduli of) vector bundles over smooth complex, projective varieties with the study of the corresponding (families of) embedded projective bundles, when the vector bundle is very ample. Inspired by the case of rank-two vector bundles over curves (see the Introduction of the paper under review and references therein) this paper takes part of a series of papers which deals with the case of rank-two vector bundles over Hirzebruch surfaces $$\mathbb{F}_e$$. As a natural continuation of previous works, the authors consider the case in which $$\mathcal{E}$$ is a vector bundle over $$\mathbb{F}_e$$ with first Chern class $$c_1(\mathcal{E})=4C_0+\lambda f$$ ($$C_0$$ stands for the minimal section and $$f$$ for the fiber of $$\mathbb{F}_e$$) which is known to be very ample.
The authors prove, see Theorem 4.2, that when $$e \leq 2$$ the corresponding ruled 3-folds are smooth points of the proper component of the Hilbert scheme, which in fact has the expected dimension. Moreover, see Theorem 4.6, these scrolls fill up either their whole component ($$e=0,1$$) or a codimension one subvariety of their component ($$e=2$$). Some interesting questions – general elements of the components of the Hilbert schemes, degenerations in terms of vector bundles – beyond the generalizations to $$e\geq 3$$, are also provided (see Section 5).
##### MSC:
 14J30 $$3$$-folds 14J26 Rational and ruled surfaces 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14C05 Parametrization (Chow and Hilbert schemes) 14D20 Algebraic moduli problems, moduli of vector bundles 14N25 Varieties of low degree 14N30 Adjunction problems
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