satisfy the lower bound $ 2c = na(1-a)$. Moreover, considering the rank one case, they allow us to obtain a rather explicit construction of the Hilbert schemes of points of the total spaces of the line bundles ${\mathcal{O}}_{\mathbb P^1}(-n)$. In both cases the results are achieved by using explicit ADHM descriptions [@new].

If not otherwise specified, by “scheme” we mean a noetherian reduced scheme of finite type over ${\mathbb{C}}$. If $X$ and $S$ are schemes, ${\mathcal{F}}$ is a sheaf of ${\mathcal{O}}_{X\times S}$-modules, and $F$ is a morphism between two such sheaves, we shall denote by ${\mathcal{F}}_s$ (resp. $F_s$) the restriction of ${\mathcal{F}}$ (resp. $F$) to the fibre of $X\times S\longrightarrow S$ over the point $s\in S$.

[**Acknowledgments.**]{} We thank A. A. Henni, B. Kenda, V. Lanza and D. Markushevich for useful discussions. We also thank the referee for the careful reading of our manuscript which helped us to considerably improve the presentation.

Monads
======

A monad $M$ on a scheme $T$ is a three-term complex of locally-free $\mathcal O_T$-modules, having nontrivial cohomology only in the middle term: $$M\,:\qquad\xymatrix{
0 \ar[r]& \mathcal{U}\ar[r]^a &
\mathcal{V}
\ar[r]^b &\mathcal{W} \ar[r] &0\,.
}
\label{eq384}$$ The cohomology of the monad will be denoted by $\mathcal{E}(M)$. It 