  We simplify Ciliberto's and Miranda's method(arXiv:0812.0032) to construct
degenerations of $\mathbb{CP}^2$ blown up in several points yielding lower
bounds of the corresponding multi-point Seshadri constants. In particular we
exploit an asymptotic result of the author (arXiv:math/0508561) which allows to
check the non-specialty of much fewer linear systems on $\mathbb{CP}^2$. We
obtain the lower bound 117/370 for the 10-point Seshadri constant on
$\mathbb{CP}^2$.
