  In this article, we study the divisor classes of del Pezzo surfaces, which
are written as the sum of distinct lines with fixed intersection according to
the inscribed simplexes and crosspolytopes in Gosset polytopes. We introduce
the k-Steiner system and cornered simplexes, and characterize the
configurations of inscribed m(<4)-simplexes with them. Higher dimensional
inscribed m(3<m)-simplexes exist in 4_{21} in the Picard group of del Pezzo
surface S_{8} of degree 1.The configurations of 4- and 7-simplexes are related
to rulings in S_{8}. And the configurations of 5- and 6-simplexes correspond
the skew 3-lines and skew 7-lines in S_{8}. In particular, the seven lines in a
6-simplex produce a Fano plane. We also study the inscribed crosspolytopes and
hypercubes in the Gosset polytopes.
