On intersection lattices of hyperplane arrangements generated by generic points

We consider hyperplane arrangements generated by generic points and study their intersection lattices. These arrangements are known to be equivalent to discriminantal arrangements. We show a fundamental structure of the intersection lattices by decomposing the poset ideals as direct products of smaller lattices corresponding to smaller dimensions. Based on this decomposition we compute the M\"obius functions of the lattices and the characteristic polynomials of the arrangements up to dimension six.