Argyres-Douglas loci, singularity structures and wall-crossings in pure N=2 gauge theories with classical gauge groups

N = 2 Seiberg-Witten theories allow an interesting interplay between the Argyres-Douglas loci, singularity structures and wall-crossing formulae. In this paper we investigate this connection by first studying the singularity structures of hyper-elliptic Seiberg-Witten curves for pure N = 2 gauge theories with SU(r + 1) and Sp(2r) gauge groups, and propose new methods to locate the Argyres-Douglas loci in the moduli space, where multiple mutually non-local BPS states become massless. In a region of the moduli space, we compute dyon charges for all 2r + 2 and 2r + 1 massless dyons for SU(r + 1) and Sp(2r) gauge groups respectively for rank r > 1. From here we elucidate the connection to the wall-crossing phenomena for pure Sp(4) Seiberg-Witten theory near the Argyres-Douglas loci, despite our emphasis being only at the massless sector of the BPS spectra. We also present 2r - 1 candidates for the maximal Argyres-Douglas points for pure SO(2r + 1) Seiberg-Witten theory.