We study spin structures on Riemann and Klein surfaces in terms of divisors. In particular, we take a closer look at spin structures on hyperelliptic and $p$-gonal surfaces defined by divisors supported on their branch points. Moreover, we study invariant spin divisors under automorphisms and antiholomorphic involutions of Riemann surfaces and count them. We generalize a formula that gives $2$-spin divisors, proved by Mumford, to the case of $m$-spin divisors for an even $m$, supported on branch points of a hyperelliptic surface.
"Spin structures and branch divisors on $p$-gonal Riemann surfaces." Rocky Mountain J. Math. 49 (6) 1769 - 1791, 2019. https://doi.org/10.1216/RMJ-2019-49-6-1769