This post is split into two parts. This post contains spoilers. I tried to give a spoiler-free title because I don't want someone getting spoiled. The post discuss the events in the 2nd season of Higurashi. Here the Oyashiro-sama's curse refers to cycle of life and death experienced by everyone in the Hinamizawa village.
Part I of post: Is it possible to avoid "Oyashiro-sama's curse" in any way? (spoilers)
In episode 6, we see that:
Rika and Hanyuu existing in a in-between dimension where Rika gets to join a new Hinamizawa world with a different scenario. We come to know that every time Rika dies in a previous world, she is reincarnated to a different world with the help of Hanyuu. So, this is a never ending loop until and unless the fate is changed i.e the culprit (Miyo Takano) is caught and Rika avoids her imminent death. (That happens in the last episode)
So, this is a never ending cycle and it can only be stopped if Rika can change her fate by avoiding her death but when and how did the cycle even started in the first place? I assume Rika was born and when July of 1983 came, she met with her fate and died which cause the triggering of the cycle? But, this leads to another question: how did Rika even born in the first place since she is the midst of the never ending cycle?