On the principles of three thousand realms in a single moment of life:
The first link is ignorance. The Dharmma analysis treasury says : "The state resulting from past delusion". That is to say, ignorance represents the earthly desires of love and crave that gave rise to in the past. Thus a son may conceive hatred for his father and love for his mother, or a daughter may conceive hatred for her mother and love for her father, as described in the ninth volume of the Dharmma analysis Treasury
Is there any more information regarding this?[
What does this mean
the earthly desires of love and crave that gave rise to in the past.
this is a basic teaching on the 12 links. "the earthly desires of love and crave that gave rise to in the past." sounds like a really choppy translation of some basic ideas on the links, which is why it's probably confusing you. i would imagine it should say: "the earthly desires of love and craving in the past that give rise to the present.". however it's possible you are working from a sutra i do not know that explains things in a way i'm not familiar with. i'll try to explain it the way i understand it according to sutras i have read: earthly desires of love and craving are desires that one never was able to let go of in his/her past life. so someone who was always looking for love in life will die and be reborn with that same craving. someone who let that craving go in life will die and not have to deal with it. ignorance is not understanding that there is no self in the first place and therefore craving is useless and should be abandoned. if one gets rid of ignorance and realizes not self, he/she is enlightened and free from rebirth having broken a link in the chain of dependent arising. the same is true for any of the links in that if one understands the link and breaks it, they are free.
for more information check out the entire book devoted to explaining the twelve links in the samyutta nikaya, the nidana vagga. much of it is available here: http://www.accesstoinsight.org/tipitaka/sn/index.html#sn12
here is a single, longer sutra, devoted to explaining the links: http://www.accesstoinsight.org/tipitaka/dn/dn.15.0.than.html