Is polygyny for women permissible and not forbidden, there being no verse that prohibits it, provided that both parties agree and lineage is documented, and is what is claimed about the spread of diseases among them incorrect?

The general rule is that the rulings of Islamic law apply to both men and women, except for what has been explicitly exempted. The Sharia has exempted men for polygyny (multiple wives), but has forbidden polyandry (multiple husbands) for women because it was a form of marriage in the جاهلية (pre-Islamic era of ignorance) that Islam abolished. Sound human nature also rejects polyandry for women due to the resulting mixing of lineages, the spread of diseases, and its classification as an abomination. The claim by some that there are no diseases among them is a fallacy, as this act is zina (fornication/adultery), which causes diseases such as AIDS. Furthermore, the fertility of the land and other worldly possessions are not a measure of a nation's righteousness, for Allah Almighty says: "So when they forgot that by which they had been reminded, We opened to them the doors of every [good] thing until, when they rejoiced in what they had been given, We seized them suddenly, and they were in despair."