What is the ruling on drawing lines with a pencil or pen on books in the university library, and what should be done when part of a book page is torn, with the fear of having to pay the full price of the book?

It is not permissible to write on books, as it distorts and corrupts the book. This is tampering with public books and may lead to guaranteeing their value if it decreases. The one who damages the paper guarantees its value unless the authorized person absolves him. If this results in a decrease in the book's value, then you are liable for the compensation for the resulting defect, not the entire value of the book.