String theorists need to know algebraic geometry , algebraic topology , moduli spaces , characteristic classes etc. How do string theorists learn these topics ? I can't find an algebraic geometry textbook for physicists. If some one wants to develop new original ideas in string theory , He presumably need to know these topics in mathematics at a deep level. Is that right ? Do one start learning algebraic geometry be reading a textbook such as Griffiths starting from the first chapter and so on ? Or just ignore the mathematics textbook and learn the required mathematics from physics sources ? I always find that knowing the general mathematical theory makes many seemingly difficult constructions and ideas in physics look much easier. Is it worth it to spend a lot lot of time reading from textbooks on pure mathematics if I'm eager to do the same kind of research that people such as Edward Witten do.