algebraic geometry 31 Rational maps

[@rosieshen4445]

19:46 Running the proof on the example, we find (1-it)^2 + (1+it)^2 = (1+t^2) ^2, so h_1(t) = 1-it and h_2(t) = 1_it do have smaller degree. The problem is that there is no h_3, so we don't get a linear relation between h_1^2 and h_2^2

[@hausdorffm]

At 4:48, f1= f2 on dense open set contained in U1, U2 ... this is maybe intersection of U1 and U2. I am not sure why the dense set is? I guess, f1 and f2 are rational function, thus they are not be defined on whole U1 but except finite poles, it is defined. And the complement of poles in U1 is dense, so dense set appeared here, isn't it??? Another reason is that variety has singular points on which rational functions dose not define. At 11:00, I give a proof of the birational between projective line P1 and affine line A1. A1 ---> P1; z ---> (1:z) P1- (0:1) ---> A1; (1:z) ---> z1 And P1 - (0:1) is dense in P1, so the above two map give birational map between P1 and A1. At 12:56, I guess, birational map between products of projective line and projective plane is P1 times P1 ----> p2 (1:z), (1:w) -----> (1:z:w). 15:59 What is omega? omega^3=1???? Uhmm, I cannot understand the factorization using omega.

[@chen4300]

I think the last example of quadratic equation has a mistake. it should be (2t)^2 not (2t^2)^2. There are two linear equations, so you can't find polynomial with small degree

[@jasiu85]

Is this all in Zariski topology?

[@analander9222]

Yes