Let f:X\rightarrow P^1 be a rational function of degree d\geq2 on a curve X. Let P

SnuluddidaPalbx4

SnuluddidaPalbx4

Answered question

2022-02-15

Let f:XP1 be a rational function of degree d2 on a curve X.
Let n2 be a divisor of d. Does there exist a curve Y with a rational function g:YP1 of degree n such that f factors through g
Can we factor f in some sense?
Note that by de Franchis

Answer & Explanation

Arlene Patel

Arlene Patel

Beginner2022-02-16Added 4 answers

Construct a cover of degree 4 such that over 0ϵPP1 there is an unramified point and a point of ramification index 3. Then this cover can't factorize through of subcover of degree 2 (edit because it would force the ramification indexes to be 1 or even). eg: k(X) given by y3(y+1)+x=0 and f given by (x,y)x. In this case X is even isomorphic to PP1!

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