Good and bad field generators

Abstract

L e t k b e a f ie ld . A fie ld generator i n tw o variables o v e r k i s a polynomial f e k [x , y ] s u c h th a t k (x , y )= k (f, g ) f o r som e rational function g E k(x, y). W e continue the investigation of field generators begun i n, w e first study in detail properties o f th e multiplicity tree at infinity of f once coordinate functions x, y h a v e been chosen that are natural for f (see [2 , 4 .7 ]). O u r o rig in a l motivation f o r th is h a d b e e n a n attem pt to show th at all field generators are good in th e sense that a complementary genera to r g c a n b e fo u n d i n k [x , y ] . However, a quite astonishing exam ple o f a b a d field generator has been constructed by C . J a n i n [1 ], a n d w e instead use th e numerical information obtained to determ ine, w ith th e help of a com puter, a ll b a d field generators o f degree < 25, th e degree o f Jan's exam ple. W e find th a t field generators a r e g o o d f o r degrees d < 2 0 a n d d=22, 23, 24, a n d that there is exactly o n e " ty p e " o f b a d field generator f o r d = 2 1 a n d d = 25 (see 2.6 fo r a m o re precise statem ent). R . Ganong helped materially with th e rather elaborate calculations needed to establish this a n d w ith t h e w riting o f an appendix in which some o f th e details are explained. A g o o d fie ld generator f appears a s p a r t o f a birational morphism cp:AZ->AZ w it h tp(a, 13)=(f(a, fi), g(a, 13)) f o r a, 13 e k. W e s h o w th a t th is is a lm o st tru e i n g e n era l. N am e ly , if f i s a f ie ld generator, a complementary generator g =a lb can alw ays be found w ith x , y ] . This m eans that th e pencil o f curves { g-pLuEk} h a s no base points at finite distance a n d th a t 9: A t -4 P , yo(cc, fl)=(I, f(a, fl), g(a, 13)), i s a birationa l morphism.

One remark on polynomials in two variables

“…It gives also a counter-example to Kaliman's theorem. In fact, from the above characterisation of bad field generator, [ (The * shows the coefficients which do not match with [8].) Prom the first remark of [5] we deduce that this polynomial has one factorisation.…”

“…In [8] the author constructs an example of bad field generator, i.e., a polynomial / G C[x, y], such that its generic fiber is rational and such that there is no polynomial g G C[#,y] verifying k(f,g) = k(x,y). This property is related with the non-existence of sections: a rational polynomial is a bad ήeld generator if and only if no horizontal component is a section.…”

Rational polynomials of simple type

“…This view allows one to see as follows a geometric proof of the result of Russell [20] that a rational polynomial f is good precisely when its resolution has at least one degree one horizontal curve. A degree one horizontal curve for f has image in P 1 × P 1 given by a (1, n) curve.…”

“…Russell [20] (correctly presented in [3]) constructed an example of a rational polynomial with no degree one horizontal curves. This is an example of a bad field generator-a polynomial that is one coordinate in a birational transformation but not in a birational morphism.…”

“…Russell [20] calls this a "field generator" and defines a good field generator to be a rational polynomial that is one coordinate of a birational morphism of the complex plane. A rational polynomial is good precisely when its resolution has at least one degree one horizontal curve, [20]. Daigle [5] studies birational morphisms C 2 → C 2 by associating to a compactification X of the domain plane a canonical map X → P 2 .…”