INTRODUCTION TO ALGEBRAIC GEOMETRY Note del corso di ...

50 MezzettiThe following rational map is called the standard quadratic map:Q : P 2 P 2 , [x 0 , x 1 , x 2 ] → [x 1 x 2 , x 0 x 2 , x 0 x 1 ].Q is regular on U := P 2 \ {A, B, C}, where A[1, 0, 0], B[0, 1, 0], C[0, 0, 1] are thefundamental points (see Fig. 2)Let a be the line through B and C: a = V P (x 0 ), and similarly b = V P (x 1 ),c = V P (x 2 ). Then Q(a) = A, Q(b) = B, Q(c) = C. Outside these three lines Q isan isomorphism. Precisely, put U ′ = P 2 \ {a ∪ b ∪ c}; then Q : U ′ → P 2 is regular,the image is U ′ and Q −1 : U ′ → U ′ coincides with Q. Indeed,[x 0 , x 1 , x 2 ] Q → [x 1 x 2 , x 0 x 2 , x 0 x 1 ] Q → [x 2 0x 1 x 2 , x 0 , x 2 1x 2 , x 0 x 1 x 2 2].So Q ◦ Q = 1 P 2 as rational map, hence Q is birational and Q = Q −1 .– Fig.2 –The set of the birational maps P 2 P 2 is a group, called the Cremonagroup. At the end of XIX century, Max Noether proved that the Cremona groupis generated by PGL(3, K) and by the single standard quadratic map above. Theanalogous groups for P n , n ≥ 3, are much more complicated and a completedescription is still unknown.Exercises to §9.1. Let φ : A 1 → A n be the map defined by t → (t, t 2 , . . ., t n ).a) Prove that φ is regular and describe φ(A 1 );b) prove that φ : A 1 → φ(A 1 ) is an isomorphism;c) give a description of φ ∗ and φ −1∗ .2. Let f : A 2 → A 2 be defined by: (x, y) → (x, xy).