INTRODUCTION TO ALGEBRAIC GEOMETRY Note del corso di ...

52 Mezzettisatisfies (*) and that w αβ ≠ 0. Then[w 00 , . . ., w ij , . . ., w nm ] = [w 00 w αβ , . . ., w ij w αβ , . . ., w nm w αβ ] == [w 0β w α0 , . . ., w iβ w αj , . . ., w nβ w αm ] == σ([w 0β , . . ., w nβ ], [w α0 , . . ., w αm ]).σ is called the Segre map and Σ n,m the Segre variety or biprojective space. Notethat Σ is covered by the affine open subsets Σ ij = Σ ∩ W ij , where W ij = P N \V P (w ij ). Moreover Σ ij = σ(U i × V j ), with U i × V j ≃ A n+m .10.2. Proposition. σ| Ui ×V j: U i × V j ≃ A n+m → Σ ij is an isomorphism ofvarieties.Proof. Assume by simplicity i = j = 0. Choose non–homogeneous coordinates onU 0 : u i = x i /x 0 and on V 0 : v j = y j /y 0 . So u 1 , . . .u n , v 1 , . . ., v m are coordinates onU 0 × V 0 . Take non–homogeneous coordinates also on W 00 : z ij = w ij /w 00 . Usingthese coordinates we have:σ| Ui ×V j:(u 1 , . . .u n , v 1 , . . ., v m ) → (v 1 , . . ., v m , u 1 , u 1 v 1 , . . ., u 1 v m , . . ., u n v m )||([1, u 1 , . . ., u n ], [1, v 1 , . . ., v m ])i.e. σ(u 1 , . . ., v m ) = (z 01 , . . ., z nm ), where⎧⎨z i0 = u i , if i = 1, . . ., n;z 0j = v j , if j = 1, . . ., m;⎩z ij = u i v j = z i0 z 0j otherwise.Hence σ| U0 ×V 0is regular.The inverse map takes (z 01 , . . ., z nm ) to (z 10 , . . ., z n0 , z 01 , . . ., z 0m ), so it is alsoregular.□□10.3. Corollary. P n × P m is irreducible and birational to P n+m .Proof. The first assertion follows from Ex.5, Ch.6, considering the covering of Σby the open subsets Σ ij . For the second assertion, by Theorem 9.15, it is enoughto note that Σ n,m and P n+m contain isomorphic open subsets, i.e. Σ ij and A n+m .□From now on, we shall identify P n × P m with Σ n,m . If X ⊂ P n , Y ⊂ P m areany quasi–projective varieties, then X × Y will be automatically identified withσ(X × Y ) ⊂ Σ.