INTRODUCTION TO ALGEBRAIC GEOMETRY Note del corso di ...

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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.
Introduction to algebraic geometry 5310.4. Proposition. If X and Y are projective varieties (resp. quasi–projectivevarieties), then X × Y is projective (resp. quasi–projective).Proof.σ(X × Y ) = ⋃ i,j(σ(X × Y ) ∩ Σ ij ) == ⋃ i,j(σ(X × Y ) ∩ (U i × V j )) == ⋃ i,j(σ((X ∩ U i ) × (Y ∩ V j ))).If X and Y are projective varieties, then X ∩U i is closed in U i and Y ∩V j is closedin V j , so their product is closed in U i × V j ; since σ| Ui ×V jis an isomorphism, alsoσ(X × Y ) ∩ Σ ij is closed in Σ ij , so σ(X × Y ) is closed in Σ, by Lemma 8.3.If X, Y are quasi–projective, the proof is similar. As for the irreducibility, seeExercise 10.1.□10.5. Example. P 1 × P 1σ : P 1 × P 1 → P 3 is given by {w ij = x i y j , i = 0, 1, j = 0, 1. Σ has only onenon–trivial equation: w 00 w 11 − w 01 w 10 , hence Σ is a quadric. The equation of Σcan be written as(∗)∣ w ∣00 w 01 ∣∣∣= 0.w 10 w 11Σ contains two families of special closed subsets parametrized by P 1 , i.e.{σ(P × P 1 )} P ∈P1 and {σ(P 1 × Q)} Q∈P 1.If P[a 0 , a 1 ], then σ(P × P 1 ) is given by the equations:⎧w 00 = a 0 y 0⎪⎨w 01 = a 0 y 1⎪⎩w 10 = a 1 y 0w 11 = a 1 y 1hence it is a line. Cartesian equations of σ(P × P 1 ) are:{a1 w 00 − a 0 w 10 = 0a 1 w 01 − a 0 w 11 = 0;they express the proportionality of the rows of the matrix (*) with coefficients[a 1 , −a 0 ]. Similarly, σ(P 1 × Q) is the line of equations{ a1 w 00 − a 0 w 01 = 0a 1 w 10 − a 0 w 11 = 0.
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Page 45 and 46: Then v n,d (X) ≃ X: it is the set
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