3264 & All That: A second course in algebraic geometry. by David Eisenbud and Joseph Harris

By David Eisenbud and Joseph Harris

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If V and W are vector spaces of dimensions r + 1 and s + 1 we may write σr,s without bases by the formula σr,s : PV × PW → P(V ⊗ W ) (v, w) → v ⊗ w. For example, the map σ1,1 is defined by the four forms a = X0 Y0 , b = X0 Y1 , c = X1 Y0 , d = X1 Y1 , and these satisfy the equation ac − bd = 0, the Segre variety Σ1,1 is the nonsingular quadric in P 3 . To compute the degree of Σr,l in general, we proceed as with the Veronese varieties. The degree of Σr,s is the number of points in which it meets the intersection of r + s hypersurfaces in P (r+1)(s+1)−1 .

For example, let Z0 , . . , Zn be homogeneous coordinates on P n and zi = Zi /Z0 , i = 1, . . , n the corresponding affine coordinates on the open set U ∼ = A n where Z0 = 0, and consider the form ϕ = dz1 ∧ dz2 ∧ · · · ∧ dzn . This is visibly regular and nonzero in U so its divisor is some multiple of the hyperplane H = V (Z0 ) at infinity. To compute the multiple, let U ⊂ P n 40 1. Overture be the open set Zn = 0, and wi = Zi /Zn , i = 0, . . , n−1, affine coordinates on U . We have 1 wi , i = 1, .

Moreover, i (Λ × Γ) ∩ ∆ ∼ =Λ∩Γ is dimensionally transverse, and is a smooth variety, so Λ × Γ intersects the diagonal ∆ transversely. It follows that ci = (δ · αi β r−i ) = # ∆ ∩ (Λ × Γ) = #(Λ ∩ Γ) =1 and thus we arrive at δ = αr + αr−1 β + · · · + αβ r−1 + β r . (This formula and its derivation will be familiar to anyone who’s had a course in algebraic topology. 3 The graph of a morphism Let f : P r → P s be a morphism given by (s + 1) homogeneous polynomials Fi of degree d that have no common zeros: f : [X0 , .

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