Download 3264 & All That - Intersection Theory in Algebraic Geometry by David Eisenbud and Joe Harris PDF

By David Eisenbud and Joe Harris

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Ys ] → [. . , Xi Yj , . . ]. The map σr,s is an immersion because on each open set where one of the Xi and one of the Yj are equal to 1 the rest of the coordinates can be recovered from the products. 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 .

Let z = z1 , . . , zn be coordinates on A n and let x0 , x1 be homogeneous coordinates of P 1 . ” If f (z) = 0 is a function 24 1. Overture vanishing on Y , then f (z/t) = f ( xx10 z) is a function vanishing on tY . We let x1 f W = V ({xdeg f ( z) | f (z) vanishes on Y }) ⊂ A n × P 1 . 0 x0 The fiber of W over the point (t, 1) ∈ P 1 is obviously tY , so the restriction of W to A n × A 1 ⊂ A n × P 1 , the open set x1 = 0, is irreducible. Its closure W dominates P 1 and has fiber Y over the point (1, 1).

Each smooth cubic surface in P 3 contains exactly 27 distinct lines. 34 is to show that any smooth cubic surface can be realized as the blowup of P 2 in 6 fairly general points, and to analyze the geometry of such a blowup in detail (see for example Manin [1986]or Reid [1988b]. The Chern class approach that we will now indicate has the 48 1. Overture advantage of applying equally to related results where no such analysis is available. For example, the Chern class method will also show that a general quintic threefold in P 4 contains exactly 2875 lines (a computation that played an important role in the discovery of mirror symmetry; see for example Morrison [1993]), or even that a general hypersurface of degree 37 in P 20 contains exactly 4798492409653834563672780605191070760393640761817269985515 lines.

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