By David Eisenbud and Joe Harris
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Extra info for 3264 & All That - Intersection Theory in Algebraic Geometry
The image Φ = Φn,d ⊂ P N of the Veronese map ν = νn,d is called the d-th Veronese variety of P n , as is any subvariety of P N projectively equivalent to it. 25. The degree of Φn,d is dn . Proof. The degree of Φ is the cardinality of its intersection with n general hyperplanes H1 , . . , Hn ⊂ P N ; since the map ν is one-to-one, this is in turn the cardinality of the intersection f −1 (H1 ) ∩ · · · ∩ f −1 (Hn ) ⊂ P n . Since the preimages of the hyperplanes Hi are n general hypersurfaces of degree d in P n , by B´ezout’s Theorem, this cardinality is dn .
This shows that A(P r × P s ) is a homomorphic image of Z[α, β]/(αr+1 , β s+1 ) = Z[α]/(αr+1 ) ⊗Z Z[β]/(β s+1 ). On the other hand, Ξr,s is a single point, so deg ϕr,s = 1. The pairing Ap+q (P r × P s ) × Ar+s−p−q (P r × P s ) → Z; p q ([X], [Y ]) → deg([X][Y ]) m n sends (α β , α β ) to 1 if p + m = r and q + n = s, because in this case the intersection is transverse and consists of one point, and to 0 otherwise, since then the intersection is empty. This shows that the monomials of bidegree (p, q), for 0 ≤ p ≤ r and 0 ≤ q ≤ s, are linearly independent over Z, proving the first statement.
The inclusion relationships, also indicated in the chart, are likewise not hard; to establish that one orbit lies in the closure of another, it suffices to exhibit a one-parameter family specializing from one type to the other. —but can also be done by focussing on the singularities of the curves. 3. While we can answer almost any question of this sort about plane cubics, the answers to analogous questions for higher degree curves or hypersurfaces of higher dimension, for example about the stratification by singularity type, remain mysterious.
3264 & All That - Intersection Theory in Algebraic Geometry by David Eisenbud and Joe Harris