Proof We know by Corollary 24 that \u03a8 \u0393 is a sum of monomials with coefficients

# Proof we know by corollary 24 that ψ γ is a sum of

This preview shows page 18 - 21 out of 54 pages.

Proof. We know by Corollary 2.4 that Ψ Γ is a sum of monomials with coefficients +1. The lemma is clear for the zero set of any polynomial with coefficients > 0. /
GRAPH POLYNOMIALS 19 Remark 7.2. (i) The assertion of the lemma is true for any graph polynomial. We do not need hypotheses about numbers of edges or loops. (ii) By Proposition 3.1, coordinate linear spaces L X Γ correspond to subgraphs G Γ such that h 1 ( G ) > 0. Proposition 7.3. Let Γ be as above. Define (7.2) η = η Γ = Ω 2 n - 1 ( A ) Ψ 2 Γ as in (5.2) . There exists a tower P = P r π r,r - 1 ---→ P r - 1 π r - 1 ,r - 2 -----→ . . . π 2 , 1 --→ P 1 π 1 , 0 --→ P 2 n - 1 ; (7.3) π = π 1 , 0 ◦ · · · ◦ π r,r - 1 where P i is obtained from P i - 1 by blowing up the strict transform of a coordinate linear space L i X Γ and such that (i) π * η Γ has no poles along the exceptional divisors associated to the blowups. (ii) Let B P be the total transform in P of the union of coordinate hyperplanes Δ 2 n - 2 : A 1 A 2 · A 2 n = 0 in P 2 n - 1 . Then B is a normal crossings divisor in P . No face (= non-empty intersection of compo- nents) of B is contained in the strict transform Y of X Γ in P . (iii) the strict transform of σ 2 n - 1 ( R ) in P does not meet Y . Proof. Our algorithm to construct the blowups will be the following. Let S denote the set of coordinate linear spaces L P 2 n - 1 which are maximal, i.e. L S, L L X Γ L = L . Define (7.4) F = { L X Γ coordinate linear space | L = \ L ( i ) , L ( i ) S } . Let F min ⊂ F be the set of minimal elements in F . Note that elements of F min are disjoint. Define P 1 π 1 , 0 --→ P 2 n - 1 to be the blowup of elements of F min . Now define F 1 to be the collection of strict transforms in P 1 of elements in F \ F min . Again elements in F 1 , min are disjoint, and we define P 2 by blowing up elements in F 1 , min . Then F 2 is the set of strict transforms in P 2 of F 1 \ F 1 , min , etc. This process clearly terminates. Note that to pass from P i to P i +1 we blow up strict transforms of coordinate linear spaces L contained in X Γ . There will exist an open set U P 2 n - 1 such that P i × P 2 n - 1 U = U and such that L U = . It follows that to calculate the pole orders of π * η Γ along exceptional divisors arising in the course of our algorithm it suffices to consider the simple blowup of a coordinate linear space L X Γ on P 2 n - 1 . Suppose L : A 1 = . . . A p = 0. By assumption, the subgraph G = { e 1 , . . . , e p } ⊂ Γ is convergent, i.e. p > 2 h 1 ( G ). As in Proposition
20 SPENCER BLOCH, H ´ EL ` ENE ESNAULT, AND DIRK KREIMER 3.5, if I = ( A 1 , . . . , A p ) K [ A 1 , . . . , A 2 n ], then Ψ Γ I h 1 ( G ) - I h 1 ( G )+1 so the denominator of η Γ contributes a pole of order 2 h 1 ( G ) along the exceptional divisor. On the other hand, writing a i = A i A 2 n , a typical open in the blowup will have coordinates a i = a i a p , i < p together with a p , . . . , a 2 n - 1 and the exceptional divisor will be defined by a p = 0.

#### You've reached the end of your free preview.

Want to read all 54 pages?