By Edoardo Ballico, Ciro Ciliberto
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Additional resources for Algebraic Curves and Projective Geometry. Proc. conf Trento, 1988
If m = , .. sL−1 em |e |δi + |e |δj + 0(εij ) λj Ωci Ωi .. |e | Bj ≤C ∞ |)ε (|ω ∞ | + |ωm εi + m 5/2 εm log5/6 ε−1 m =i + ε 5/2 √ εij j + ε3is ∞ (ω ∞ | + |ωm |)ε m +ε +εij 5/2 m log5/6 ε−1 m ∞ . (|ωi∞ | + |ωm |)εim + εim log5/6 ε−1 im 5/2 m=i ε3is for = i; Proof. Observe that Ωc |e |δi = O(ε i ) for = i, O i that Ωi |e |δj = O(εi ) if = i since either λ ≤ λi and δ ≤ cδi on 24/5 6/5 5/6 Ωi , Ωi |e |δj ≤ c Ωi δ 4 δi δj ≤ C Ωi δ δi ≤ Cεi ; or λ ≥ λi , = i and the estimate follows from the deﬁnition of Ωi after the use of H¯ older.
The ﬁrst contribution comes from Q∗ (∆J ( αj ωj )). It is estimated in Lemma 6. We have: Q∗ ∆J αj ωj w = O(Γi )|w|H01 . Next, we have the contribution of O ω 4 (|v k | + |hk | + |k ∗ |) + |v k |5 + |hk |5 + |k ∗ |5 k=i which, by Lemma 8, is o(Γi )|w|H01 . Next, we have the contribution of hi which we trace back to Lemmas 9– 10 and Lemma 11. It is o(Γi )|w|H01 except for ωi4 |hi ||w| which yielded a contribution equal to 1 √ λi |w|H01 o(Γi ) + |w|H01 0 Max |hi | . Bi /2 −Bj We revisit this estimate using Lemma 12.
Lemma 28 follows. Lemma 29 ω 4 |ωm | ≤ C Ωj ∞ |ε |ωm O(ε2 ) √ m + √m λ λm c λj Max εjs ω 4 |ωm | ≤ Ωj (a) 5/2 εjt if , m = j. (b) Proof. (b) follows in a straightforward way from 1. of Lemma 13 and H¯ older. To prove (a), we expand O(δ 2 ) ∞ δm + √ m ωm = cωm λm so that Ωj ∞ ω 4 |ωm | ≤ c|ωm | We split Ωj 1 ω 4 δm + √ λm Ωj ω 4 δm . Ωj ω 4 δm into two pieces A and B. t |x − am | ≤ 14 |a − am |}. On B, ∞ c|ωm | Ωj ε2 2 ∞ ω 4 δm ≤ C √ m +c|ωm | λm ω 4 δm ≤ Ωj |x−am | |a −am | ≥ 14 . We then have ∞ c|ωm 1 | · − am |4 λ5/2 m λ2 |a +√ +√ ∞ | c|ωm λm |a − am | Ωj ∞ | c|ωm inf λm |a − am | r≤ λ4m |a −am | r2 dr √ 1 + r2 ∞ |ε m c|ωm ω4 ≤ √ λ λ |a − am | 1 1 , λj λ .
Algebraic Curves and Projective Geometry. Proc. conf Trento, 1988 by Edoardo Ballico, Ciro Ciliberto