Now ﬁx an element Y =
miEi ∈ E
and suppose that T , which, for
convenience, we will assume has the form T =
Ei, is a Tjurina component for
Y . Let
b = bc(OB(Y )), b = bc(OB(Y + Z(T ))), b = bc(OB(Z(T )))
and let ki = −Y · Ei, ki = −(Y + Z(T )) · Ei for i = 1, . . . , n. Then
b − b =
(ki − ki) =
(ki − ki) = b −
where ti is the number of non-Tjurina components intersecting Ei. Here the last
equality follows from the arguments given in the proof of Lemma 3.2. Now let
g = genus(OB(Y )), g = genus(OB(Y + Z(T ))), g = genus(OB(Z(T )))
It then follows, using Lemma 3.2, that
(g + b ) − (g + b) = g + b − 1,
which is always nonnegative.
Now suppose that Ei is a non-Tjurina component for Y which does not intersect
any Tjurina component for Y . Let
β = bc(OB(Y + Ei)), γ = genus(OB(Y + Ei)),
and let κj = −(Y + Ei) · Ej for j = 1, . . . , n. Then
β − b =
(κj − kj )
= (κi − ki) +
(κj − kj ) +
(κj − kj )
where the last equality follows from the arguments given in the proof of Lemma 3.3.
It then follows, using Lemma 3.3, that
(γ + β) − (g + b) = ki − Ei
which is always positive since ki ≥ 1 and Ei 2 ≤ −1. This completes the proof.
We wish to thank Sergey Finashin for prompting us to prove Theorem 1.2.
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