MINIMAL PAGE-GENUS OF MILNOR OPEN BOOKS 9
Now fix an element Y =
∑n
i=1
miEi E
+
and suppose that T , which, for
convenience, we will assume has the form T =
l
i=1
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 =
n
i=1
(ki ki) =
l
i=1
ki +
n
i=l+1
(ki ki) = b
l
i=1
ti,
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 =
n
j=1
(κj kj )
= (κi ki) +
i−1
j=1
(κj kj ) +
n
j=i+1
(κj kj )
= −Ei
2
vi,
where the last equality follows from the arguments given in the proof of Lemma 3.3.
It then follows, using Lemma 3.3, that
+ β) (g + b) = ki Ei
2
1,
which is always positive since ki 1 and Ei 2 −1. This completes the proof.
Acknowledgement
We wish to thank Sergey Finashin for prompting us to prove Theorem 1.2.
References
[1] M. Artin, On isolated rational singularities of surfaces, American J. Math. 88 (1) (1966),
129–136.
[2] M. Bhupal, Open book decomposition of links of simple surface singularities, preprint, 2007.
[3] C. Caubel and P. Popescu-Pampu On the contact boundaries of normal surface singularities,
C.R. Acad. Sci. Paris, Ser. I 339 (2004), 43–48.
[4] C. Caubel, A. emethi and P. Popescu-Pampu, Milnor open books and Milnor fillable contact
3-manifolds, Topology 45 (2006), 673–689.
[5] T. Etg¨ u and B. Ozbagci, Explicit horizontal open books on some plumbings, Int. Jour. Math.
17 (9) (2006), 1013–1031.
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(2004), 4255–4267.
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