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

(γ + 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. N´ 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.

[6] J. Etnyre, Planar open book decompositions and contact structures, Int. Math. Res. Not. 79

(2004), 4255–4267.

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

(γ + 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. N´ 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.

[6] J. Etnyre, Planar open book decompositions and contact structures, Int. Math. Res. Not. 79

(2004), 4255–4267.

9