MINIMAL PAGE-GENUS OF MILNOR OPEN BOOKS 7
Proof. Assume, without loss of generality, that the Tjurina component T for
Y is given by T = ∪i=1Ei. l Also assume, as before, that the irreducible curves
Ei of E are ordered so that Ei intersects some curve Eφ(i) in {E1, . . . , Ei−1} for
each i = 2, . . . , n. Let Z(T ) =
∑l
i=1
ziEi. Define nonnegative integers ki, for
i = 1, . . . , n, by (3.1). Since T is Tjurina, note that ki = 0 for i = 1, . . . , l. Also
define nonnegative integers ki by replacing (m1, . . . , mn) by (m1 + z1, . . . , mn + zn)
in (3.1), where, for convenience, we define zl+1 = ··· = zn = 0.
Consider now the open books OB(Y + Z(T )) and OB(Y ) associated to the
divisors Y + Z(T ) and Y respectively. By Lemma 3.1, the respective page-genera
are given by
g(Y + Z(T )) = 1 +
n
i=1
(vi 2)(mi + zi) + (mi + zi 1)ki
2
g(Y ) = 1 +
n
i=1
(vi 2)mi + (mi 1)ki
2
.
Subtracting we find
g(Y + Z(T )) g(Y ) =
n
i=1
(vi + ki 2)zi + (mi 1)(ki ki)
2
.
Consider now the equation defining the ki:
(m1 + z1, . . . , mn + zn)IE = (−k1, . . . , −kn).
Comparing this with (3.1) we obtain
(3.2) (z1, . . . , zn)IE = (k1 k1, . . . , kn kn).
Multiplying the left and right hand sides of (3.2) on the right by −(m1, . . . , mn)t
we obtain
0 =
n
i=1
kizi =
n
i=1
mi(ki ki).
Substituting this into the expression for the difference g(Y +Z(T ))−g(Y ) we obtain
g(Y + Z(T )) g(Y ) =
n
i=1
(vi + ki 2)zi (ki ki)
2
=
l
i=1
(vi + ki 2)zi ki
2

n
i=l+1
ki ki
2
.
(3.3)
We now compute the page-genus of the open book OB(Z(T )) associated to
Z(T ), the fundamental cycle of the Tjurina component T =
l
i=1
Ei. For this, let
wi denote the number of irreducible curves Ej , j = i, in T intersecting Ei and
define nonnegative integers ki , for i = 1, . . . , l, by
(z1, . . . , zl)IT = (−k1 , . . . , −kl ),
where IT denotes the intersection matrix of T . Since IT is the submatrix of IE
obtained by taking the first l rows and first l columns of IE , it follows from (3.2)
7
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