MINIMAL PAGE-GENUS OF MILNOR OPEN BOOKS 3

may be read on a resolution as the fact that the arithmetic genus of the fundamental

cycle is zero. Via the adjunction formula, the latter can be seen to be equivalent

to the conditions in the deﬁnition above; see, for example, [17] for more details.

From now on we assume that the germ (X, x) has a rational singularity at x.

Definition 2.5. A Tjurina component of E for an element Y ∈ E

+

is a max-

imal connected set T of irreducible components of E such that Y · Ei = 0 for all

irreducible components Ei of T . A non-Tjurina component of E for an element

Y ∈ E

+

is an irreducible component Ei of E such that Y · Ei 0. A Tjurina

component T of E for an element Y ∈ E

+

is said to be attached to a non-Tjurina

component Ei of E for Y if there is an irreducible component Ej of T such that

Ej · Ei = 1.

Theorem 2.6. The set E

+

is precisely the set that is generated if we consider

the set of all elements one obtains by starting with the fundamental cycle Z of E

and then applying one of the following two operations, at each step, a ﬁnite number

of times:

(T) Y → Y + Z(T ), where T is a Tjurina component for Y ;

(NT) Y → Y + Ei, where Ei is a non-Tjurina component for Y that does not

intersect any Tjurina component for Y .

Proof. The proof given below is based on a theorem of Pinkham ([14], §14,

Proposition). See also [18].

By deﬁnition, any element Y ∈ E

+

is greater than or equal to the fundamental

cycle Z of E. If Y = Z, there is nothing to prove. So suppose that Y = Z. Then

there is an irreducible component Ej1 of E which has diﬀerent multiplicities in Y

and Z.

Now either Ej1 · Z = 0 or Ej1 · Z 0. Suppose that Ej1 · Z = 0. Set

A1 = {D ∈ E + | D ≥ Z + Ej1 } and deﬁne Z1 to be the least element of A1.

By Pinkham’s Theorem, Z1 is well-deﬁned and Z1 = Z + Z(T1), where T1 is the

Tjurina component for Z that contains Ej1 . Since Y ∈ A1, Y ≥ Z1. If Ej1 · Z 0,

that is Ej1 is non-Tjurina for Z, then either there is a Tjurina component T2 for

Z attached to Ej1 or there is no Tjurina component for Z attached to Ej1 . In the

former case there is an irreducible component Ej2 in T2, satisfying Ej2 · Ej1 = 1,

which has diﬀerent multiplicities in the divisors Y and Z. Arguing as above, one

can now show that Y ≥ Z2 = Z + Z(T2). If there is no Tjurina component for Z

attached to Ej1 , then Y ≥ Z = Z + Ej1 , where Z ∈ E +. Thus, in all cases, we

obtain a divisor Z ∈ E + satisfying Y ≥ Z Z by applying the operation (T) or

(NT) to the fundamental cycle Z. If Y = Z , then we repeat the above process

replacing Z by Z . Since Y ≤ MZ for M ∈ Z suﬃciently large, it is clear that we

will eventually reach Y after a ﬁnite number of steps.

2.3. Milnor open books. We assume that the reader is familiar with the

basic topology of Milnor open books (see, for example, the lectures of Pichon in

this School and Workshop, or [2]); however, for the sake of setting up notation, we

give some elementary deﬁnitions here.

Let (X, x) denote a germ of a normal complex analytic surface having a sin-

gularity at x. Fix a local embedding of (X, x) in (CN , 0). The link of (X, x) is

the 3-manifold MX = X ∩ S2N−1 obtained by intersecting X with a suﬃciently

small euclidean sphere centred at the origin in CN . The link MX carries a natural

3

may be read on a resolution as the fact that the arithmetic genus of the fundamental

cycle is zero. Via the adjunction formula, the latter can be seen to be equivalent

to the conditions in the deﬁnition above; see, for example, [17] for more details.

From now on we assume that the germ (X, x) has a rational singularity at x.

Definition 2.5. A Tjurina component of E for an element Y ∈ E

+

is a max-

imal connected set T of irreducible components of E such that Y · Ei = 0 for all

irreducible components Ei of T . A non-Tjurina component of E for an element

Y ∈ E

+

is an irreducible component Ei of E such that Y · Ei 0. A Tjurina

component T of E for an element Y ∈ E

+

is said to be attached to a non-Tjurina

component Ei of E for Y if there is an irreducible component Ej of T such that

Ej · Ei = 1.

Theorem 2.6. The set E

+

is precisely the set that is generated if we consider

the set of all elements one obtains by starting with the fundamental cycle Z of E

and then applying one of the following two operations, at each step, a ﬁnite number

of times:

(T) Y → Y + Z(T ), where T is a Tjurina component for Y ;

(NT) Y → Y + Ei, where Ei is a non-Tjurina component for Y that does not

intersect any Tjurina component for Y .

Proof. The proof given below is based on a theorem of Pinkham ([14], §14,

Proposition). See also [18].

By deﬁnition, any element Y ∈ E

+

is greater than or equal to the fundamental

cycle Z of E. If Y = Z, there is nothing to prove. So suppose that Y = Z. Then

there is an irreducible component Ej1 of E which has diﬀerent multiplicities in Y

and Z.

Now either Ej1 · Z = 0 or Ej1 · Z 0. Suppose that Ej1 · Z = 0. Set

A1 = {D ∈ E + | D ≥ Z + Ej1 } and deﬁne Z1 to be the least element of A1.

By Pinkham’s Theorem, Z1 is well-deﬁned and Z1 = Z + Z(T1), where T1 is the

Tjurina component for Z that contains Ej1 . Since Y ∈ A1, Y ≥ Z1. If Ej1 · Z 0,

that is Ej1 is non-Tjurina for Z, then either there is a Tjurina component T2 for

Z attached to Ej1 or there is no Tjurina component for Z attached to Ej1 . In the

former case there is an irreducible component Ej2 in T2, satisfying Ej2 · Ej1 = 1,

which has diﬀerent multiplicities in the divisors Y and Z. Arguing as above, one

can now show that Y ≥ Z2 = Z + Z(T2). If there is no Tjurina component for Z

attached to Ej1 , then Y ≥ Z = Z + Ej1 , where Z ∈ E +. Thus, in all cases, we

obtain a divisor Z ∈ E + satisfying Y ≥ Z Z by applying the operation (T) or

(NT) to the fundamental cycle Z. If Y = Z , then we repeat the above process

replacing Z by Z . Since Y ≤ MZ for M ∈ Z suﬃciently large, it is clear that we

will eventually reach Y after a ﬁnite number of steps.

2.3. Milnor open books. We assume that the reader is familiar with the

basic topology of Milnor open books (see, for example, the lectures of Pichon in

this School and Workshop, or [2]); however, for the sake of setting up notation, we

give some elementary deﬁnitions here.

Let (X, x) denote a germ of a normal complex analytic surface having a sin-

gularity at x. Fix a local embedding of (X, x) in (CN , 0). The link of (X, x) is

the 3-manifold MX = X ∩ S2N−1 obtained by intersecting X with a suﬃciently

small euclidean sphere centred at the origin in CN . The link MX carries a natural

3