New York Journal of Mathematics

New York J. Math. 1 (1994) 1–9.

Heinrich’s Counterexample to Azevedo’s Conjecture

Robert W. Berger

Abstract. Johannes Heinrich, a student of the University of Saarbrücken, found by computer calculation an example of a plane algebroid curve whose module of differentials has a bigger torsion than that of the canonical branch in the same equisingularity class. That contradicts a conjecture of Azevedo.

Consider an irreducible plane algebroid curve C over an algebraically closed field k of characteristic zero. C has a local ring 𝔬 = k[[x, y]] = k[[X, Y]]/(f(X, Y)) where f(X, Y) is an irreducible power series with coefficients in k. Let Ω(𝔬/k) denote the universally finite differential module of 𝔬 over k, T = τ(Ω(𝔬/k)) its torsion submodule, and 𝔬̄ = k[[t]] the integral closure of 𝔬.

In his paper [5] Zariski shows that for the lengths as 𝔬-modules one has ℓ(T) ≤ 2·ℓ(𝔬̄/𝔬), and equality holds if and only if, after a suitable change of the variables, the curve C can be represented by an equation f(X, Y) = Y^{n} −X^{m} with gcd(m, n) = 1. We say in this case that “C has maximal torsion”. (For a generalization of this notion to not necessarily plane curves see [4].) Equivalently this means, that C has a parametric representation of the form

x = t^{n}, y = t^{m} .

Now any non regular plane algebroid curve C has a parametric representation of the form

(1)  \begin{aligned}x & = & t^{n} \\ y & = & ∑_{i=m}^{∞} a_{i}t^{i}, a_{m} ≠ 0,  n < m, n ∤ m , \end{aligned}

the Puiseux Expansion of C. Another way of writing this is:

y = ∑_{i=m}^{∞} a_{i}x^{\frac{i}{n}} . (2)

\begin{matrix}c \\ ◯ \end{matrix}1994 State University of New York ISSN 1076-9803/94

Received March 7, 1994

Mathematics Subject Classification. 14H20, 13N05.

Key words and phrases. plane curves, torsion of differential module.

Among the exponents \frac{i}{n} in (2) certain exponents ε_{k} are called the characteristic exponents of C. They are defined as follows ([6]):

ε_{1} = \frac{m}{n} .

If (m, n) = 1 , this is the only characteristic exponent.

In any case write \frac{m}{n} = \frac{m_{1}}{n_{1}} with (m_{1}, n_{1}) = 1. Since k((t)) = k((x, y)) the greatest common divisor of n and all the i with a_{i} ≠ 0 is 1. Therefore, if n_{1} < n there is a first i with a_{i} ≠ 0 such that \frac{i}{n} cannot be written as a fraction with denominator n_{1}. Define:

ε_{2} = \frac{i}{n} is the first exponent in (2) with a_{i} ≠ 0 and which cannot be written as a fraction with denominator n_{1}. Write ε_{2} = \frac{m_{2}}{n_{1}n_{2}} with (m_{2}, n_{2}) = 1.

As before, if n_{1}n_{2} < n there exists a first i with a_{i} ≠ 0 and such that \frac{i}{n} cannot be written as a fraction with denominator n_{1}n_{2}. Define:

ε_{3} = \frac{i}{n} is the first exponent in (2) with a_{i} ≠ 0 and which cannot be written as a fraction with denominator n_{1}n_{2}. Write ε_{3} = \frac{m_{3}}{n_{1}n_{2}n_{3}} with (m_{3}, n_{3}) = 1.

etc. After finitely many steps the process stops with an exponent

ε_{g} = \frac{m_{g}}{n_{1}n_{2}n_{3}···n_{g}} with (m_{g}, n_{g}) = 1 and n_{1}n_{2}n_{3} ⋯ n_{g} = n. Putting in evidence the characteristic exponents one can rewrite the expansion (2) in the form

y = b_{1}x^{\frac{m_{1}}{n_{1}}} + ⋯ + b_{2}x^{\frac{m_{2}}{n_{1}n_{2}}} + ⋯ + b_{3}x^{\frac{m_{3}}{n_{1}n_{2}n_{3}}} + ⋯ + b_{g}x^{\frac{mg}{n_{1}n_{2}n_{3}···ng}} + …,
b_{k} ≠ 0, k = 1, …, g .

Note that the characteristic exponents ε_{k} uniquely determine the pairs (m_{k}, n_{k}) and vice versa. The pairs (m_{k}, n_{k}), k = 1 … g are called the characteristic pairs of the curve C. Two plane algebroid curves over k are called equisingular if and only if they have the same characteristic exponents (or characteristic pairs). Equisingularity obviously is an equivalence relation. Of all curves in an equisingularity class (i.e. with a given set of characteristic exponents ε_{1} … ε_{g} ) there is one called the canonical branch of this class:

y = x^{ε_{1}} + x^{ε_{2}} + ⋯ + x^{ε_{g}}

or in parametric expansion:

x = t^{n}
y = t^{β_{1}} + t^{β_{2}} + t^{β_{3}} + ⋯ + t^{β_{g−1}} + t^{β_{g}}

with β_{i} := m_{i}n_{i+1} ⋯ n_{g} for i = 1, …, g. Now the theorem of Zariski cited above can be reformulated as follows

An irreducible singular algebroid curve C has maximal torsion (i.e. ℓ(T) = 2ℓ(ō/o)) if and only if it is a canonical branch with exactly one characteristic pair.

In his thesis [1] Azevedo conjectures the following generalization of this result:

Of all irreducible plane algebroid curves in an equisingularity class the canonical branch has the biggest torsion, i.e. if C is an arbitrary irreducible plane algebroid curve and C′ is the canonical branch in the equisingularity class of C then for the torsions T and T′ of the differential modules of C and C′ respectively holds ℓ(T) ≤ ℓ(T′) .

In [1] this conjecture is backed up by examples and also proved for certain curves with two characteristic pairs. There is also an example in [1] that, contrary to the case of only one characteristic pair, other curves in the same equisingularity class can have the same length of the torsion of the differential module as the canonical branch.

Now Heinrich shows in his Diplomarbeit [3] that the above conjecture is false by giving a counterexample:

Example: Let C be the curve :

x := t^{6}
y := t^{9} + 2t^{10} − 2t^{11}.

Then the canonical branch C′ in the class of C is

x := t^{6}
y := t^{9} + t^{10}.

But for the torsions T and T′ of C and C′ respectively one has:

ℓ(T) = 36 > 35 = ℓ(T′).

Proof ([3]): Let us denote by: ν the natural valuation of Quot(𝔬), ν(𝔬) the value semigroup of 𝔬, 𝔠 the conductor of 𝔬̄ into 𝔬, c := ν(𝔠) the value of the conductor, D the universally finite derivation of 𝔬̄ over k. In the usual manner we extend the valuation ν to a valuation of 𝔬̄ D 𝔬̄ = {∑_{i=0}^{∞} a_{i} · t^{i} D t | a_{i} ∈ k} by

ν (∑_{i=0}^{∞} a_{i} · t^{i} D t) := ν (∑_{i=0}^{∞} a_{i} · t^{i}) .

Then obviously for every z ∈ 𝔪̄ := 𝔬̄ · t we have ν( D z) = ν(z) − 1. By Korollar 2 to Satz 8 of [2], which is valid also in the algebroid case, one has for every plane algebroid curve:

ℓ(T) = ℓ(𝔬̄ D 𝔬̄/𝔬 D 𝔬) + ℓ(𝔬̄/𝔬).

Now for any 𝔬-module M of finite length we have ℓ(M) = dim_{k}(M), since k is algebraically closed. Therefore ℓ(𝔬̄/𝔬) = \frac{1}{2} · ℓ(𝔬̄/𝔠) = \frac{1}{2} · dim_{k}(𝔬̄/𝔠) = \frac{1}{2} · c . So we get:

ℓ(T) = dim_{k} (𝔬̄ D 𝔬̄/𝔬 D 𝔬) + \frac{c}{2} .

Because of 𝔬 ⊇ 𝔠 we have the inclusions 𝔬̄ D 𝔬̄ ⊇ 𝔬 D 𝔬 ⊇ D 𝔠 and therefore

𝔬̄ D 𝔬̄/𝔬 D 𝔬 ≅ (𝔬̄ D 𝔬̄/ D 𝔠) / (𝔬 D 𝔬/ D 𝔠) .

From 𝔠 = {z | z ∈ 𝔬̄, ν(z) ≥ c} we obtain

D 𝔠 = {( ∑_{i=c−1}^{∞} a_{i} · t^{i})· D t | a_{i} ∈ k} = 𝔬̄ · t^{c−1} D t .

It follows that the residue classes modulo 𝔬̄ · t^{c−1} D t

\{ \overline{D t}, \overline{t D t}, \overline{t^{2} D t} …, \overline{t^{c−2} D t}\}

are a basis of 𝔬̄ D 𝔬̄/ D 𝔠 as a k-vector space, and therefore

dim_{k} (𝔬̄ D 𝔬̄/ D 𝔠) = c − 1 .

On the other hand the residue classes modulo 𝔬̄ · t^{c−1} D t

\{\overline{x^{i}y^{j} D x} | i, j ∈ ℕ_{0}, ν(x^{i}y^{j} D x) < c − 1\} ∪ \{\overline{x^{i}y^{j} D y} | i, j ∈ ℕ_{0}, ν(x^{i}y^{j} D y) < c − 1\}

form a set of generators of 𝔬 D 𝔬/ D 𝔠 as a k-vector space. Denote these generators by ω_{1}, …, ω_{s} in any order, represent them by the above basis of 𝔬̄ D 𝔬̄/ D 𝔠:

ω_{i} = ∑_{j=0}^{c−2} a_{ji} · \overline{t^{j} D t},  a_{ji} ∈ k,  i = 1, …, s ,

and define

A = (a_{ji})_{\left\{{j=0,...,c−2 \atop i=1,...,s}\right\}} .

Then

dim_{k} (𝔬̄ D 𝔬̄/𝔬 D 𝔬) = dim_{k} (𝔬̄ D 𝔬̄/ D 𝔠) − dim_{k} (𝔬 D 𝔬/ D 𝔠) = c − 1 − rank A,

and consequently

ℓ(T) = \frac{3c}{2} − 1 − rank A . (3)

We will now compute ℓ(T) and ℓ(T′) for the example using formula (3): Both curves have the same characteristic pairs

(m_{1}, n_{1}) = (3, 2) (m_{2}, n_{2}) = (10, 3)

and consequently the same value group. We obtain β_{1} = m = 9, β_{2} = m_{2} = 10. In order to compute c we use formula (3.14) from Chapter II of [7]:

c = β_{2}(e_{1} − 1) + β_{1}(n − e_{1}) − n + 1 ,

where e_{1} := gcd(n, m) = 3. (Loc. cit. definition 3.2.) So we get:

c = 10(3 − 1) + 9(6 − 3) − 6 + 1 = 42 .

Heinrich’s Counterexample to Azevedo’s Conjecture 5

For both curves one has ν(x) = 6, ν(y) = 9, ν( D x) = 6, ν( D y) = 9. Therefore the set {ω_{1}, …, ω_{s}} for both curves consists of the following elements:

ω_{1} = \overline{D x},  ω_{2} = \overline{y D x},  ω_{3} = \overline{y^{2} D x},  ω_{4} = \overline{y^{3} D x},
ω_{5} = \overline{x D x},  ω_{6} = \overline{xy D x},  ω_{7} = \overline{xy^{2} D x},  ω_{8} = \overline{xy^{3} D x},
ω_{9} = \overline{x^{2} D x},  ω_{10} = \overline{x^{2}y D x},  ω_{11} = \overline{x^{2}y^{2} D x},
ω_{12} = \overline{x^{3} D x},  ω_{13} = \overline{x^{3}y D x},
ω_{14} = \overline{x^{4} D x},  ω_{15} = \overline{x^{4}y D x},
ω_{16} = \overline{x^{5} D x},
ω_{17} = \overline{D y},  ω_{18} = \overline{y D y},  ω_{19} = \overline{y^{2} D y},  ω_{20} = \overline{y^{3} D y},
ω_{21} = \overline{x D y},  ω_{22} = \overline{xy D y},  ω_{23} = \overline{xy^{2} D y},
ω_{24} = \overline{x^{2} D y},  ω_{25} = \overline{x^{2}y D y},  ω_{26} = \overline{x^{2}y^{2} D y},
ω_{27} = \overline{x^{3} D y},  ω_{28} = \overline{x^{3}y D y},
ω_{29} = \overline{x^{4} D y},
ω_{30} = \overline{x^{5} D y}.

Now one can compute the matrices A′ and A for the curves C′ and C respectively. The matrix A′ for the curve x = t^{6}, y = t^{9} + t^{10} is (dots representing zeros):

ω1 ω2 ω3 ω4 ω5 ω6 ω7 ω8 ω9 ω10 ω11 ω12 ω13 ω14 ω15 ω16 ω17 ω18 ω19 ω20 ω21 ω22 ω23 ω24 ω25 ω26 ω27 ω28 ω29 ω30
\overline{t0 D t} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
\overline{t1 D t} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
\overline{t2 D t} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
\overline{t3 D t} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
\overline{t4 D t} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
\overline{t5 D t} 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
\overline{t6 D t} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
\overline{t7 D t} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
\overline{t8 D t} . . . . . . . . . . . . . . . . 9 . . . . . . . . . . . . .
\overline{t9 D t} . . . . . . . . . . . . . . . . 10 . . . . . . . . . . . . .
\overline{t10 D t} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
\overline{t11 D t} …. 6 . . . . . . . . . . . . . . . . . . . . . . . . .
\overline{t12 D t} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
\overline{t13 D t} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
\overline{t14 D t} . 6 . . . . . . . . . . . . . . . . . . 9 . . . . . . . . .
\overline{t15 D t} . 6 . . . . . . . . . . . . . . . . . . 10 . . . . . . . . .
\overline{t16 D t} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
\overline{t17 D t} . . . . . . . . 6 . . . . . . . . 9 . . . . . . . . . . . .
\overline{t18 D t} . . . . . . . . . . . . . . . . . 19 . . . . . . . . . . . .
\overline{t19 D t} . . . . . . . . . . . . . . . . . 10 . . . . . . . . . . . .
\overline{t20 D t} . . . . . 6 . . . . . . . . . . . . . . . . . 9 . . . . . .
\overline{t21 D t} . . . . . 6 . . . . . . . . . . . . . . . . . 10 . . . . . .
\overline{t22 D t} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 R. W. Berger

\overline{t23 D t} . . 6 . . . . . . . . 6 . . . . . . . . . 9 . . . . . . . .
\overline{t24 D t} . . 12 . . . . . . . . . . . . . . . . . . 19 . . . . . . . .
\overline{t25 D t} . . 6 . . . . . . . . . . . . . . . . . . 10 . . . . . . . .
\overline{t26 D t} . . . . . . . . . 6 . . . . . . . . 9 . . . . . . . 9 …
\overline{t27 D t} . . . . . . . . . 6 . . . . . . . . 28 . . . . . . . 10 …
\overline{t28 D t} . . . . . . . . . . . . . . . . . . 29 . . . . . . . . . . .
\overline{t29 D t} . . . . . . 6 . . . . . . 6 …. 10 . . . . . 9 . . . . .
\overline{t30 D t} . . . . . . 12 . . . . . . . . . . . . . . . . . 19 . . . . .
\overline{t31 D t} . . . . . . 6 . . . . . . . . . . . . . . . . . 10 . . . . .
\overline{t32 D t} … 6 . . . . . . . . 6 . . . . . . . . . 9 . . . . . 9 .
\overline{t33 D t} … 18 . . . . . . . . 6 . . . . . . . . . 28 . . . . . 10 .
\overline{t34 D t} … 18 . . . . . . . . . . . . . . . . . . 29 . . . . . . .
\overline{t35 D t} … 6 . . . . . . 6 …. 6 … 9 . . 10 …. 9 . .
\overline{t36 D t} . . . . . . . . . . 12 . . . . . . . . 37 . . . . . . . 19 . .
\overline{t37 D t} . . . . . . . . . . 6 . . . . . . . . 57 . . . . . . . 10 . .
\overline{t38 D t} . . . . . . . 6 . . . . . . 6 …. 39 . . . . . 9 … 9
\overline{t39 D t} . . . . . . . 18 . . . . . . 6 …. 10 . . . . . 28 … 10
\overline{t40 D t} . . . . . . . 18 . . . . . . . . . . . . . . . . . 29 ….

After deleting all rows that consist of zeros alone, deleting row \overline{t^{9} D t} because it depends of row \overline{t^{8} D t}, deleting row \overline{t^{18} D t} because it depends of row \overline{t^{19} D t}, and some permutations of the columns one obtains the following matrix with the same rank:

ω1 ω17 ω5 ω2 ω21 ω9 ω18 ω6 ω24 ω12 ω3 ω22 ω27 ω10 ω19 ω14 ω7 ω25 ω13 ω29 ω4 ω23 ω11 ω28 ω20 ω15 ω8 ω26 ω16 ω30
\overline{t5 D t} 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
\overline{t8 D t} . 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
\overline{t11 D t} . . 6 . . . . . . . . . . . . . . . . . . . . . . . . . . .
\overline{t14 D t} … 6 9 . . . . . . . . . . . . . . . . . . . . . . . . .
\overline{t15 D t} … 6 10 . . . . . . . . . . . . . . . . . . . . . . . . .
\overline{t17 D t} . . . . . 6 9 . . . . . . . . . . . . . . . . . . . . . . .
\overline{t19 D t} . . . . . . 10 . . . . . . . . . . . . . . . . . . . . . . .
\overline{t20 D t} . . . . . . . 6 9 . . . . . . . . . . . . . . . . . . . . .
\overline{t21 D t} . . . . . . . 6 10 . . . . . . . . . . . . . . . . . . . . .
\overline{t23 D t} . . . . . . . . . 6 6 9 . . . . . . . . . . . . . . . . . .
\overline{t24 D t} . . . . . . . . . . 12 19 . . . . . . . . . . . . . . . . . .
\overline{t25 D t} . . . . . . . . . . 6 10 . . . . . . . . . . . . . . . . . .
\overline{t26 D t} . . . . . . . . . . . . 9 6 9 . . . . . . . . . . . . . . .
\overline{t27 D t} . . . . . . . . . . . . 10 6 28 . . . . . . . . . . . . . . .
\overline{t28 D t} . . . . . . . . . . . . . . 29 . . . . . . . . . . . . . . .
\overline{t29 D t} . . . . . . . . . . . . . . 10 6 6 9 . . . . . . . . . . . .
\overline{t30 D t} . . . . . . . . . . . . . . . . 12 19 . . . . . . . . . . . .
\overline{t31 D t} . . . . . . . . . . . . . . . . 6 10 . . . . . . . . . . . .
\overline{t32 D t} . . . . . . . . . . . . . . . . . . 6 9 6 9 . . . . . . . .
\overline{t33 D t} . . . . . . . . . . . . . . . . . . 6 10 18 28 . . . . . . . .
\overline{t34 D t} . . . . . . . . . . . . . . . . . . . . 18 29 . . . . . . . .
\overline{t35 D t} . . . . . . . . . . . . . . . . . . . . 6 10 6 9 9 … 6 .
\overline{t36 D t} . . . . . . . . . . . . . . . . . . . . . . 12 19 37 . . . . .

Heinrich’s Counterexample to Azevedo’s Conjecture 7

\overline{t37 D t} . . . . . . . . . . . . . . . . . . . . . . 6 10 57 . . . . .
\overline{t38 D t} . . . . . . . . . . . . . . . . . . . . . . . . 39 6 6 9 . 9
\overline{t39 D t} . . . . . . . . . . . . . . . . . . . . . . . . 10 6 18 28 . 10
\overline{t40 D t} . . . . . . . . . . . . . . . . . . . . . . . . . . 18 29 . .

One can easily see that all the 27 rows are linearly independent and so

rank A′ = 27 ,
ℓ(T′) = \frac{3 · 42}{2} − 1 − 27 = 35 .

Now we compute the matrix A for the curve x = t^{6}, y = t^{9} + 2t^{10} − 2t^{11} (we omit the \overline{t^{0} D t}, …, \overline{t^{40} D t} in front of the rows in order to save space):

ω1 ω2 ω3 ω4ω5 ω6 ω7ω8ω9 ω10 ω11ω12 ω13ω14 ω15ω16 ω17 ω18 ω19 ω20 ω21 ω22 ω23 ω24 ω25ω26 ω27 ω28 ω29 ω30
. . . . . . . . . . . . . . . . . .  .  …  . . . . . . . .
. . . . . . . . . . . . . . . . . .  .  …  . . . . . . . .
. . . . . . . . . . . . . . . . . .  .  …  . . . . . . . .
. . . . . . . . . . . . . . . . . .  .  …  . . . . . . . .
. . . . . . . . . . . . . . . . . .  .  …  . . . . . . . .
6 . . . . . . . . . . . . . . . . .  .  …  . . . . . . . .
. . . . . . . . . . . . . . . . . .  .  …  . . . . . . . .
. . . . . . . . . . . . . . . . . .  .  …  . . . . . . . .
. . . . . . . . . . . . . . . . 9 .  .  …  . . . . . . . .
. . . . . . . . . . . . . . . . 20 .  .  …  . . . . . . . .
. . . . . . . . . . . . . . . .−22 .  .  …  . . . . . . . .
…. 6 . . . . . . . . . . . . .  .  …  . . . . . . . .
. . . . . . . . . . . . . . . . . .  .  …  . . . . . . . .
. . . . . . . . . . . . . . . . . .  .  …  . . . . . . . .
. 6 . . . . . . . . . . . . . . . .  .  . 9 .  . . . . . . . .
. 12 . . . . . . . . . . . . . . . .  .  . 20 .  . . . . . . . .
.−12 . . . . . . . . . . . . . . . .  .  .−22 .  . . . . . . . .
. . . . . . . . 6 . . . . . . . . 9  .  …  . . . . . . . .
. . . . . . . . . . . . . . . . . 38  .  …  . . . . . . . .
. . . . . . . . . . . . . . . . . 0  .  …  . . . . . . . .
. . . . . 6 . . . . . . . . . . .−84  .  …  . 9 . . . . . .
. . . . . 12 . . . . . . . . . . . 44  .  …  . 20 . . . . . .
. . . . .−12 . . . . . . . . . . . .  .  …  .−22 . . . . . .
. . 6 . . . . . . . . 6 . . . . . .  .  . . 9  . . . . . . . .
. . 24 . . . . . . . . . . . . . . .  .  . . 38  . . . . . . . .
. . 0 . . . . . . . . . . . . . . .  .  . . 0  . . . . . . . .
. .−48 . . . . . . 6 . . . . . . . .  9  . .−84  …. 9 …
. . 24 . . . . . . 12 . . . . . . . . 56  . . 44  …. 20 …
. . . . . . . . .−12 . . . . . . . . 58  …  ….−22 …
. . . . . . 6 . . . . . . 6 ….−160  …  . . 9 . . . . .
. . . . . . 24 . . . . . . . . . . .−124  …  . . 38 . . . . .
. . . . . . 0 . . . . . . . . . . . 256  …  . . 0 . . . . .
… 6 . .−48 . . . . . 6 . . . . . −88  …  9 .−84 … 9 .
… 36 . . 24 . . . . . 12 . . . . .  .  … 56 . 44 … 20 .
… 36 . . . . . . . .−12 . . . . .  .  … 58 . . . . .−22 .
…−96 . . . . . . 6 …. 6 . .  .  9 . .−160 …. 9 . .
…−72 . . . . . . 24 . . . . . . .  . 74 . .−124 …. 38 . .
… 144 . . . . . . 0 . . . . . . .  . 152 . . 256 …. 0 . .
…−48 … 6 . .−48 … 6 …  .−156 . . −88 . . 9 .−84 . 9
. . . . . . . 36 . . 24 … 12 …  .−560 . .  … 56 . 44 . 20
. . . . . . . 36 . . . . . .−12 …  . 328 . .  … 58 …−22

After deleting all rows that consist of zeros alone, deleting rows \overline{t^{9} D t} and \overline{t^{10} D t} because they depend of row \overline{t^{8} D t}, and some permutations of the columns one obtains the following matrix with the same rank:

( \begin{matrix}B & O_{1} \\ O_{2} & D \end{matrix} ) ,

8 R. W. Berger

where the matrices O_{1}, O_{2} contain zeros only, and:

B :

ω_{1} ω_{17} ω_{5} ω_{2} ω_{21} ω_{9} ω_{18} ω_{6} ω_{24}
\overline{t^{5} D t} 6  . .  .  . .  .  .  .
\overline{t^{8} D t} .  9 .  .  . .  .  .  .
\overline{t^{11} D t} .  . 6  .  . .  .  .  .
\overline{t^{14} D t} .  . . 6 9 .  .  .  .
\overline{t^{15} D t} .  . . 12 20 .  .  .  .
\overline{t^{16} D t} .  . . −12 −22 .  .  .  .
\overline{t^{17} D t} .  . .  .  . 6 9  .  .
\overline{t^{18} D t} .  . .  .  . . 38  .  .
\overline{t^{20} D t} .  . .  .  . . −84 6 9
\overline{t^{21} D t} .  . .  .  . . 44 12 20
\overline{t^{22} D t} .  . .  .  . .  . −12 −22

D :

ω_{12} ω_{3} ω_{22} ω_{10} ω_{27} ω_{19} ω_{14} ω_{7} ω_{25} ω_{13} ω_{29} ω_{4} ω_{23} ω_{16} ω_{11} ω_{28} ω_{20} ω_{8} ω_{15} ω_{26} ω_{30}
\overline{t^{23} D t} 6 6 9  .  .  . .  .  .  .  .  .  . .  .  .  . .  . .  .
\overline{t^{24} D t} . 24 38  .  .  . .  .  .  .  .  .  . .  .  .  . .  . .  .
\overline{t^{26} D t} . −48 −84 6 9  9 .  .  .  .  .  .  . .  .  .  . .  . .  .
\overline{t^{27} D t} . 24 44 12 20 56 .  .  .  .  .  .  . .  .  .  . .  . .  .
\overline{t^{28} D t} .  .  . −12 −22 58 .  .  .  .  .  .  . .  .  .  . .  . .  .
\overline{t^{29} D t} .  .  .  .  . −160 6 6 9  .  .  .  . .  .  .  . .  . .  .
\overline{t^{30} D t} .  .  .  .  . −124 . 24 38  .  .  .  . .  .  .  . .  . .  .
\overline{t^{31} D t} .  .  .  .  . 256 . 0 0  .  .  .  . .  .  .  . .  . .  .
\overline{t^{32} D t} .  .  .  .  . −88 . −48 −84 6 9 6  9 .  .  .  . .  . .  .
\overline{t^{33} D t} .  .  .  .  .  . . 24 44 12 20 36 56 .  .  .  . .  . .  .
\overline{t^{34} D t} .  .  .  .  .  . .  .  . −12 −22 36 58 .  .  .  . .  . .  .
\overline{t^{35} D t} .  .  .  .  .  . .  .  .  .  . −96 −160 6 6 9  9 .  . .  .
\overline{t^{36} D t} .  .  .  .  .  . .  .  .  .  . −72 −124 . 24 38 74 .  . .  .
\overline{t^{37} D t} .  .  .  .  .  . .  .  .  .  . 144 256 . 0 0 152 .  . .  .
\overline{t^{38} D t} .  .  .  .  .  . .  .  .  .  . −48 −88 . −48 −84 −156 6 6 9 9
\overline{t^{39} D t} .  .  .  .  .  . .  .  .  .  .  .  . . 24 44 −560 36 12 56 20
\overline{t^{40} D t} .  .  .  .  .  . .  .  .  .  .  .  . .  .  . 328 36 −12 58 −22

It can easily be seen that the matrix B has rank 9 (rows \overline{t^{16} D t} and \overline{t^{22} D t} can be eliminated, the rest is linearly independent), while the the matrix D consists of only 17 rows, so that its rank is ≤ 17. Therefore the rank of the entire matrix is ≤ 26, giving

ℓ(T) ≥ \frac{3 · 42}{2} − 1 − 26 = 36 > ℓ(T′),

which is enough to prove the counterexample. A more detailed analysis (e.g. by using a computer or computing the rank modulo 5) shows that the rank of the matrix D is equal to 17, so that indeed

ℓ(T) = 36 .

Final Remarks: Heinrich found this counterexample by computing the matrix A

Heinrich’s Counterexample to Azevedo’s Conjecture 9

for a “generic” curve with the characteristic pairs (3, 2) and (10, 3)

x = t^{6} ,
y = t^{9} + a_{10}t^{10} + a_{11}t^{11} + a_{13}t^{13} + a_{14}t^{14} + a_{16}t^{16} + a_{17}t^{17} +

a_{20}t^{20} + a_{22}t^{22} + a_{23}t^{23} + a_{26}t^{26} + a_{29}t^{29} + a_{32}t^{32} + a_{35}t^{35} + a_{41}t^{41} ,

and indeterminate coefficients a_{i} (where one can even omit the two terms of degree 22 and 41 by [1], Chapter 3, Proposition 1) and transforming it to a staircase form with the help of MAPLE. From the result he got conditions on the coefficients to lower the rank of the matrix.

For this general computation, but also for a little bit more complicated special curves, the use of a computer algebra system seems to be indispensable.

References

1. Alberto Azevedo, The Jacobian Ideal of a Plane Algebroid Curve, Ph.D. thesis, Purdue University, 1967.

2. Robert Berger, Differentialmoduln eindimensionaler lokaler Ringe, Math.Z. 81 (1963), 326– 354.

3. Johannes Heinrich, Über eine Vermutung von Azevedo, Diplomarbeit, Universität des Saarlandes, 1994.

4. Thomas Pohl, Differential modules with maximal torsion, Arch. Math 57 (1991), 438–445.

5. Oscar Zariski, Characterization of plane algebroid curves whose module of differentials has maximum torsion, Proc. Nat. Acad. Sci. 56 (1966), 781–786, Erratum p. 1927.

6. , Studies in equisingularity III. Saturation of local rings and equisingularity, Amer. J. Math. 90 (1968), 961–1023.

7. , Le Problème des Modules pour les Branches Planes, Hermann, Paris, 1986, Cours donné au Centre de mathématique de l’École Polytechnique, octobre–novembre 1973. Nouvelle édition revue par l’auteur. Rédigé par François Kmety et Michele Merle. Avec un Appendice de Bernard Teissier.

Fachbereich Mathematik, Universität des Saarlandes, D-66041 Saarbrücken, Germany

e-mail: rberger@math.uni-sb.de