Ae -codimension of germs of analytic curves

manuscripta math. 124, 237–246 (2007)

© Springer-Verlag 2007

M. E. Hernandes · M. E. Rodrigues Hernandes · M. A. S. Ruas

Ae -codimension of germs of analytic curves Received: 21 March 2007 / Revised: 18 May 2007 Published online: 14 July 2007 Abstract. This paper deals with the local study of invariants of analytic curve singularities. For monomial curves we obtain a numerical description of the Ae -codimension of parametrized curves in terms of classical invariants of the theory of curves, like the delta invariant, the Tjurina number and the Cohen-Macaulay type of its local ring.

1. Introduction Recent results in singularity theory relate the Ae -codimension of an analytic mapgerm φ : (Cn , 0) → (C p , 0) and the number of 0-dimensional singularities appearing in a stable perturbation φt of φ (see [5, 10, 11]). The number of isolated stable singularities of a given type of φt is an analytic invariant of the germ φ, named by D. Mond, 0-stable invariant. In [11], he studies the topology of a stable perturbation of an A-finitely determined germ φ : (Cn , S) → (Cn+1 , 0) where S is a finite set of points, and its relation with the Ae -codimension of φ. When n = 1, that is, φ is a germ of analytic curve, there is only one 0-stable invariant, namely, the invariant δ, measuring the number of double points in a stable perturbation of φ. In this case, Mond proves that if φ is 1 − 1 outside S then Ae cod(φ) ≤ δ −r + 1, where r is the number of branches of the curve, and the equality holds if and only if the map-germ φ is quasihomogeneous (see [10]). This geometric approach does not hold for curves φ : (C, S) → (Cn+1 , 0) with n ≥ 2, since in these dimensions φ is stable if and only if it has no singularities. Our purpose is to study parametrized multi-germs of curves φ : (C, S) → (Cn+1 , 0), n ≥ 1 with an algebraic approach, to obtain estimates for the Ae -codimension of φ in terms of classical invariants of the theory of curves. Our main result determines the Ae -codimension of a monomial parametrization φ : (C, 0) → (Cn+1 , 0) in terms of the delta invariant and the Cohen-Macaulay type of the local ring of the curve. As a consequence of this result, we obtain a characterization of Gorensteinness in terms of the Ae -codimension and its relation with the Tjurina number. In particular, for monomial curves in C3 , the property M. E. Hernandes (B) · M. E. Rodrigues Hernandes: Universidade Estadual de Maringá, Av. Colombo 5790, Maringá-PR 87020-020, Brazil. e-mail: [email protected] M. A. S. Ruas: Universidade de São Paulo, ICMC, Caixa Postal 668, São Carlos SP 13560-970, Brazil. Mathematics Subject Classification (2000): 32S10, 14H20; Second 32S05, 14H50

DOI: 10.1007/s00229-007-0116-0

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of complete intersection follows from the parity of the Ae -codimension of its parametrization.

2. Notation We denote by C{X } = C{X 1 , . . . , X n } the ring of absolutely convergent power series in some neighborhood of the origin. Let (C, 0) ⊂ (Cn , 0) be the germ of a reduced curve with isolated singularity at the origin and ρ : (C, 0) → (C, 0) its normalization, where (C, 0) is the multi-germ (C, ρ −1 (0)). Denote by O = O(C ,0) the ring associated to the curve given by C{X }/I , where I is the generating ideal of the curve and O = ρ∗ O(C ,0)  ⊕ri=1 C{ti }, where r is the number of branches of the curve. We denote by ΩC = ΩC1 ,0 the module of (Kähler) holomorphic 1-forms on

n−1 (C, 0), and by ωC ,0 the dualising module of Grothendieck, that is, ωC ,0 = E xtO (O, n n n n ΩCn ,0 ), being ΩCn ,0 the module of holomorphic n-forms on (C , 0) and On the local ring of function germs f : (Cn , 0) → C, with maximal ideal Mn . One can think of ωC ,0 , via the so-called class map cl : ΩC → ωC ,0 as consisting of certain meromorphic differential forms on the curve. The composition of the differential operator d : O → ΩC with the class map gives a map which we also denote by d : O → ωC ,0 . The Milnor number of a reduced curve (C, 0) is defined as µ = dim C ωC ,0 /dO. The main property of this invariant is that it generalizes the Milnor formula for plane curves, µ = 2δ − r + 1, where δ is the delta invariant of (C, 0) given by δ = dimC O/O (see Buchweitz and Greuel [2] for more details). 1 (Ω , O) be the space of first order infinitesimal deformations of Let T 1 = E xtO C (C, 0), whose complex dimension τ is called the Tjurina number. Now we fix some further notation. Let O S (n, p) be the space of map-germs f : (Cn , S) → C p , where S ⊂ Cn is a finite set of points. The set of germs f : (Cn , 0) → C p denoted by O(n, p) is a free On -module of rank p. The submodule of germs f ∈ O(n, p) such that f (0) = 0 is denoted by Mn O(n, p). The set of map-germs f : (Cn , S) → (C p , 0) is isomorphic to a direct sum of p copies of Mn O(n, p). For the Mather groups A and K (of left-right and of contact equivalence, respectively) acting on the space of multi-germs we can define the tangent spaces to the A and K-orbits as follows. Let θ ( f ) = {σ : (Cn , S) → T C p ; germs of vector fields along f }. Similarly we define θCn ,S = θ (idCn ) (respectively, θC p ,0 = θ (idC p )) the space of germs at S (respectively, at 0) of vector fields on Cn (respectively, C p ). Let t f : θCn ,S → θ ( f ) such that t f (s) = d f (s), where d f is the differential of f and w f : θC p ,0 → θ ( f ) such that w f (η) = η ◦ f . The extended tangent space to the orbit of a multi-germ f : (Cn , S) → (C p , 0) by the action of the group A (respectively, K) is defined by T Ae ( f ) = t f (θCn ,S )+w f (θC p ,0 ) (respectively, T Ke ( f ) = t f (θCn ,S )+ f ∗ (M p )θ ( f ), where f ∗ : O p → O S (n, 1) is such that f ∗ (η) = η ◦ f ). For G = A or K, we define the Ge -codimension of f , denoted by Ge cod( f ), as the C-dimension of the quotient θ ( f )/T Ge ( f ). If f −1 (0) is an analytic subvariety of complete intersection, then τ = Ke cod( f ) (see [13]). In this case, the Tjurina number is the minimum number of parameters in a Ke -versal deformation of f .

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Similarly the Ae -codimension of a parametrization φ : (C, S) → (Cn , 0) is the minimum number of parameters in an Ae -versal unfolding of φ (see [9]). A basic reference for these concepts is the survey on determinacy [14] by Wall. 3. Ae -codimension of parametrized curves In this section we present an upper bound for the Ae -codimension of a multi-germ parametrising reduced curves. Let φ : (C, S) → (Cn+1 , 0) with S = r be an A-finitely determined multigerm at S given by φ = [φ 1 , . . . , φ r ] where φ i : (C, 0) → (Cn+1 , 0), φ i = (φ0i , φ1i , . . . , φni ) is a branch of φ at 0 for all i ∈ {1, . . . , r }. Notice that (w0 , w1 , . . . , wn ) ∈ T Ae (φ) if and only if there exist  ∈ θC,S and η j ∈ On+1 , with j = 0, 1, . . . , n such that w j = dφ j () + η j (φ) where φ j : (C, S) → (C, 0) is given by φ j = [φ 1j , . . . , φ rj ]. Consider φ = (φ0 , ψ) where ψ : (C, S) → (Cn , 0) is such that ψ = [ψ 1 , . . . , ψ r ] where ψ i = (φ1i , . . . , φni ), for i = 1, . . . , r . It is easy to see that θ (φ)  θ (φ0 ) ⊕ θ (ψ). We can consider the following subspaces T0 = {w0 ∈ θ (φ0 ); w0 = dφ0 () + η0 (φ), η0 ∈ On+1 ,  ∈ θC,S } T = {(w1 , . . . , wn ) ∈ θ (ψ); (0, w1 , . . . , wn ) ∈ T Ae (φ)}. The codimensions of T0 and T are defined by cod T0 = dimC θ (φ0 )/T0 and cod T = dimC θ (ψ)/T , respectively. If (w0 , w1 , . . . , wn ) ∈ θ (φ), we denote by [(w0 , w1 , . . . , wn )] the class of this element in the quotient space θ (φ)/T Ae (φ), similarly for θ (φ0 ) and θ (ψ). Proposition 1. If φ : (C, S) → (Cn+1 , 0) is an A-finitely determined multi-germ at S, given by φ = (φ0 , ψ). Then (i) Ae cod(φ) = cod T0 + cod T, (ii) Ae cod(φ) ≤ nδ + Ae cod(φ0 ). Moreover, if n > 1 and φ0 is regular or has a singular point of Morse type, then Ae cod(φ) ≤ nδ. Proof. (i) In fact, the result follows observing that the sequence 0 −→

θ (φ) θ (ψ) i ∗ π ∗ θ (φ0 ) −→ −→ −→ 0, T T Ae (φ) T0

is exact, where i ∗ and π ∗ are defined by i ∗ ([(w1 , . . . , wn )]) = [(0, w1 , . . . , wn )] and π ∗ ([(w0 , w1 , . . . , wn )]) = [w0 ]. (ii) The tangent space to the A-orbit of φ0 : (C, S) → (C, 0) is given by T Ae (φ0 ) = {w0 ∈ θ (φ0 ); w0 = dφ0 () + ρ0 (φ0 ), ρ0 ∈ O1 ,  ∈ θC,S } and thus
n cod T0 ≤ Ae cod(φ0 ). Moreover, θ (ψ)  i=1 O, thus θ (ψ)/T is isomor
n phic to a subspace of i=1 O/O and therefore cod T ≤ nδ. Finally, if n > 1 and φ0 is regular or has a singular point of Morse type, then φ0 is stable, hence 

Ae cod(φ0 ) = 0.

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3.1. Analytic irreducible curves Let (C, 0) ⊂ (Cn+1 , 0) be a germ of singular analytic irreducible curve, M the maximal ideal of the local ring O and O  C{t}. We can consider a particular change of coordinates to obtain a parametrization φ of the form ⎧ x0 = t u 0  ⎪ ⎪ ⎪ ⎨ x1 = t u 1 + i>u ai 1 t i 1 φ: (1) .. ⎪ . ⎪ ⎪  ⎩ xn = t u n + i>u n ai n t i , with u 0 < u 1 < · · · < u n . The integer u 0 is the multiplicity of the curve. A parametrization is primitive if it cannot be reparametrized by a power of a new variable. In what follows we consider only primitive parametrizations. For irreducible curve singularity we can obtain another upper bound for the Ae -codimension of φ as follows. Corollary 1. If (C, 0) is a germ of analytic irreducible curve in Cn+1 , with parametrization φ and multiplicity u 0 then Ae cod(φ) ≤ nδ + u 0 − 2. Proof. Let φ given as in (1). In this case codT0 = u 0 − 2. Therefore the result is a consequence of Proposition 1. 

In what follows we show that the Ae -codimension of φ is closely related with the module of Kähler differentials of the curve. Proposition 2. Let φ : (C, 0) → (Cn+1 , 0) be a parametrization of an analytic irreducible curve (C, 0) as in (1). 1. If (0, w1 , .. . , wn ) ∈ T Ae (φ) then there exist η0 ∈ Mn+1 and η j ∈ On+1 such  that w j = (η j ◦ φ)x0 − (η0 ◦ φ)x j /x0 for all j = 1, . . . , n;   2. Let w j = (η j ◦ φ)x0 − (η0 ◦ φ)x j /x0 with η0 ∈ Mn+1 and η j ∈ On+1 , j = 1, . . . , n. Then (0, w1 , . . . , wn ) ∈ T Ae (φ). Proof. (1) If (0, w1 , . . . , wn ) ∈ T Ae (φ) then the system ⎧ x0 (t) · (t) + η0 (φ(t)) = 0 ⎪ ⎪ ⎪ ⎨ x (t) · (t) + η1 (φ(t)) = w1 (t) 1 .. ⎪ . ⎪ ⎪ ⎩ xn (t) · (t) + ηn (φ(t)) = wn (t),

(2)

has a solution, for some  ∈ O and η j ∈ On+1 , j = 0, 1, . . . , n. If η0 = 0 it follows that  = 0, and w j = η j ◦ φ ∈ O for all j = 1, . . . , n. Otherwise  = − η0x◦φ ∈ O and thus η0 ∈ Mn+1 \ {0} and w j = ((η j ◦ φ)x0 − (η0 ◦ φ)x j )/x0 0 for all j = 1, . . . , n. Item (2) follows similarly. 

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4. Monomial curves Monomial curves have been considered by several authors in different situations. Our main purpose in this section is to determine the Ae -codimension of monomial curves in terms of the Cohen-Macaulay type of the local ring and the delta invariant of it. In this case we need to introduce one more invariant: the semigroup of the curve. Let (C, 0) ⊂ (Cn+1 , 0) be a germ of analytic irreducible curve and consider a primitive parametrization φ(t) = (t v0 , φ1 (t), . . . , φn (t)) of (C, 0), we notice that O  C{t v0 , φ1 (t), . . . , φn (t)}. The semigroup of values of (C, 0) is defined by Γ = {ν(h); h ∈ O \ {0}}, where ν(h) = ν(h(t)), with h(t) = h ◦ φ(t) and ν is the valuation of C{t}, that is, the order in t of h(t). The semigroup Γ is finitely generated and if {v0 , v1 , . . . , vg } is the minimal set of generators of Γ, we write Γ = v0 , v1 , . . . , vg . A semigroup Γ generated by natural numbers v0 , . . . , vg is called a numerical semigroup if GC D(v0 , . . . , vg ) = 1 (the semigroup of a primitive parametrization satisfies this condition), this is equivalent to say that q + N ⊂ Γ for some q. The smallest such integer c is called the conductor of Γ . Therefore, the set L = N \ Γ is finite and it is called the set of gaps of the semigroup Γ . Moreover we have that

L = dim C O/O = δ. Let φ(t) = (t v0 , t v1 , . . . , t vn ) be a monomial parametrization of (C, 0) ⊂ Cn+1 , such that the semigroup Γ of the curve is generated by {v0 , v1 , . . . , vn } with v0 < v1 < · · · < vn . We introduce for each i = 1, . . . , n, the following sets: i = {k ∈ L; k = α + vi − v0 , with α ∈ Γ \ {0}} , Λi = {α + vi − v0 ∈ L; α ∈ Γ \ {0} and α + v j − v0 ∈ Γ, ∀ j < i} and ∆i = L \ (i ∪ Λi ). Notice that ∆i is the set of gaps k > vi such that for every representation k = α + vi − v0 with α ∈ Γ \ {0}, we have that α + v j − v0 ∈ L for some j = 1, . . . , i − 1. For each i = 1, . . . , n it follows that ˙ ∆i ∪ ˙ Λi L = i ∪

(3)

where ∆1 is empty. We can determine the Ae -codimension of φ in terms of these sets as we show in the following theorem. Theorem 1. If φ is a monomial parametrization of the curve (C, 0) in Cn+1 with multiplicity v0 then Ae cod(φ) +

n

Λi = nδ + v0 − 2.

i=1 θ(φ) Proof. To determine a monomial basis to the normal space N = T A it is e (φ) k sufficient to find elements of the form (0, 0, . . . , t , . . . , 0) with k ∈ L. If k ∈ Λi ,

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since φ is monomial, there exists tα ∈ M \ {0}  η0 ∈ On+1 , where  η0 (φ(t))  α+v =  −v 0 j ∈ Γ , that is, such that k = ν (η0 ◦ φ)xi /x0 and ν (η0 ◦ φ)x j /x0 = ν t (η0 ◦φ)x j /x0 = η j ◦φ ∈ O for all j < i and some η j ∈ On+1 . Take  = (η0 ◦φ)/x0 and ηi ∈ On+1 , i = 0, 1, . . . , n conveniently (see system (2), Proposition 2) such that (0, . . . , 0, t k , 0, . . . , 0) ∈ T Ae (φ)+Mk+1 1 θ (φ), where the non zero coordinate appears in the (i + 1)-position. If k ∈ ∆i (respectively, k ∈ i ) it is easy to see (respectively, by Proposition 2) that (0, . . . , 0, t k , 0, . . . , 0) ∈ T Ae (φ). Hence by the Proposition 1(i), ⎧⎛ k ⎞ t 0 ⎪ ⎪ ⎪ ⎨⎜ 0 ⎟ ⎜ ⎟ β = ⎜ . ⎟, ⎪ ⎝ .. ⎠ ⎪ ⎪ ⎩ 0

⎛

0

⎞

⎛

⎜ ⎜ t k1 ⎟ ⎜ ⎜ ⎟ ⎜ .. ⎟ , . . . , ⎜ ⎝ ⎝ . ⎠ t 0

⎫ ⎪ ⎪ ⎬ ⎟ k0 = 1, . . . , v0 − 2, ⎪ ⎟ ⎟; ki ∈ i ∪˙ ∆i , ⎪ ⎠ ⎪ ⎪ i = 1, . . . , n ⎭ kn

0 0 .. .

⎞

n n is a basis for N and Ae cod(φ) = v0 − 2 + i=1 i + i=2 ∆i . Adding the number of elements of the n decompositions in (3) the result follows. 

The sets i , ∆i and Λi for i = 1, . . . , n can also be described in terms of the Apéry set of the semigroup Γ of the curve (C, 0). Let Γ ⊆ N be a numerical semigroup and s ∈ Γ with s = 0. The s-standard basis of Γ is the generating set Bs = {a0 , a1 , . . . , as−1 } such that a0 = 0 and for i > 0, ai is the least integer in Γ having s-residue distinct from those of a0 , . . . , ai−1 . The set Bs is called the Apéry set of Γ with respect to s. In this work we consider the Apéry set with respect to the multiplicity v0 of the curve and in this case we denote Bv0 by B. It is a known fact that the set of gaps L of the semigroup Γ is given by L = {a j −lv0 > 0 ; a j ∈ B \ {0}, and l ∈ N \ {0}}.   Example 1. Let φ(t) = t 6 , t 9 , t 17 , t 25 . Its semigroup is Γ = 6, 9, 17, 25 and its conductor is c = 29. We can determine the sets Λi ’s analyzing the gaps of the first column of the Table 1. Notice that 20 = v1 + v2 − v0 and 28 = v1 + v3 − v0 = 2v2 − v0 , in this Λ2 = {20} and Λ3 = ∅. Since δ = 16 way it is easy to verify that Λ1 = {20, 28}, 3

Λi = 48 + 4 − 3 = 49 and the we have that Ae cod(φ) = 3δ + v0 − 2 − i=1 Table 1. Apéry set and the set of gaps of Γ = 6, 9, 17, 25 B 0 v1 = 9 v2 = 17 v3 = 25 v1 + v2 = 26 v1 + v3 = 2v2 = 34

L 3 11 19 20 28

5 13 14 22

7 8 16

1 2 10

4

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following set is a basis for the normal space N : ⎧⎛ k ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎫ 0 0 0 t 0 ⎪ ⎪ ⎪ ⎨⎜ ⎟ ⎜ k1 ⎟ ⎜ ⎟ ⎜ ⎟ k0 = 1, . . . , 4, ⎪ ⎬ 0 ⎟ ⎜t ⎟ ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ . β = ⎝ ⎠ , ⎝ ⎠ , ⎝ k2 ⎠ , ⎝ ⎠ ; ki ∈ L \ Λi , 0 0 t 0 ⎪ ⎪ ⎪ ⎪ i = 1, 2, 3 ⎩ ⎭ 0 t k3 0 0 Remark that the representation of a gap in terms of the Apéry set is important to determine the sets Λi ’s and consequently the Ae cod(φ). Now we shall show how the Cohen-Macaulay type of the local ring of a monomial curveis related with its Ae -codimension. More precisely we express the n

Λi in terms of the multiplicity of the curve and the CohenA-invariant i=1 Macaulay type of its local ring. Notice that every element in Λi can be expressed by a j − v0 with a j ∈ B \ {0}, thus we can omit −v0 in the description of the elements in this set, because this does not change the cardinality of Λi . We use the same notation to denote this set. Thus, Λi = {α + vi ∈ B; α ∈ Γ \ {0} and α + v j ∈ B, ∀ j < i}. For each i = 1, . . . , n, let Λi = {a ∈ B \ {0}; a + v j ∈ B, ∀ j < i and a + vi ∈ B}. It is easy to check that the sets Λi are pairwise disjoint and there exists a bijection between Λi and Λi . In fact, it is sufficient to consider ϕi : Λi → Λi defined by ϕi (x) = x − vi . We also consider the following subset of B,  B = {b ∈ B \ {0}; b + ai ∈ B, ∀ ai ∈ B \ {0}}. This is a non-empty set because the biggest element of B, that is av0 −1 , belongs to it. The number of elements of  B is denoted by t B and it is an A-invariant that admits an interesting interpretation in the case of monomial curves. Let (A, M, k) be a d-dimensional Cohen-Macaulay local ring with maximal ideal M and k a field. The Cohen-Macaulay type of A is the number r (A) = dimk E xt Ad (k, A). If r (A) = 1 then A is a Gorenstein ring. Cavaliere and Niesi in ([4], Proposition 2.7) proved that if the semigroup Γ =

v0 , v1 , . . . , vn  is the semigroup of a curve whose local ring is O  C{t v0 , t v1 , . . . , t vn } then the Cohen-Macaulay type of O is given by r (O) = t B . Thus we have the following result. Theorem 2. Let φ be a monomial parametrization of the curve (C, 0) in Cn+1 with B then tB =  Ae cod(φ) = nδ + t B − 1. n Proof. By Theorem n 1, itis sufficient to show that i=1 Λi = v0 − t B − 1.  B = ∅. Now let α ∈ B \ {0} and suppose that α ∈ Λi Remark that i=1 Λi for all i = 1, . . . , n. Given ak ∈ B \ {0}, there exists al ∈ B such that ak = vi + al for some vi . In this way α + ak = (α + vi ) + al ∈ B, otherwise α + vi ∈ B α∈  B. Since B = v0 , then and this is a contradiction, because α ∈ Λi . Hence, n n  

B − 1 = i=1 Λi + B and we conclude that i=1 Λi = v0 − t B − 1.

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It is a well known fact that an analytic irreducible curve is Gorenstein if its local ring O is Gorenstein or equivalently if its semigroup is symmetric and r (O) = 1 if and only if O is Gorenstein. Thus we have the following corollary: Corollary 2. Let Γ be a semigroup of a monomial curve in Cn+1 parametrized by φ, then Γ is symmetric if and only if Ae cod(φ) = nδ. For monomial curves in C3 , Cavaliere and Niesi determined the CohenMacaulay type t B of the local ring O. Proposition 3 ([3]). Let Γ = v0 , v1 , v2  be a semigroup of a monomial curve, then  1 if Γ is symmetric tB = 2 if Γ is not symmetric. In this way, the following result is immediate. Corollary 3. Let φ be a monomial parametrization of the curve (C, 0) in C3 . Then 1. (C, 0) is Gorenstein if and only if Ae cod(φ) = 2δ. 2. (C, 0) is non-Gorenstein if and only if Ae cod(φ) = 2δ + 1. A germ of an analytic irreducible curve (C, 0) in C3 is Gorenstein if and only if it is a complete intersection (see [12]), then the above corollary shows that the parity of the Ae -codimension of the parametrization of a curve in C3 determines whether or not the curve is a complete intersection. Using the Corollaries 2 and 3 we can add, to the classification of simple singularities of germs of irreducible curves, obtained by Gibson and Hobbs [6], the information of which curves in the list are complete intersection. An interesting question in this context is: what is the relation between the invariants τ and Ae cod(φ)? Greuel ([7], Corollary 2.5) presented the following result: Proposition 4. Let (C, 0) be a germ of reduced smoothable curve in Cn with isolated singularity at the origin and r (O) the Cohen-Macaulay type. 1. If (C, 0) is quasihomogeneous then τ ≥ µ + r (O) − 1. 2. The equality holds if and only if (C, 0) is quasihomogeneous and unobstructed. A quasihomogeneous irreducible curve admits a monomial parametrization, then the next proposition follows as a consequence of Theorem 2 and Greuel’s result. Proposition 5. Let (C, 0) be a germ of analytic irreducible smoothable curve in Cn . If (C, 0) is quasihomogeneous and unobstructed then ⎧ ⎨ τ > Ae cod(φ), if n = 2 τ = Ae cod(φ), if n = 3 ⎩ τ < Ae cod(φ), if n > 3, where φ is a monomial parametrization of (C, 0).

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For reduced plane curves, not necessarily quasihomogeneous, we know that Ae cod(φ) = τ − δ (see [8], Proposition 2.30(3)), thus in this case, the inequality τ > Ae cod(φ) is always true. It is also known that if (C, 0) is a quasihomogeneous curve in C3 then it is smoothable and unobstructed (see [1], for example) and therefore in this case τ = Ae cod(φ). Moreover, complete intersection curves are smoothable and unobstructed and the previous result can be applied to this case as well. 4.1. Ae -codimension of deformations of monomial curves Let φ be a monomial parametrization of an irreducible curve in Cn+1 , given by φ(t) = (t v0 , t v1 , . . . , t vn ) with v0 < v1 < · · · < vn and semigroup Γ =

v0 , v1 , . . . , vn . Considering Φ : C × Cl , 0 → Cn+1 , 0 with Φ(t, u) = (Φ0 (t, u), . . . , Φn (t, u)), Φ(t, 0) = φ(t), a deformation of φ, such that or dt (Φi (t, u) − t vi ) > vi , for i = 0, 1, . . . , n. For each u ∈ Cl , set Φu (t) = Φ(t, u), and Γu the corresponding semigroup. An important property of the Ae -codimension is that it is upper-semicontinuous and thus Ae cod(Φu ) ≤ Ae cod(φ), for all sufficiently small values of the parameter u ∈ Cl . In this way the following result is immediate. Proposition 6. With the above notation, Ae cod(Φu ) ≤ nδ + t B − 1, where δ and t B are, respectively, the delta invariant and the Cohen-Macaulay type of the local ring of φ. The following example illustrates the above result: Example 2. (Curves with maximal embedding dimension) A curve (C, 0) in Cn+1 is called (a curve) with maximal embedding dimension, if it admits a parametrization ⎧ x0 = t n+1  ⎪ ⎪ ⎪ ⎨ x1 = t v1 + i>v ai 1 t i 1 φ: .. ⎪ . ⎪ ⎪  ⎩ xn = t vn + i>vn ai n t i , such that its semigroup of values is Γ = n + 1, v1 , . . . , vn . We can check by an easy direct calculation that such curves satisfy the following properties: 1. if n ≥ 2 then (C, 0) is non-Gorenstein; B = B \ {0} = n; 2. the Apéry set is B = {0, v1 , . . . , vn }. Thus, t B =  Notice that the semigroup of φ and φ0 (t) = (t n+1 , t v1 , . . . , t vn ) are equal and Ae cod(φ) ≤ nδ + t B − 1 = n(δ + 1) − 1. Acknowledgements. The authors gratefully acknowledges financial support from CAPES/ PROCAD and CNPq. The third author was also partially supported by FAPESP.

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