THE DEGREE OF THE SECANT VARIETY AND THE JOIN OF MONOMIAL CURVES

K. RANESTAD^{∗}

Abstract. A monomial curve is a curve parametrized by monomials. The degree of the secant variety of a monomial curve is given in terms of the sequence of exponents of the monomials defining the curve. Likewise, the degree of the join of two monomial curves is given in terms of the two sequences of exponents.

Contents

1. Introduction 1 2. The multiplicity sequence of a plane curve singularity 2 2.1. Algorithm computing the multiplicity sequence 3 3. The degree of the secant variety of a monomial curve 4 4. The degree of the join of two monomial curves 6 4.1. Intersection multiplicity algorithm 8 References 9

1. Introduction

A monomial curve C is the image of an injective morphism of f : ℙ^{1} → ℙ^{r} defined by monomials. After ordering the monomials by ascending degree it is therefore given by

(s : t) ↦ (s^{d} : s^{d−a_{1}}t^{a_{1}} : …: s^{d−a_{r−1}}t^{a_{r−1}} : t^{d})

where a_{1} < a_{2} < … < a_{r} = d. So this latter sequence completely determines C. We define the first secant variety SecC to be the closure of the union of lines that meet C in two distinct points. The first aim of this note is to compute the degree of this secant variety as a subvariety of ℙ^{r}. A simple wellknown argument using a general projection π : C → \overline{C} ⊂ ℙ^{2} shows that this degree is given by the formula

degSecC = \binom{d − 1}{2} − δ_{p} − δ_{q}

where δ_{p} and δ_{q} are the genus contributions or equivalently, 2δ_{p} and 2δ_{q} are the Milnor numbers of the cusps at p = π([1 : 0 …: 0]) and q = π([0 : …: 0 : 1]) on \overline{C}. To compute δ_{p} and δ_{q} given C, we find the characteristic terms of the Puiseux expansion of \overline{C} at the cusps. From the characteristic terms of the Puiseux expansion the genus contribution is

Date: May 27, 2004.

1991 Mathematics Subject Classification. Primary 14; Secondary 14.

∗ Supported by MSRI.

2 K. RANESTAD

computed by an algorithm due to Chisini and Enriques, eventually refined and given a closed form by Casas-Alvero.

Given two curves C and D in ℙ^{r} we define their join Join(C, D) to be the closure of the union of lines that meet C and D in two distinct points. We consider the join of two monomial curves C and D: In the notation of the previous section we ask that the two curves are defined by

C : (s : t) ↦ (s^{d_{C}} : s^{d_{C}−a_{1}}t^{a_{1}} : …: s^{d_{C}−a_{r−1}}t^{a_{r−1}} : t^{d_{C}})

where a_{1} < a_{2} < … < a_{r} = d_{C}, and

D : (s : t) ↦ (s_{D}^{d} : s^{d_{D}−b_{1}}t^{b_{1}} : …: s^{d_{D}−b_{r−1}}t^{b_{r−1}} : t^{d_{D}})

where b_{1} < b_{2} < … < b_{r} = d_{B}. Again the two sequences

a_{1} < a_{2} < … < a_{r} = d_{C}, b_{1} < b_{2} < … < b_{r} = d_{B}

determine the two curves completely, and our second goal is to compute the degree of the join of C and D as a subvariety of ℙ^{r}. In this case the general projection of the two curves to ℙ^{2} gives the formula

degJoin(C, D) = d_{C} · d_{D} − I_{s}(C, D),

where I_{s}(C, D) is the sum of the intersection multiplicities in ℙ^{2} of the two curves at the images of intersection points between the two curves in ℙ^{r}. An algorithm computing the sum of intersection multiplicity I_{s}(C, D) is given in section 4.

A closed formula for the degree of the join in terms of the sequences a_{1} < a_{2} < … < a_{r} = d_{C} and b_{1} < b_{2} < … < b_{r} = d_{B}, like the formula for the degree of the secant variety, would have been prefered. So far we can only give an explicit algorithm for its computation.

The author thanks MSRI for excellent working conditions, Bernd Sturmfels for posing the problem and Eduardo Casas-Alvero for giving the solution a nicer form.

2. The multiplicity sequence of a plane curve singularity

A crucial ingredient in the two algorithms below is the multiplicity sequence of a plane curve singularity. Given a point p in the plane and a sequence of blowups at simple points (p = p_{0}, p_{1}, p_{2}, ..., p_{s}), such that all exceptional divisors lie over p, i.e. is mapped to p by the natural map to the original plane, and such that the strict transform of the curve is smooth. The multiplicities m_{0}(C), (resp. m_{i}(C), i > 0) of C at p (respectively its strict transforms at p_{i}), form the multiplicity sequence of C at p with respect to the sequence of blowups. Equivalently, the multiplicity sequence coincides with the sequence of intersection numbers of the strict transform of the C with the exceptional divisor of each blow up. The multiplicity sequence may contain 1’s, but these would not appear in a blowup that provides a minimal resolution of the singularity. In the latter case we say that the multiplicity sequence is minimal. Note that by the unicity of a minimal resolution of a plane curve singularity, the minimal multiplicity sequence is unique. Both minimal and nonminimal cases will however occur in our setting.

THE DEGREE OF THE SECANT VARIETY AND THE JOIN OF MONOMIAL CURVES 3

We shall use the multiplicity sequence of plane singularities with given Puiseux series. Consider the parameterized affine plane curve

C : t ↦ t^{m}, t^{k_{1}} + t^{k_{2}} + · · · .

This plane curve has a cusp at the origin, where t = 0. The multiplicity sequence is computed from the sequence m, k_{1}, k_{2}, … as described in a result of Enriques and Chisini [1] Theorem 8.4.12. We give this algorithm. The genus contribution may be computed from the multiplicity sequence. Casas-Alvero have found a closed form for this genus contribution which we present below.

2.1. Algorithm computing the multiplicity sequence. Consider the strictly increasing sequence

m < k_{1} < k_{2} < …

Step 1. The gcd-sequence and characteristic terms. (This step is not neccessary to compute the multiplicity sequence, but clearifies the role of the different terms k_{i}.) Let g_{0} = m and g_{i} = gcd{m, k_{1}, …, k_{i}} for i > 0. The g_{i} form the gcd-sequence of m, k_{1}, …, k_{r}:

g_{0} ≥ g_{1} ≥ g_{2} ≥ g_{3} ….

Clearly, in the gcd-sequence, g_{i} = 1 for some i, since otherwise the parametrization is not 1 : 1. The characteristic terms in the sequence t^{k_{1}} + t^{k_{2}} + ⋯ are the terms

k_{i_{1}}, ..., k_{i_{s}}

where i_{1} = min{i|g_{i} < m}, i_{2} = min{i|g_{i} < g_{i_{1}}} etc. Thus m = g_{0} > g_{i_{1}} and

g_{i_{1}} > ... > g_{i_{s}} = 1.

In particular the number of characteristic terms is finite and bounded by the number of prime factors in m.

Step 2. The multiplicity sequence. Given the sequence

m < k_{1} < k_{2} < …

let κ_{i} = k_{i} − k_{i−1} where k_{0} = 0 and i = 1, 2, .... We call

κ_{1}, κ_{2}, …

the difference sequence of the cusp. We will only need a finite number of terms in this sequence. In fact the difference sequence of the finite sequence of characteristic terms form will do. So we assume we have a difference sequence with s terms.

Apply the Euclidean algorithm successively to the elements of the difference sequence: Let

κ_{i} = e_{i,1}r_{i,1} + r_{i,2}
r_{i,1} = e_{i,2}r_{i,2} + r_{i,3}
...
r_{i,w(i)−1} = e_{i,w(i)}r_{i,w(i)}

with 0 ≤ r_{i,j+1} < r_{i,j} and r_{1,1} = m, r_{i,1} = r_{i−1,w(i−1)}, i > 1. Note that

r_{i+1,1} = g_{i} = gcd(m, k_{1}, ..., k_{i}).

4 K. RANESTAD

The multiplicity sequence is e_{i,j} times the multiplicity r_{i,j}, with 1 ≤ i ≤ s and 1 ≤ j ≤ w(i).

As a common convention we write the sequence in the order it is computed, and with repetitions in stead of the numbers e_{i,j}. To separate the output between each subroutine i above, we sometimes use a semicolon. Note that the overall sequence anyway is nonincreasing. The genus contribution or δ-invariant of the sequence is given by

δ = ∑_{i,j} e_{i,j}\binom{r_{i,j}}{2}.

This sum is given a closed form in terms of the original sequence and its gcd-sequence by the following result due to Casas-Alvero:

Proposition 2.1. [2] Given the sequence

m < k_{1} < k_{2} < …

Let

g_{0} ≥ g_{1} ≥ g_{2} ≥ g_{3} …

be its gcd-sequence. Then the δ-invariant of the sequence is

δ = \frac{1}{2}(∑_{i≥1} k_{i}(g_{i−1} − g_{i}) − m + 1.

Proof. See [2] page 194 exercise 5.6. □

3. The degree of the secant variety of a monomial curve

Let C ⊂ ℙ^{r} be a monomial curve defined by the sequence of positive integers a_{1} < a_{2} < … < a_{r} = d as above. Consider the secant variety SecC of C. This is a threefold, so its degree is counted by the intersection of this variety with a general codimension three subspace, or equivalently by the number of ordinary double points of the general projection π : C → ℙ^{2}. For a general projection the only other singularities on \overline{C} = π(C) are possible cusps at the image of the points π(p) and π(q) where p = (1 : …: 0) and q = (0 : …: 1) in ℙ^{r}. The formula for the arithmetic genus of a plane curve of degree d and the computation of the genus contribution at these cups provides a formula for the degree of SecC.

Proposition 3.1. Let C ⊂ ℙ^{r} be a monomial curve defined by the sequence of positive integers a_{1} < a_{2} < … < a_{r} = d. Let b_{i} = d − a_{r−i}, for i = 1, ..., r − 1 and b_{r} = d. Let g_{i} = gcd(a_{1}, ..., a_{i}) and h_{i} = gcd(b_{1}, ..., b_{i}), then

degSecC = \binom{d − 1}{2} − \frac{1}{2}(∑_{i} a_{i+1}(g_{i} − g_{i+1}) − a_{1} + ∑_{i} b_{i+1}(h_{i} − h_{i+1}) − b_{1}) − 1.

Proof. The arithmetic genus p(C) for a curve C on a smooth surface S is given by the adjunction formula [3] on the surface:

2p(C) − 2 = C · C + C · K_{S}

where K_{S} is the canonical divisor on S. If C has multiplicity m at a point q on S, and S′ → S is the blowup of S at q, then the adjunction formula on S′ says

2p(C′) − 2 = C′ · C′ + C′ · K_{S′} =
= (C^{∗} − mE) · (C^{∗} − mE) + (C^{∗} − mE) · (K_{S} + E) = 2p(C) − 2 − m^{2} + m

where E is the exceptional divisor and C^{∗} is the total transform and C′ is the strict transform of C(cf. [3] chapter V). Thus

p(C′) = p(C) − \binom{m}{2}.

After resolving all singularities on \overline{C} ⊂ ℙ^{2} by a series of blow ups centered at singular points of \overline{C} or its strict transform, the arithmetic genus of the strict transform C′ is 0 since it is a rational curve. At the ordinary double points the difference between the arithmetic genus of the curve and its strict transform after blowing up the point is \binom{2}{2} = 1. The points π(p) and π(q) are the only other singularities on \overline{C}. The contribution δ_{p} is by definition the difference between the arithmetic genus of \overline{C} and a smooth strict transform C′ that is isomorphic to \overline{C} outside the point p. Likewise for δ_{q}. Since K_{ℙ^{2}} ≅ −3L, where L is a line in the plane, the arithmetic genus of \overline{C} is given by 2p(\overline{C}) − 2 = d_{C}(d_{C} − 3). Adding all genus contributions we get the formula:

degSecC = \binom{d − 1}{2} − δ_{p} − δ_{q}.

It remains therefore to give an explicit computation of the genus contributions at p and q. As explained above the genus contribution of a plane curve singularity is determined by the multiplicity sequence of the singularity. The first term in this sequence is the multiplicity of the curve in the singular point, the next term is the multiplicity of the strict transform at the singular point on the exceptional divisor (if the strict transform is not already smooth) etc. The algorithms 2.1 computes this multiplicity sequence from the exponents of the Puiseux expansion, so the genus contribution is nothing but the δ invariant of the sequence of exponents in the Puiseux expansion.

The parametrization of the cusp at p is given by

x = t^{a_{1}} + b_{13}t^{a_{3}} + ... + b_{1r}t^{a_{r}}, y = t^{a_{2}} + b_{23}t^{a_{3}} + ... + b_{2r}t^{a_{r}}.

To compute the multiplicity sequence of the Puiseux expansion we need only the characteristic terms in a Puiseux expansion of our curve at π(p).

Lemma 3.2. The characteristic terms in the Puiseux expansion of \overline{C} at p coincides with the characteristic terms in the Puiseux expansion

x = t^{a_{1}}, y = t^{a_{2}} + b_{23}t^{a_{3}} + ... + b_{2r}t^{a_{r}}.

Proof. To start with, the exponents a_{i} are coprime. The ideal in k[t] generated by x and y therefore has finite codimension as a vectorspace, and there is an N_{0} such that t^{N} is in the ideal when N ≥ N_{0}. So it is enough to prove the statement of the lemma modulo

6 K. RANESTAD

t^{N_{0}}. We therefore reparameterize \overline{C} by substituting t with t + ut^{a_{3}−a_{1}+1} for suitable u to cancel the coefficient of t^{a_{3}}. In the new parametrization we get:

x = t^{a_{1}} + b′_{14}t^{a′_{4}} + ... + b′_{1r}t^{a′_{r′}}, y = t^{a_{2}} + b_{23}t^{a_{3}} + ... + b_{2r}t^{a_{r}} + c_{1}t^{b_{1}} + ....

where a′_{4} > a_{3}, and all new exponents appearing are of the form a_{i} + k(a_{3} − a_{1}) for some positive integer k. Compare the greatest common divisors g_{i} of a_{1} and the i lowest exponents of t occuring in y, before and after the reparametrization. The only difference is a possible repetition of some terms, so the characteristic terms remain the same. Now, we may reparameterize until x has only one term with exponent less than N and we are done. □

Since non-characteristic terms do not contribute to the δ-invariant the proposition follows from 2.1. □

Example 3.3. Consider the monomial curve C given by the sequence (0, 30, 45, 55, 78). At p = (1 : 0), we may compute the δ-invariant from the Puiseux expansion with exponents m = 30, (a_{3}, a_{4}, a_{5}) = (45, 55, 78) The gcd-sequence is (30, 15, 5, 1) and the δinvariant is

δ_{p} = \frac{1}{2}(45(30 − 15) + 55(15 − 5) + 78(5 − 1) − 30 + 1) = 754.

At q = (0 : 1) we compute the δ-invariant from the Puiseux expansion with exponents m = 23 and (a_{3}, a_{4}, a_{5}) = (33, 48, 78). Since m is prime and coprime to 33, the only characteristic term is 33 with gcd-sequence (23, 1). The δ-invariant is

δ_{q} = \frac{1}{2}(33(23 − 1) − 23 + 1) = 352

The degree of the secant variety of C is

degSecC = \binom{77}{2} − δ_{p} − δ_{q} = 2926 − 754 − 352 = 1820

4. The degree of the join of two monomial curves

Consider the join of two monomial curves C and D in ℙ^{r} defined by

C : (s : t) ↦ (s^{d_{C}} : s^{d_{C}−a_{1}}t^{a_{1}} : …: s^{d_{C}−a_{r−1}}t^{a_{r−1}} : t^{d_{C}})

where a_{1} < a_{2} < … < a_{r} = d_{C}, and

D : (s : t) ↦ (s_{D}^{d} : s^{d_{D}−b_{1}}t^{b_{1}} : …: s^{d_{D}−b_{r−1}}t^{b_{r−1}} : t^{d_{D}})

where b_{1} < b_{2} < … < b_{r} = d_{B}. The two sequences

a_{1} < a_{2} < … < a_{r} = d_{C}, b_{1} < b_{2} < … < b_{r} = d_{B}

therefore determine the two curves completely. For the parametrizations to be 1 − 1 onto the image, we ask that the a_{i} have no common factor, and likewise for the b_{i}. The join is a threefold, so its degree coincides with the number of lines meeting the two curves in distinct points that also meet a given codimension 3 linear space L in ℙ^{r}. But this number equals the number of new intersection points obtained by projecting the union of the two

THE DEGREE OF THE SECANT VARIETY AND THE JOIN OF MONOMIAL CURVES 7

curves from L to a plane. Denote by π_{L} the projection from L, and let \overline{C} = π_{L}(C) and \overline{D} = π_{L}(D) be the images of C and D respectively. The total intersection number

\overline{C} · \overline{D} = d_{C} · d_{D}

by Bezout’s theorem, so to get the degree we have to subtract the intersection multiplicity at the points of π_{L}(C ∩ D). In our special situation there certainly are points in C ∩ D:

{p = (1 : 0 …: 0), q = (0 : ⋯: 0 : 1), u = (1 : …: 1)} ⊂ C ∩ D.

Furthermore, if

β = gcd(b_{1} − a_{1}, b_{2} − a_{2}, ..., b_{r} − a_{r}),

then all roots of t^{β} −1 define intersection points. This is, however, all as is easily checked. Namely, we may assume that the first coordinate is 1 at an intersection point and that t_{1}^{a_{i}} = t_{2}^{b_{i}} for i = 1, .., r, for some t_{1}, t_{2} in the ground field. Then we may assume (over ℂ) that there is an α such that t_{1}^{α} = t_{2}. Then we get t_{1}^{a_{i}} = t_{1}^{α·b_{i}}, i.e. t_{1} = 1 or a_{i} = α · b_{i} for i = 1, ..., r, or there is some integer β such that β · a_{i} = α · b_{i}. Therefore α is a rational number. Furthermore the b_{i} have no common factor, so α must be an integer. Since the a_{i} also have no common factor, we may assume that α = 1, and we have precisely the intersection points described above.

The intersections at the roots of t^{β}−1 are always transversal, i.e. with distinct tangents: The tangent direction is given by the derivatives (a_{1}t^{a_{1}−1}, ..., a_{r}t^{a_{r}−1}) and (b_{1}t^{b_{1}−1}, ..., b_{r}t^{b_{r}−1}). Here the powers of t coincide, but the coefficients are not proportional, so the tangent directions are distinct. Therefore the intersection multiplicity is 1 at the image of these points by π_{L}.

For the points π_{L}(p) and π_{L}(q) the intersection multiplicity is at least two, since the two curves have the same tangent(cone) at those points. In fact, since the curves are unibranched, there is a unique tangent direction at the point, i.e. if they are singular they have a cusp there. The intersection multiplicity at these points is determined by a procedure similar to the one given in the previous section. More precisely consider say the point π_{L}(p). Blow it up and let p_{1} be the common intersection point of the strict transforms of the two curves on the exceptional divisor. There is a unique such intersection point, since the two curves are unibranched and the tangents to \overline{C} and \overline{D} at π_{L}(p) coincide. Now blow up in the point p_{1}. If the strict transforms meet on the new exceptional divisor, then denote it by p_{2} and blow up in this point. Continue, until the strict transforms do not intersect on the exceptional divisor. Thus we get a finite sequence p_{0} = π_{L}(p), p_{1}, …, p_{k}, and together with it the multiplicities of the strict transforms of the two curves at each p_{i}. We denote these multiplicity sequences by m_{0}(C), …, m_{k}(C) and m_{0}(D), …, m_{k}(D). The intersection multiplicity between the two curves at the point π_{L}(p) is:

∑_{i=0}^{k} m_{i}(C)m_{i}(D).

The multiplicity sequences m_{0}(C), …, m_{k}(C) and m_{0}(D), …, m_{k}(D) are decreasing and similar to the multiplicity sequences constructed in the previous section. There are however a main difference in that the new ones may contain 1’s. Because of the unibranch property these 1’s could only be added to the end of the sequence though. Thus the new

8 K. RANESTAD

sequences coincides with part of the old one, extended possibly with 1’s only in case it contains all of the old one.

The problem is how to compute these sequences from sequences a_{1}, ..., a_{r} and b_{1}, ..., b_{r} of the curves C and D. In this case the non-characteristic terms are as important as the characteristic ones, since the intersection point of the strict transforms with the exceptional divisor is crucial. Some special cases may illustrate the issue:

Example 4.1. Consider monomial curves C : (1, 2, 3, 4) and D : (1, 2, 3, 5). Then the two curves separate after four blowups and the multiplicity sequences are m_{i}(C) : 1, 1, 1, 1 and m_{i}(D) : 1, 1, 1, 1. The intersection multiplicity at π_{L}(p) is 1 + 1 + 1 + 1 = 4.

Example 4.2. The monomial curves C : (1, 2, 3, 4) and D : (2, 4, 6, 9) separate after 4 blowups starting at p (t = 0), the multiplicity sequences are m_{i}(C) : 1, 1, 1, 1 and m_{i}(D) : 2, 2, 2, 2. The intersection multiplicity at π_{L}(p) is 2 + 2 + 2 + 2 = 8. Similarly if b_{i} = ea_{i}, i = 1, …, s, then m_{i}(D) = em_{i}(C) for i = 1, ..., k where k is the length of the multiplicity sequence for a curve D′ : (b_{1}, …, b_{s}, b_{s+1}).

Example 4.3. For C : (1, 2, 3, 4) and D : (b_{0}, b_{1}, b_{2}, b_{3}) where b_{0} > 1 and b_{1} ≠ 2b_{0}, then k = 2 and m_{i}(C) = (1, 1) and m_{i}(D) = (b_{0}, min{(b_{1} − b_{0}), b_{0}}).

With these examples in mind we formulate the algorithm computing the degree of the join.

4.1. Intersection multiplicity algorithm. Given two monomial curves C and D defined by the sequences

a_{1} < a_{2} < … < a_{r} = d_{C}, b_{1} < b_{2} < … < b_{r} = d_{D}

respectively, and assume that for some j ≥ 1 b_{i} ≥ a_{i} for i < j while b_{j} > a_{j}. The following three steps computes the intersection multiplicity of the general projection of the two curves to a plane in the image of the origin.

Step 1. Let α = \frac{b_{1}}{a_{1}} . If α is not an integer, then set k = 0, otherwise let

k = max{i|b_{i} = αa_{i}}

Let m_{1}, m_{2}, …, m_{s} be the multiplicity sequence, the outcome of the algorithm 2.1, of the sequence (b_{1}, b_{2}, …, b_{k}).

Set

δ_{k} = \frac{1}{α}(m_{1}^{2} + ⋯ + m_{s}^{2}),

and let g = gcd(b_{1}, ..., b_{k}).

Step 2. Apply the multiplicity algorithm 2.1 to the sequences (\frac{g}{α}, a_{k+1}−a_{k}) and (g, b_{k+1}− b_{k}), with outcome

(e_{1}, r_{1}), (e_{2}, r_{2}), ..., (e_{m}, r_{m}) and (e′_{1}, r′_{1}), ..., (e′_{n}, r′_{n})

respectively. Let l = min{j|e_{j} = e′_{j}} and f = min{e_{l+1}, e′_{l+1}} and let

ϵ = ∑_{j}^{l} e_{j} · r_{j}r′_{j} + f · r_{l+1}r′_{l+1}.

Step 3. The intersection multiplicity between the curves C and D at the origin is

I(C, D) = δ_{k} + ϵ.

Proof. To start we project C and D into the plane and may choose coordinates in the plane such that π(C) and π(D) have the parametrizations

π(C) : x = t^{a_{1}} + c_{1,3}t^{a_{3}} + ... + c_{1,r}t^{a_{r}}, y = t^{a_{2}} + c_{2,3}t^{a_{3}} + ... + c_{2,r}t^{a_{r}}

and

π(D) : x = t^{b_{1}} + c_{1,3}t^{b_{3}} + ... + c_{1,r}t^{b_{r}}, y = t^{b_{2}} + c_{2,3}t^{b_{3}} + ... + c_{2,r}t^{b_{r}}.

By assumption a_{1} < a_{2} and b_{1} < b_{2}, so both curves are tangent along the x-axis. Now, we blow up the plane in the origin. The strict transforms of these curves on the blowup intersect the exceptional curve in the x-chart (with coordinates (x, xy)). In this chart the strict transforms π(C)′ and π(D)′ have local parameterizations:

π(C)′ : x = t^{a_{1}}+c_{1,3}t^{a_{3}}+...+c_{1,r}t^{a_{r}}, y = t^{a_{2}−a_{1}}+c_{2,3}t^{a_{3}−a_{1}}+...+c_{2,r}t^{a_{r}−a_{1}}−c_{1,3}t^{a_{2}+a_{3}−2a_{1}}+…

and

π(D)′ : x = t^{b_{1}}+c_{1,3}t^{b_{3}}+...+c_{1,r}t^{b_{r}}, y = t^{b_{2}−b_{1}}+c_{2,3}t^{b_{3}−b_{1}}+...+c_{2,r}t^{b_{r}−b_{1}}−c_{1,3}t^{b_{2}+b_{3}−2b_{1}}+….

The tangent at the origin is y = 0 if a_{1} < a_{2} − a_{1}, it is x = 0 if a_{1} − a_{2} < a_{1} and it is x = y if a_{1} = a_{2} − a_{1}.

Notice, that the terms of order less than a_{k} − a_{1} and b_{k} − b_{1} respectively, have the same coefficients and differ only by the factor t^{α_{p}}. Therefore, if k > 0 the two curves π(C)′ and π(D)′ have the same tangent direction at the origin, and their strict transform on the blow up in the origin intersect. Proceeding we need to know after how many blowups, the strict transforms does not intersect, and keep track of the multiplicities of the two strict transforms up to that point. Computing the number of blowups needed to separate the two curves, comes down to keeping track of first terms of the parametrizations of the strict transforms after successive blowups. The tangent direction decides the parametrization of the strict transform: If the tangent direction is y = 0 then the strict transform is parametrized by x, \frac{y}{x}, if the tangent direction is x = 0, then the strict transform is parametrized by \frac{x}{y}, y, and if the tangent direction is x = y, then the strict transform is parametrized by x, \frac{y−x}{x}. Now, the multiplicities of the strict transforms at the origin form the multiplicity sequence obtained by the algorithm 2.1, but keeping track of which of the tangent directions at each point, we actually also control the intersection between the two curves. The change from y = 0 to x = 0 of tangent direction correspond to going from (i, j) to (i, j + 1) in the Euclidean algorithm, while the third kind of tangent corresponds to going from (i, w(i)) to (i+1,1) or to non-characteristic terms. In this algorithm, as long as i ≤ k, the leading terms of the parametrizations differ only by a factor of t^{α}. So the corresponding tangent directions coincide. When i = k + 1 and j = 1 we have parametrizations t^{g} + ..., t^{a_{k+1}−a_{k}} + ... and t^{α+g} + ..., t^{b_{k+1}−b_{k}} + .... To see when these two curves separate, we apply again the Euclidean algorithm. So here we compare the coefficients e_{k,j} for the two curves. The curves split after e_{k,1} +e_{k,2} +...+e_{k,s−1} +ϵ blowups if e_{k,j} = e′_{k,j} for j < s, while ϵ = min{e_{k,s}, e′_{k,s}}. □

Proposition 4.4. Given two monomial curves C and D defined by the sequences

a_{1} < a_{2} < … < a_{r} = d_{C}, b_{1} < b_{2} < … < b_{r} = d_{D}

respectively, then the intersection multiplicity I_{p} of C and D at p is computed by the algorithm 4.1. Likewise the intersection multiplicity I_{q} of C and D is computed.

Let β = gcd(b_{1} − a_{1}, ...b_{r} − a_{r}), then the degree of the join of C and D is

degJoin(C, D) = d_{C} · d_{D} − I_{p} − I_{q} − β.

References

[1] E. Brieskorn and H. Knörrer: Plane algebraic curves. Birkhäuser, 1986

[2] E. Casas-Alvero: “Plane curve singularities.” London Math. Soc. Lecture Notes 276, Cambridge University Press, 2000

[3] R. Hartshorne: Algebraic Geometry. GTM, Springer Verlag, New York, 1977

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