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\begin{document}



\title[The degree of the secant variety and the join of monomial 
curves]
{The degree of the secant variety and the join of monomial curves}

\author[ K. Ranestad]{ K. Ranestad$^{*}$}



\address{Matematisk Institutt, Universitetet i Oslo P.O.Box 1053,
Blindern,
N-0316 Oslo,  NORWAY } \email{ranestad@math.uio.no}

\date{\today}
\thanks{ $^{*}$ Supported by MSRI}


\subjclass{Primary 14; Secondary 14}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{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.
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\maketitle

\tableofcontents

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \section{Introduction} \label{intro}

A monomial curve $C$ is the image of an injective morphism of $f:\PP^1\to \PP^r$
defined by monomials.  After ordering the monomials by ascending
degree it is therefore given by
$$(s:t)\mapsto
(s^d:s^{d-a_1}t^{a_1}:\ldots:s^{d-a_{r-1}}t^{a_{r-1}}:t^d)$$
where $a_1<a_2<\ldots<a_r=d$.  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 $\PP^r$.  A simple wellknown argument using a general projection $\pi:C\to \overline{C}\subset \PP^2$ shows that this degree is given by the formula 
$${\rm deg}SecC={{d-1}\choose 2}-\delta_p-\delta_q $$
where $\delta_p$ and $\delta_q$ are the genus contributions or equivalently, $2\delta_p$ and $2\delta_q$ are the
Milnor numbers of the cusps at $p=\pi([1:0\ldots :0])$ and $q=\pi([0:\ldots :0:1])$ on $\overline C$.  To compute $\delta_p$ and $\delta_q$ 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 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 $\PP^r$ 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
$$C: (s:t)\mapsto
(s^{d_C}:s^{d_C-a_1}t^{a_1}:\ldots:s^{d_C-a_{r-1}}t^{a_{r-1}}:t^{d_C})$$
where $a_1<a_2<\ldots<a_r=d_C$, and

$$D: (s:t)\mapsto
(s^d_D:s^{d_D-b_1}t^{b_1}:\ldots:s^{d_D-b_{r-1}}t^{b_{r-1}}:t^{d_D})$$
where $b_1<b_2<\ldots<b_r=d_B$.  Again the two sequences
$$a_1<a_2<\ldots<a_r=d_C, \quad b_1<b_2<\ldots<b_r=d_B$$
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
$\PP^r$.  In this case the general projection of the two curves to $\PP^2$ gives the formula   

$${\deg}Join(C,D)=d_C\cdot d_D-I_s(C,D),$$
where $I_s(C,D)$ is the sum of the intersection multiplicities in $\PP^2$ of the two curves at the images of intersection points between the two curves in $\PP^r$.  An algorithm computing the sum of intersection multiplicity $I_s(C,D)$ is given in section \ref{joindegree}. 


A  closed formula for the degree of the join
in terms of the sequences $a_1<a_2<\ldots<a_r=d_C$ and
$b_1<b_2<\ldots<b_r=d_B$, 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.

\vskip 4mm 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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\section{The multiplicity sequence of a plane curve 
singularity}\label{ms}
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.

We shall use the multiplicity sequence of plane
singularities with given Puiseux series. 
Consider the parameterized affine plane curve 
$$C: t\mapsto t^m, t^{k_1}+t^{k_2}+\cdots .$$
This plane curve has a cusp at the origin, where $t=0$.
The multiplicity sequence is  computed from the sequence 
$m,k_{1},k_{2},\ldots$ as described in a
result of Enriques and Chisini \cite {BrieskornKnorrer} 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.

\subsection{Algorithm computing the multiplicity sequence}\label{alg}
\medskip
Consider the strictly increasing sequence $$m<k_{1}< k_2< \ldots $$
\par

{\it Step 1. The {\it 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 $g_i=gcd\{m,k_1,\ldots , k_i\}$ for $i>0$. The $g_i$ form the {\it gcd}-sequence of $m,k_{1},\ldots,k_{r}$: 
$$ g_0\geq g_1\geq g_2\geq g_3\ldots .$$

Clearly, in the {\it gcd}-sequence,
$g_i=1$ for some $i$, since otherwise the parametrization is not $1:1$.  The {\it characteristic } terms in the sequence $t^{k_1}+t^{k_2}+\cdots$ are the terms $$k_{i_1},..., k_{i_s}$$ where $i_1={\rm min}\{i|g_i<m\}$, $i_2={\rm min}\{i|g_i<g_{i_1}\}$ 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 $m$. 
%The essential
%(characteristic) terms determine the multiplicity sequence
%completely.  The addition of non-essential terms do
%not change this sequence (cf. cite {Brieskorn-Knorrer} section 8.3
%for details). 

\medskip



{\it Step 2. The multiplicity  sequence.}
Given the sequence
$$m<k_{1}< k_2< \ldots $$

 let $\kappa_i=k_i-k_{i-1}$ where $k_0=0$ and $i=1,2,...$.   We
 call $$\kappa_{1},\kappa_{2},\ldots $$ the {\it 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. 

\medskip

Apply the Euclidean algorithm successively to the 
 elements of the difference sequence:
Let
$$\kappa_i=e_{i,1}r_{i,1}+r_{i,2}$$
$$r_{i,1}=e_{i,2}r_{i,2}+r_{i,3}$$
$$...$$
$$r_{i,w(i)-1}=e_{i,w(i)}r_{i,w(i)}$$
with $0\leq r_{i,j+1}<r_{i,j}$ and $r_{1,1}=m, 
r_{i,1}=r_{i-1,w(i-1)}, i>1$.
Note that $$r_{i+1,1}=g_i={\rm gcd}(m,k_1,...,k_i).$$
The {\it multiplicity sequence} is
$e_{i,j}$ times the multiplicity $r_{i,j}$, with $1\leq i\leq s$ and 
$1\leq j\leq w(i)$. \par
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  $\delta$-invariant of the sequence is given by 
$$\delta =\sum_{i,j}e_{i,j}{r_{i,j}\choose 2}.$$

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:

\medskip

\begin{proposition}\label{deltainvariant}\cite{Casas}
Given the sequence
$$m<k_{1}< k_2< \ldots $$
Let 
$$ g_0\geq g_1\geq g_2\geq g_3\ldots $$
be its {\it gcd}-sequence.  Then the $\delta$-invariant of the sequence is
$$\delta={1\over 2}(\sum_{i\geq 1}k_i(g_{i-1}-g_i)-m+1.$$
\end{proposition}

\begin{proof}  See \cite{Casas} page 194 exercise 5.6.\end{proof}

\section{The degree of the  secant variety of a monomial
curve}\label{secdegree}

%We will compute the degree of the first secant variety of a monomial 
%curve.
% (and describe an approach to find its equations.)

Let $C\subset\PP^r$ be a monomial curve defined by the sequence of
positive integers $a_1<a_2<\ldots<a_r=d$ 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 $\pi: C\to\PP^2$.  For a general
projection the only other singularities on $\overline C=\pi(C)$ are
possible cusps at the
image of the points $\pi(p)$ and $\pi(q)$ where $p=(1:\ldots :0)$ and
$q=(0:\ldots:1)$ in $\PP^r$.
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 $SecC$.
\begin{proposition}Let $C\subset\PP^r$ be a monomial curve defined by the sequence of
positive integers $a_1<a_2<\ldots<a_r=d$. 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 
$${\rm deg}SecC={{d-1}\choose 2}-{1\over 2}(\sum_i a_{i+1}(g_i-g_{i+1})-a_1+\sum_i b_{i+1}(h_i-h_{i+1})-b_1)-1.$$
\end{proposition}

\begin{proof}  The arithmetic genus $p(C)$ for a curve $C$ on a smooth surface $S$ is given by the adjunction formula \cite{Hartshorne} on the surface:
$$2p(C)-2=C\cdot C+C\cdot K_S$$
where $K_S$ is the canonical divisor on $S$.  If $C$ has multiplicity $m$ at a point $q$ on $S$, and $S'\to S$ is the blowup of $S$ at $q$, then the adjunction formula on $S'$ says
$$2p(C')-2=C'\cdot C'+C'\cdot K_{S'}=$$
$$=(C^*-mE)\cdot (C^*-mE)+(C^*-mE)\cdot (K_S+E)=2p(C)-2-m^2+m$$ where $E$ is the exceptional divisor and $C^*$ is the total transform and $C'$ is the strict transform of $C$(cf. \cite{Hartshorne} chapter V). Thus $$p(C')=p(C)-{m\choose 2}.$$  After resolving all singularities on $\overline C\subset\PP^2$ 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 ${2\choose 2}=1$. The points $\pi(p)$ and $\pi(q)$ are the only other singularities on $\overline C$.  The contribution $\delta_p$ 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 $\delta_q$.  Since $K_{\PP^2}\cong -3L$, 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:

$${\rm deg}SecC={{d-1}\choose 2}-\delta_p-\delta_q.$$


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 {\it 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 \ref{alg} computes this multiplicity sequence from the exponents of the Puiseux expansion, so the genus contribution is nothing but the $\delta$ 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 $\pi(p)$.

\begin{lemma}  The characteristic terms in the Puiseux expansion of
$\overline C$ at $p$ coincides with the characteristic terms in the
Puiseux expansion
 $$x=t^{a_1}, y=t^{a_2}+b_{23}t^{a_3}+...+b_{2r}t^{a_r}.$$\end{lemma}

\begin{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 $N\geq N_0$.  So it is enough to prove the
statement of the lemma modulo $t^{N_0}$.   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:
$$x=t^{a_1}+b'_{14}t^{a'_4}+...+b'_{1r}t^{a'_{r'}},
y=t^{a_2}+b_{23}t^{a_3}+...+b_{2r}t^{a_r}+c_1t^{b_1}+....$$
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.
\end{proof}



Since non-characteristic terms do not contribute to the $\delta$-invariant the proposition follows from \ref{deltainvariant}.
\end{proof}


\begin{example} Consider the monomial curve $C$ given by the sequence
$(0,30,45,55,78)$. At $p=(1:0)$, we may compute the $\delta$-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 $\delta$-invariant is $$\delta_p={1\over 2}(45(30-15)+55(15-5)+78(5-1)-30+1)=754.$$

%The difference sequence is $(45,10,23)$ and the multiplicity sequence
%is
%$$(30,15,15;10,5,5;5,5,5,5,3,2,1,1).$$  
 At
$q=(0:1)$ we compute the $\delta$-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 $\delta$-invariant is $$\delta_q={1\over 2}(33(23-1)-23+1)=352$$ 
%and the multiplicity sequence is
%$(23,10,10,3,3,3,1,1,1).$  
The degree of the secant variety of $C$ is
 $${\rm deg}SecC={77\choose 2}-\delta_p-\delta_q= 2926-754-352=1820$$
\end{example}


% Of course $1$'s in the multiplicity sequence does not contribute.
%Likewise if  $\kappa_i=1$ in the difference sequence, then the
%multiplicities $r_{n,j}=1$ for $n\geq i$.


\section{The degree of the join of two monomial
curves}\label{joindegree}

 Consider the join of two monomial curves $C$ and
$D$ in $\PP^r$ defined by
$$C: (s:t)\mapsto
(s^{d_C}:s^{d_C-a_1}t^{a_1}:\ldots:s^{d_C-a_{r-1}}t^{a_{r-1}}:t^{d_C})$$
where $a_1<a_2<\ldots<a_r=d_C$, and

$$D: (s:t)\mapsto
(s^d_D:s^{d_D-b_1}t^{b_1}:\ldots:s^{d_D-b_{r-1}}t^{b_{r-1}}:t^{d_D})$$
where $b_1<b_2<\ldots<b_r=d_B$.  The two sequences
$$a_1<a_2<\ldots<a_r=d_C, \quad b_1<b_2<\ldots<b_r=d_B$$
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 $\PP^r$.  But this number equals the number
of new intersection points obtained by projecting the union of the
two curves from $L$ to a plane.  Denote by $\pi_L$ the projection
from $L$, and let $\overline C=\pi_L(C)$ and $\overline D=\pi_L(D)$ be
the images of $C$ and $D$ respectively.  The total intersection
number
$$\overline C\cdot \overline D=d_C\cdot d_D$$
by Bezout's theorem, so to get the degree we have to subtract the intersection
multiplicity at the points of $\pi_L(C\cap D)$.
In our special situation there certainly are points in $C\cap D$:
$$\{p=(1:0\ldots :0),q=(0:\dots :0:1),u= (1:\ldots :1)\}\subset C\cap
D.$$
Furthermore,
if $$\beta=gcd (b_1-a_1,b_2-a_2,...,b_r-a_r),$$ then all
roots of $t^{\beta}-1$ 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 $\CC$) that there is an $\alpha$ such that $t_1^\alpha=t_2$.  Then we get $t_1^{a_i}=t_1^{\alpha\cdot b_i}$, i.e. $t_1=1$ or 
$a_i=\alpha\cdot b_i$ for $i=1,...,r$, or there is some integer $\beta$ such that $\beta\cdot a_i=\alpha\cdot b_i$.  Therefore $\alpha$ is a rational number.  Furthermore the $b_i$ have no common factor, so $\alpha$ must be an integer.  Since the $a_i$ also have no common factor, we may assume that $\alpha=1$, and we have precisely the intersection points described above. 

  The
intersections at the roots of $t^{\beta}-1$ are always transversal, i.e. with distinct
tangents:  The tangent direction is given by the
derivatives $(a_1t^{a_1-1},...,a_rt^{a_r-1})$ and
$(b_1t^{b_1-1},...,b_rt^{b_r-1})$.  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 $\pi_L$.  

For the
points $\pi_L(p)$ and $\pi_L(q)$ 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 $\pi_L(p)$.  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 $\pi_L(p)$ 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
$p_0=\pi_L(p), p_1,\ldots, p_k$, and together with it the
multiplicities of the strict transforms of the two curves at each
$p_i$.  We denote these multiplicity sequences by
$m_0(C),\ldots, m_k(C)$ and  $m_0(D),\ldots, m_k(D)$.   The
intersection multiplicity between the two curves at the point
$\pi_L(p)$ is:
$$\sum_{i=0}^km_i(C)m_i(D).$$


The multiplicity sequences $m_0(C),\ldots, m_k(C)$ and
$m_0(D),\ldots, m_k(D)$ 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 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:

\begin{example}\label{first}  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 $\pi_L(p)$ is $1+1+1+1=4$. \end{example}

\begin{example}\label{second} 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 $\pi_L(p)$ is
$2+2+2+2=8$.   Similarly if $b_i=ea_i, i=1,\ldots, s$, then
$m_i(D)=em_i(C)$ for $i=1,...,k$ where $k$ is the length of the
multiplicity sequence for a curve $D':(b_1,\ldots, b_s, b_{s+1})$.
\end{example}

\begin{example}\label{third}
 For $C:(1,2,3,4)$ and $D:(b_0,b_1,b_2,b_3)$ where
$b_0>1$ and $b_1\not= 2b_0$, then $k=2$ and $m_i(C)=(1,1)$ and
$m_i(D)=(b_0,min\{(b_1-b_0),b_0\})$.\end{example}




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

\subsection{Intersection multiplicity algorithm}\label{ima}
Given two monomial curves $C$ and $D$ defined by the sequences
$$a_1<a_2<\ldots<a_r=d_C, \quad b_1<b_2<\ldots<b_r=d_D$$
respectively, and assume that for some $j\geq 1$
$b_{i}\geq a_{i}$ 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. \par
{\it Step 1}. Let
$\alpha={b_{1}\over a_{1}}$.  If
$\alpha$
is not an integer, then set $k=0$, otherwise let
$$k=max\{i|b_{i}=\alpha a_{i}\}$$


Let $m_{1}, m_{2},\ldots,
m_{s}$ be the multiplicity sequence,  the outcome of the algorithm 
\ref{alg}, of the sequence
$(b_{1},b_{2},\ldots,b_{k})$. 

Set
$$\delta_k={1\over\alpha}(m_{1}^2+\cdots +m_{s}^2),$$ and let
$g={\rm gcd}(b_1,...,b_k)$.

\medskip

{\it Step 2}.  Apply the multiplicity algorithm \ref{alg} to the sequences
$({g\over \alpha}, a_{k+1}-a_k)$ and $(g, b_{k+1}-b_k)$, with outcome 
$$(e_1,r_1),(e_2,r_2),...,(e_m,r_m)\quad {\rm and }\quad (e'_1,r'_1),...,(e'_n,r'_n)$$ respectively.  Let $l={\rm min}\{j|e_j=e'_j\}$ and $f={\rm min}\{e_{l+1},e'_{l+1}\}$ and let
$$\epsilon=\sum^l_je_j\cdot r_jr'_j+f\cdot r_{l+1}r'_{l+1}.$$

{\it Step 3}.
The intersection multiplicity between the curves $C$ and $D$ at the origin is
$$I(C,D)=\delta_k+\epsilon.$$
\medskip

\begin{proof}
To start  we  project $C$ and $D$ into the plane and may choose coordinates in the plane such that $\pi(C)$ and $\pi(D)$ have the parametrizations
$$\pi(C): x=t^{a_1}+c_{1,3}t^{a_3}+...+c_{1,r}t^{a_r},y= t^{a_2}+c_{2,3}t^{a_3}+...+c_{2,r}t^{a_r}$$
and
$$\pi(D): x=t^{b_1}+c_{1,3}t^{b_3}+...+c_{1,r}t^{b_r}, y=t^{b_2}+c_{2,3}t^{b_3}+...+c_{2,r}t^{b_r}.$$
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 $\pi(C)'$ and $\pi(D)'$  have local parameterizations:
$$\pi(C)': x=t^{a_1}+c_{1,3}t^{a_3}+...+c_{1,r}t^{a_r}, y=t^{a_2-a_1}+c_{2,3}t^{a_3-a_1}+...+c_{2,r}t^{a_r-a_1}-c_{1,3}t^{a_2+a_3-2a_1}+\ldots$$
and
$$\pi(D)':x= t^{b_1}+c_{1,3}t^{b_3}+...+c_{1,r}t^{b_r}, y=t^{b_2-b_1}+c_{2,3}t^{b_3-b_1}+...+c_{2,r}t^{b_r-b_1}-c_{1,3}t^{b_2+b_3-2b_1}+\ldots.$$
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 $t^{\alpha_p}$. 
Therefore, if $k>0$ the two curves $\pi(C)'$ and $\pi(D)'$ 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 $x,{y\over x}$, if the tangent direction is $x=0$, then the strict transform is parametrized by ${x\over y}, y$, and if the tangent direction is $x=y$, then the strict transform is parametrized by $x,{{y-x}\over x}$. 
Now, the multiplicities of the strict transforms at the origin form the multiplicity sequence obtained by the algorithm \ref{alg}, 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 $i\leq k$,
 the leading terms of the parametrizations differ only by a factor of $t^{\alpha}$.  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^{\alpha +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 $e_{k,1}+e_{k,2}+...+e_{k,s-1}+\epsilon$ blowups if $e_{k,j}=e'_{k,j}$ for $j<s$, while $\epsilon ={\rm min}\{e_{k,s}, e'_{k,s}\}$. 
\end{proof}



\begin{proposition}
    Given two monomial curves $C$ and $D$ defined by the sequences
$$a_1<a_2<\ldots<a_r=d_C, \quad b_1<b_2<\ldots<b_r=d_D$$
respectively, then the intersection
multiplicity $I_{p}$ of $C$ and $D$ at $p$ is computed by the algorithm \ref{ima}.  
Likewise the intersection multiplicity $I_q$ of $C$ and $D$ is computed.

Let $\beta=gcd(b_1-a_1,...b_r-a_r)$, then the degree of the join of
$C$ and $D$ is  $${\rm deg} Join(C,D)=d_C\cdot
d_D-I_p-I_q-\beta.$$
\end{proposition}

\begin{thebibliography}{999}


\bibitem{BrieskornKnorrer} E. Brieskorn and H. Kn\"orrer: Plane
algebraic curves.  Birkh\"auser, 1986
\bibitem{Casas} E. Casas-Alvero:  ``Plane curve singularities.'' London Math. Soc. Lecture Notes 276, Cambridge University Press, 2000
\bibitem{Hartshorne} R. Hartshorne: Algebraic Geometry. GTM, Springer Verlag, New York, 1977 

\end{thebibliography}


\end{document}





