\section{A Very Free Rational Curve of Degree 9}

\noindent
We consider the morphism $\varphi = (G_0,...,G_5): \mathbb{P}^1 \rightarrow \mathbb{P}^5$  into the fermat hypersurface given by
\begin{align*}
G_0 &= S^5T^4 + S^8T + S^9\\
G_1 &= ST^8 + S^2T^7 + S^3T^6 + S^4T^5 + S^8T + S^9\\
G_2 &= T^9 +ST^8 +S^2T^7 + S^3T^6 + S^5T^4 + S^9\\ 
G_3 &= S^4T^5\\
G_4 &= ST^8 +S^4T^5 +S^9\\ 
G_5 &= T^9 +ST^8 + S^2T^7 +S^3T^6 +S^5T^4 + S^6T^3 +S^7T^2.
\end{align*}

\noindent
Let $e_1,...,e_5$ denote the splitting type of $\Omega (\varphi)$, where we have that $e_i \le -9$ $(1 \le i \le 5)$.  Note that $\dim(\Omega(\varphi)_9) \ne 0 \Leftrightarrow$ the $G_i$ $(0 \le i \le 5)$ satisfy a $k$-linear relation.  We examine the matrix of coefficients with columns ordered by increasing degree of S:  

\[ \left( \begin{array}{cccccccccc}
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1\\
0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1\\
1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1\\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1\\
1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 \end{array} \right)  \]

\noindent
It is easily seen via row reduction that this matrix has rank $6$, hence $\dim_k(\Omega(\varphi)_9) = 0$.  Next we claim that $\tilde{\varphi} : R_1^{\bigoplus6} \rightarrow R_{10}$ is surjective.  In fact it can be checked that the $\tilde{\varphi}(b_i)$ $(1 \le i \le 12)$, where the $b_i$ are distinct basis elements of $R_1^{\bigoplus6}$, span $R_{10}$.  It follows that $\tilde{\varphi} : R(-9)_n^{\bigoplus6} \rightarrow R_n$ is surjective for $n \ge 10$.  Hence 
\begin{align*}
\dim_k(\Omega(\varphi)_{10}) &= \dim_k(R_1^{\bigoplus6}) - \dim_k(R_{10}) = 1\\
\dim_k(\Omega(\varphi)_{11}) &= \dim_k(R_2^{\bigoplus6}) - \dim_k(R_{11}) = 6.
\end{align*}

\noindent
After reordering this yields $e_1,...,e_5 = -11,-11,-11,-11,-10$, which corresponds to the splitting type $1,1,1,1,5$ of $E_X(\varphi)$, showing that $\varphi$ is very free. 

\medskip \noindent
As an alternative to the preceding computations, one can execute the following PARI/GP code in order to calculate both the splitting type of $\Omega(\varphi)$ and of $E_X(\varphi)$:    

\scriptsize
\begin{verbatim}
\\takes as input the 6 polynomials that define the morphism and the 
\\degree of the morphism d.

find_splittingTypeE(G0,G1,G2,G3,G4,G5,d) =
{
return(
4 *(find_splittingTypeOmega(G0,G1,G2,G3,G4,G5,d) + [d,d,d,d,d]) + [d,d,d,d,d])
}

find_splittingTypeOmega(G0,G1,G2,G3,G4,G5,d) =
{
local(r,i,m,e);
e = vector(5);
m = d; r = 0; i = 0;
while(r < 5,
     rTemp = r;
     r = rTemp + hilbert_function(G0,G1,G2,G3,G4,G5,d,m) - i;
     j = 0;
     while(j < r - rTemp,
         e[rTemp + 1 + j] = -m;
         j++);
     i = i + rTemp + 2 *(r-rTemp);
     m++);
return(e)
}

hilbert_function(G0,G1,G2,G3,G4,G5,d,m) =
{
local(G,A,i,j,k);
G = [G0,G1,G2,G3,G4,G5];
A = matrix(m+1,(6) * (-d + m + 1));
i = 0;
j = 1;
k= 0;
while(i <= 5,
    while(j <= -d + m + 1,
         while(k <= d,
                  A[j + k,i*(-d + m + 1) + j] = 
                        polcoeff(polcoeff(G[i+1],d-k,S),k,T);
         k++);
     j++;k=0);
i++;j=1);
return((6 * (-d + m + 1) - matrank(A)))
}
\end{verbatim}
 