- \   V(I) \textrm{ is finite and locally a complete intersection.}
\end{array}\right\}$$

Matrix representations of $\phi$
--------------------------------

The approximation complex of cycles associated to the sequence of polynomials $f_0,f_1,f_2,f_3$ is of the form (see e.g. [@BuJo03 §4] for an introduction to these complexes in this context) $$\label{eqZComplex}
 {\mathcal Z}_\bullet: \qquad 0\to {\mathcal Z}_3 \to {\mathcal Z}_2 \to {\mathcal Z}_1 \stackrel{M(\phi)}{{\longrightarrow}} {\mathcal Z}_0 \to 0.$$ The map $M(\phi)$ sends a 4-tuple $(g_0,g_1,g_2,g_3)$ to the form $\sum_{i=0}^{3}g_ix_i$, with the condition that $(g_0,g_1,g_2,g_3) \in {\mathcal Z}_1$ if and only if $\sum_{i=0}^{3}g_if_i=0$. The complex ${\mathcal Z}_\bullet$ inherits a grading from the canonical grading of the polynomial ring $S$. Thus, the notation $M(\phi)_\nu$ stands for the matrix in $({\mathcal Z}_\bullet)_\nu$ that corresponds to the degree $\nu$ part of $M(\phi)$.

Under our hypothesis, the symmetric algebra ${\mathcal S}_I:=\operatorname{Sym}_S(I)$ of the ideal $I\subset S$ is projectively isomorphic to the Rees algebra ${\mathcal R}_I:=\operatorname{Rees}_S(I)$. In other words, the graph $\Gamma \subset {\mathbb P}^2_k\times {\mathbb P}^3_k$ of the parameterization $\phi$ is scheme defined by ${\mathcal S}_I$. As a consequence, the approximation complex of cycles ${\mathcal Z}_\bullet$ can be used to determine an implicit representation of the closed image of $\phi$, i.e. the surface ${\mathscr S}$ (recall that $H_0({\mathcal Z}_\bullet)=\operatorname{Sym}_S(I)$).

