There seems to be some terminology drift here. I would say that "order" would be called degree in modern terminology, for example.

Here is the way I see it, and please someone correct me if I am wrong.

Take $l(l-1)/2$ lines through the origin in $\mathbb C^3$, i.e. ideals $I_i\subset\mathbb C[X_1,X_2,X_3]$ of the form $\langle X_2-\alpha_iX_1,X_3-\beta_i X_1\rangle $ with $1\leq i\leq l(l-1)/2$ and $\alpha_i$ and $\beta_i$ generic (so that no degree $l-2$ homogeneous polynomial in $X_1,X_2,X_3$ lies in all of the above ideals $I_i$).

Consider a polynomial $F_1$ which is homogeneous of degree $l$ and is contained in each $I_i$. Containment in $I_i$ is a codimension one condition, and there is a dimension $(l+1)(l+2)/2$ space of degree $l$ polynomials, so such $F_1$ exists. I assume that such $F_1$ is chosen generically (this is the "cone").

Consider a polynomial $F_2$ which does not have to be homogeneous, but rather is a sum of polynomials of degrees $\leq l$ which is contained in all $I_i$ (and presumably is generic with respect to this property).

Then the ideal in question is the radical of $\langle F_1,F_2\rangle$.

I don't claim to know how the actual argument proceeds, but this looks like the algebraic translation you are after.

There seems to be some terminology drift here. I would say that "order" would be called degree in modern terminology, for example.

Here is the way I see it, and please someone correct me if I am wrong.

Take $l(l-1)/2$ lines through the origin in $\mathbb C^3$, i.e. ideals $I_i\subset\mathbb C[X_1,X_2,X_3]$ of the form $\langle X_2-\alpha_iX_1,X_3-\beta_i X_1\rangle $ with $1\leq i\leq l(l-1)/2$ and $\alpha_i$ and $\beta_i$ generic (so that no degree $l-2$ homogeneous polynomial in $X_1,X_2,X_3$ lies in all of the above ideals $I_i$).

Consider a polynomial $F_1$ which is homogeneous of degree $l$ and is contained in each $I_i$. Containment in $I_i$ is a codimension one condition, and there is a dimension $(l+1)(l+2)/2$ space of degree $l$ polynomials, so such $F_1$ exists. I assume that such $F_1$ is chosen generically (this is the "cone").

Consider a polynomial $F_2$ which does not have to be homogeneous, but rather is a sum of polynomials of degrees $\geq l$ which is contained in all $I_i$ (and presumably is generic with respect to this property).

Then the ideal in question is the radical of $\langle F_1,F_2\rangle$.

I don't claim to know how the actual argument proceeds, but this looks like the algebraic translation you are after.