Macaulay's example of prime ideals in $\mathbb C[X_1,X_2,X_3]$ having large number of generators

Question

There is a famous example of Macaulay which shows that there are prime ideals of height two in $\mathbb C[X_1,X_2,X_3]$ having at least $l$ generators for any $l\ge 3$.

In Macaulay's words, the example is constructed as follows: "Consider $l(l-1)/2$ straight lines through the origin $O$ in $3$-dimensional space, not lying on any cone of order $l-2$. Draw a cone of order $l$ and a surface (not a cone) of order $l$ through the $l(l-1)/2$ lines so as to intersect again in an irreducible curve of order $l(l+1)/2$ with $l(l-1)/2$ tangents at $O$. Then no basis of the prime module determined by this curve can have less than $l$ members, where $l$ is a number which can be chosen as high as we please."

I'd be grateful if someone can "translate" the above in an algebraic language.

(In this paper one can find more details and a proof, but the language is also geometric.)

Answer 1

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.