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.*" 



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> I'd be grateful if someone can "translate" the above in an algebraic language.
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(In [this paper](http://www.math.unipd.it/~frank/ALGANT/2010/projects/macaulay-examplex.pdf) one can find more details and a proof, but the language is also geometric.)



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


