Number of generators of prime ideals in $K[x_1,x_2,...,x_n]$

Question

Is there any bound for the number of generators of prime ideals in $K[x_1,x_2,...,x_n]$? (For example in $K[x,y]$.)

We know that maximal ideals of $K[x_1,x_2,...,x_n]$ have $n$ generators.

Answer 1

1. For $K[X,Y]$ the prime ideals are well known, and they have at most two generators.

2. There is a famous example of Macaulay which shows that there are prime ideals in $\mathbb C[X_1,X_2,X_3]$ having at least $n$ generators for any $n\ge 1$. For more details and a proof see this paper.

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."

Maybe it's worthwhile to mention another famous result which says the following:
If $\mathfrak p$ is a prime ideal in $K[X_1,\dots,X_n]$ such that $K[X_1,\dots,X_n]/\mathfrak p$ is regular, then $\mathfrak p$ is generated by (at most) $n$ elements. (Forster)