I am trying to do exercise 2.6 of Hassett's "Introduction to algebraic geometry":
i) Give an example of a monomial ideal $I\subseteq\mathbb{C}[x,y]$ with a minimal set of generators consisting of five elements.
ii) Is there any bound on the number of generators of a monomial ideal in $\mathbb{C}[x,y]?$
It is on the chapter of Gröbner Basis, so it must have something to do with it, but I am not able to find an answer.
i) Thanks to Thomas Andrews' comment, I can guess that $J:=\langle x^{4}, x^{3}y, x^{2}y^{2}, xy^{3}, y^{4}\rangle$ is an ideal of the kind we are looking for.
If I am not wrong, $f$ is in the ideal if and only if each monomial of $f$ is divisible by one of those generators of $J$. From here, I do not know how to prove that there is not a set of $4$ elements that generate $J$.
ii) If i) is correct, the answer to ii) is no, because we can consider
$$
\langle x^{n}, x^{n-1}y,\ldots, xy^{n-1}, y^{n}\rangle
$$
for any $n\in\mathbb{N}-\{0\}$.
Best Answer
It is enough to show $\dim_K(J_n/\mathfrak mJ_n)=n+1$, where $K$ is a field ($K$ replaces $\mathbb C$ which is not necessary in this story), $J_n=\langle x^{n}, x^{n-1}y,\ldots, xy^{n-1}, y^{n}\rangle$ and $\mathfrak m=(x,y)$. In order to do this we show that the residue classes of $x^{n-i}y^i$ for $i=0,1,\dots,n$ are linearly independent over $K$. Suppose that $a_0x^n+a_1x^{n-1}y+\cdots+a_ny^n\in\mathfrak mJ_n$ with $a_i\in K$. By using the homogeneity we conclude that $a_i=0$ for all $i=0,1,\dots,n$.