Note: This is a much simplified version of my earlier question with the same title. I hope the simpler version has a better chance of generating answers, which I maybe then can use to find the answer to the original question myself. But answers there are of course still welcome too.
Let $V$ be a projective algebraic set, meaning that $V$ is the set of common zeroes of a finite set of homogeneous polynomials:
$$V = Z(f_1, f_2, \ldots, f_n) \tag{1}\label{f}$$
Taking the geometric object $V$ at the left hand side of (\ref{f}) as fixed , I define $\mu$ to be the smallest possible value of $n$ that could be taken in (\ref{f}) for which we can still find $f_1, \ldots, f_\mu$ making the equation true.
Question: Can $\mu$ be expressed in terms of geometric attributes of $V$, e.g. its number of irreducible components, their dimension and co-dimension, perhaps their degrees?
I admit that my notion of 'geometric' is a bit vague. I like to think about it as 'things we can just see when looking at $V$', but in reality I only can see these things if the ambient projective space has dimension as most 3 and the ground field is $\mathbb{R}$. For this question however I am interested in the case where the dimension of the space is $n$ and the ground field is $\mathbb{C}$. So the real meaning of 'geometric' then becomes 'things that a hypothetical $2n$-dimensional being can just see when looking at $V$', which is probably even more vague than 'geometric'.
Second question. Let $\mathcal{I}(V)$ be (as always) the set of all homogeneous polynomials that vanish on $V$ and let $m$ be the (minimum) number of generators of $\mathcal{I}(V)$ as an ideal. How do $\mu$ and $m$ relate to each other?
I must say my intuition is torn here. On one hand it seems that $\mu$ and $m$ are the only two numbers of its kind (i.e. possible values of $n$ in (\ref{f})) that are uniquely defined in terms of $V$. This suggest they might be equal.
On the other hand $\mu$ is defined as the smallest of its kind, while $\mathcal{I}(V)$ is the biggest ideal that vanishes on $V$. So that would rather suggest that $m > \mu$ generically. But then perhaps their difference has a nice geometric interpretation?