Some terminology involving homogeneous ideals in polynomial rings I was thinking about a problem that (as I only realized today) can be translated into the language of projective ideals in polynomial rings $-$ i.e. the stuff that defines projective varieties.
I am now hoping to make use of the vast body of already existing knowledge on projective algebraic geometry, but I find it hard to find what I need because I don't speak this language very well. So I have a number of questions about what things are called. I hope someone can point me to the right terminology?

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*Suppose $I \subset k[x_0, \ldots, x_N]$ is an ideal, generated by a finite set of homogeneous polynomials, where moreover all these homogeneous generators are homogeneous of the same degree $n$. Does this number $n$ have a name?

Intuitively I would expect it to be the degree of $I$, but the geometric object $V(I)$, consisting of the set of points in $\mathbb{P}^N$ on which all elements of $f$ are zero, apparently also has a degree which is defined in term of intersections with hyperplanes. It does not seem obvious to me that the two notions of degree are the same or even related. If they are not, what is my number $n$ called then?


*Let $I$ and $n$ be as above. Let $H^n$ be the ($\binom{N + n -1}{n}$-dimensional) vectorspace of all homogeneous degree $n$ polynomials. (Actually, if there is a more standard notation for $H^n$ I would also like to know.) What is the number $\dim (H^n \cap I)$ called?

It is a dimension, but obviously it is not the dimension of the corresponding geometric object $V(I)$.
The space $H^n \cap I$ is what generates $I$, by assumption. Keeping in mind Wikipedia's view of $I$ as being 'generated by a finite set of homogeneous polynomials' (rather than a vectorspace of homogeneous polynomials), we note that a set $S$ of homogeneous polynomials is a set generating $I$ which is minimal w.r.t. inclusion among all sets with that property, if and only if $S$ is a basis of $H^n \cap I$. So from that perspective $\dim H^n \cap I$ is perhaps just called the number of generators of $I$? But the term 'number of generators' feels a bit ambiguous. It sounds like it depends on some chosen set of generators, that need moreover not be minimal w.r.t. inclusion. Maybe it is some special value of the Hilbert polynomial?


*The common set of zeros $V(I)$ of $I$ is apparently only called a variety if $I$ is a prime ideal, which, if I recall correctly, happens if and only if $V(I)$ is Zariski-closed. However, when $I$ is not prime, $V(I)$ is still a geometric object $-$ a well defined subset of $\mathbb{P}^N$. So what is this geometric object called if it is not (necessarily) a variety? A 'shape'? A 'blob'? A 'bunch of points that together exhibit a remarkably smooth and rigid structure, as if they really try to be a variety but then fail at the last moment'?


*Common sense tells me $V(I)$ must have a Zariski-closure $\overline{V(I)}$ and that associated to that thing there is some ideal, which I now will call $\overline{I}$ by analogy, consisting of all polynomials that vanish on $\overline{V(I)}$. The question is of course what is $\overline{I}$ called, and what is the standard notation for it.
But now that I type this am also wondering what is the relation between $I$ and $\overline{I}$ is. On one hand being zero on a bigger set is a more restrictive condition and I expect $\overline{I} \subset I$. On the other hand, making the innocuous sounding assumption that polynomials are continuous, we would have that every element of $I$ vanishes on all of $\overline{V(I)}$ and hence $I \subset \overline{I}$. Together this implies $I = \overline{I}$. But this would break the relation I thought I remembered between $V(I)$ being closed or not and $I$ being prime or not. So somewhere I am making a mistake, but where? (I am pretty sure I knew the answer to this conundrum 20 years ago, but now I am at a loss. I hope someone can help me)
 A: *

*I don't think there is a standard name for this number. It is the degree of $V$ if you have only one polynomial, but already for larger sets of polynomials it is smaller than the degree: for instance, the two quadratic polynomials $x^2$ and $y^2$ in $\mathbf{C}[x,y]$ together generate an ideal whose common zero set is simply the origin $(0,0)$, but as a scheme it is a fat point of multiplicity $4 >2$, since a basis of the quotient by this ideal is given by the polynomials $1,x,y$, and $xy$.


*Under your hypothesis that $I$ is generated in degree $n$, the dimension of $H^n \cap I$ is called the zeroth Betti number of $I$, also known as the number of minimal generators (which is, of course, independent of the specific set chosen).


*Most people would indeed refer to $V(I)$ as a (not necessarily irreducible) variety, even when $I$ is not a prime ideal. What's true is that $V(I)$ is irreducible if and only if the ideal $I(V(I))$ of functions that vanish on it (which might be bigger than $I$, and by the Nullstellensatz is exactly the radical of $I$) is prime. Some people do assume that variety means irreducible, but then it's kind of annoying to always be checking whether or not some ideal is prime (which can be computationally challenging in concrete examples).


*$V(I)$ is already Zariski closed, so this bit doesn't really apply.
