In particular, I am wondering:

  1. Historically, what was the impetus for the development of the connection between the algebra of polynomial rings and the geometry of algebraic varieties? Concretely, what problems were solved by Hilbert's work?

  2. From a modern perspective, what are the most illustrative motivating examples which demonstrate the utility of the Nullstellensatz for studying the geometry of polynomial curves?

I am studying elementary algebraic geometry from a variety of introductory texts, and I find the Nullstellensatz to be a very deep theorem, but either I'm missing something, or these texts are not actually telling me what you can do with the Nullstellensatz which is of direct relevance to the study of particular algebraic varieties! A very compelling dictionary between algebra and geometry is created, but can someone please give me an example of where this dictionary is used to say something interesting about geometry?

  • $\begingroup$ You are entitled to opinions concerning what is interesting, relevant, motivating. But all of that is subjective, and irrelevant for mathematics. $\endgroup$
    – user436658
    Feb 14 '18 at 8:25
  • $\begingroup$ There is a bunch of theorems that use that dictionary. When working on elementary algebraic geometry (I am not entitled to say anything about more modern algebraic geometry), it is very usefull to work with the algebraic meaning, because sometimes problems are easier with those tools. $\endgroup$
    – Javi
    Feb 14 '18 at 8:27
  • 1
    $\begingroup$ I'd say that any application of algebraic geometry is an application of the Nullstellensatz as this theorem allows one to use algebraical techniques to study geometry. In other words, the Nullstellensatz is what puts algebra in algebraic geometry. $\endgroup$ Feb 14 '18 at 8:27
  • 3
    $\begingroup$ The Nullstellensatz tells you that varieties have points. That should be important to you. $\endgroup$
    – KReiser
    Feb 14 '18 at 8:29

Which polynomials $P(X,Y)\in \mathbb C[X,Y]$ vanish on every point in $\mathbb C^2$ of the complex circle $X^2+Y^2-1=0$ ?
Well there is $P(X,Y)=X^2+Y^2-1$, duh!
Is that all? No: any polynomial of the form $P(X,Y)=Q(X,Y)(X^2+Y^2-1)$ where $Q(X,Y)$ is a completely arbitrary polynomial obviously also vanishes on all points of our circle.
Is that all? $$\operatorname {YES \;!!}$$ And this (suitably generalized) is the content of the Nullstellensatz!


Here are two elementary and important consequences/rephrasements of HNS. They should be included in whatever textbook you are using.

  1. Every maximal ideal in $k[x_{1},...,x_{n}]$ (with $\bar{k}=k$) has the form $$ (x_{1}-a_{1},...,x_{n}-a_{n}) $$ for some $a_{1},...,a_{n}\in k$.

  2. There is a one-to-one correspondence between radical ideals in $k[x_{1},...,x_{n}]$ (with $\bar{k}=k$) and affine varieties in $k^{n}$. Moreover, the ideal is prime if and only if the variety is irreducible.

I think that the importance of these two results should be pretty clear even at first glance. In particular the second one is especially important (in my opinion). It is in the very heart of algebraic geometry: what we use to go from algebra to geometry and viceversa is the equivalence of categories (reversing arrows) between affine varieties and their coordinate rings (reduced finitely generated $k$-algebras).

In fact, the introduction of schemes in algebraic geometry generalizes this equivalence in a vast way: affine schemes (with arrows reversed) are equivalent to commutative rings as categories.

  • $\begingroup$ Hi Pedro, I am familiar with and understand those consequences, but instead of merely saying that they are what you use to go between algebra and geometry, can you provide an example of such a usage? I am particularly interested in applications which were intractable or very difficult with pre-NSS techinques but greatly simplified upon the introduction of this correspondence. $\endgroup$ Feb 14 '18 at 17:00

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