[Math] Why the name “variety” and the notation “V” for zeroes of polynomials

ag.algebraic-geometryterminology

The following questions came to my mind while preparing the notes for the first class of (my first) course on algebraic geometry.

Question 1: Is there any motivation for choosing the term "variety" for zeroes of polynomials? For example, I can sort of guess/understand the logic behind the term "manifold": 2-fold, 3-fold, … $\rightarrow$ many-fold. For the term "variety" I don't see any such clear explanation.

Question 2: Why write $V(f_1, \ldots, f_k)$ for zero sets of polynomials $f_1, \ldots, f_k$? Is it because of the term "variety"? Some (newer) texts use $Z(f_1, \ldots, f_k)$ – I thought $Z$ was meant to convey "zeroes". Does $V$ stand for "zeroes" in some other languages (a cursory look at the German and French Wikipedia pages for algebraic variety did not help)?

Best Answer

Also in Italian "varietà" is the term for both.

Starting from this, I looked at the Italian Wikipedia webpage for varieties which, at the end, has this remark about the origin of the term:

In italiano si traduce con varietà il termine tedesco Mannigfaltigkeit, che compare per la prima volta nella tesi di dottorato del 1851 di Bernhard Riemann, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. Riemann si pone il problema di introdurre delle "grandezze molteplicemente estese", aventi cioè "più dimensioni", e le definisce usando quel termine.
Analizzando il termine come parola composta, Mannig-faltig-keit, si riconosce in essa un parallelo con il termine latino multi-plic-itas, sicché lo si potrebbe tradurre letteralmente come 'molteplicità'.

That can be approximatively translated as

In Italian we translate with "varietà" the German word "Mannigfaltigkeit", which appears for the first time in the 1851 doctoral thesis of Bernhard Riemann, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. Riemann introduced some "dimensions with multiple extensions", having "more dimensions", and he defines them using that term.
Analiyzing the term as a compound word, "Mannig-faltig-keit", we can identify a parallelism with the Latin word "multi-plic-itas", so that it can be literally translated as "multiplicity".

So, at least in Italian, the term comes from "variety" and "multiplicity" in the sense of "diversity".

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