# Why the name "variety" and the notation "V" for zeroes of polynomials?

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)?

• In French and Dutch the word for manifold and variety is the same: “variété” and “variëteit”. This might or might not have anything to do with an answer to your question.
– jmc
Commented Mar 11, 2014 at 13:18
• I have always assumed V stands for "vanishing", as in, the vanishing set of the polynomials. Commented Mar 11, 2014 at 13:25
• Your guess is backwards: the terms 2-fold, 3-fold, and $n$-fold came after manifold, not before. The usual older term for 2-fold is... surface. Commented Mar 11, 2014 at 13:37
• I always assumed that the word "manifold" came from engineering, wherein a manifold is pipe which has several tubes connecting to it in order to collect several different gases or fluids into one place. Commented Mar 11, 2014 at 15:37
• @KConrad: in German there is the term "Dreifaltigkeit", which roughly translates literally to three-foldedness and actually means trinity. There is also the term "mannigfaltig", which approximately means of great variety. Both terms seem to be much older than the "Mannifaltigkeit", which apparently has been introduced by Riemann; so in English the term 'manifold' as a noun seems to be newer than 'manifold' as a property. I am however no linguist. Commented Mar 12, 2014 at 8:03

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".

• According to this link, the term was first used by E. Beltrami in 1869.
– abx
Commented Mar 11, 2014 at 13:57
• Thanks :) To me this makes sense, Beltrami knew the works of Riemann and so maybe he was the one who first translated to the Italian "varietà" the German "Mannigfaltigkeit". Commented Mar 11, 2014 at 14:04
• Following your advice and after a bit of research, I can also point out that few years later (in the 1880's and 1890's) the word "varietà" was already widely used in the works (and titles) of Fano, Enriques, del Pezzo. Commented Mar 11, 2014 at 15:08