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)?

aftermanifold, not before. The usual older term for 2-fold is... surface. $\endgroup$3more comments