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)?