I may know that the term "projective" relates to kind of virtual added points at infinity of the variety.
I may know that the term "non-singular" relates to the fact that the variety has no singular points and that it might have a link with an idea of possible differentiation.
I may know what "algebraic" means, but it can refer to algebraic topology, or algebraic number theory. I would not really know how to have enough discernment here.
The term variety is, to me, today, too large, to explain. I don't really know what it is in mathematics.
$\mathbb{C}$ is the set of complex numbers, we can write like $a+ib$ where $i^2 = -1$.
"over $\mathbb{C}$" is also a mystery to me. Could it mean "made by/of" complex numbers? or "in complex numbers related spaces"?
Is it possible to write the question with symbols and quantifiers? at least a certain number of?