I'm currently trying to learn some complex and projective geometry. There is one issue bugging me again and again, from different perspectives, and I just can't get my head around it. One incarnation of my problem:
In the book by Griffiths and Harris, they define the degree of a variety in chapter 1.3. One of their definitions is:
In case [the variety] $V \subset \mathbb P^n$ is a hypersurface, we have seen that it may be given in terms of homogeneous coordinates $X_0 \dots X_n$ as the locus $V = \big( F(X_0 \dots X_n) = 0 \big)\;$ of a homogeneous polynomial $F$. If $F$ has degree $d$, then [...] $V$ has degree $d$.
A consequence would be for example that the canonical bundle of $V$ is $\mathcal O(d - n - 1) \big|_V$.
My problem is: As far as I understand it, the object $V$ here is just the hypersurface as a geometrical object (an algebraic variety was defined as the locus of a collection of polynomials and nothing more). Hence $F$ and $F^2$ would define the same object $V$. But using $F^2$ instead of $F$ gives us a different degree / a different canonical bundle, which doesn't make sense...