Help in this notation in Fulton's Algebraic Curves book

Question

I'm reading Fulton's Algebraic Curves book, I'm stuck in the following proposition (page 105):

In fact, what I didn't understand is the following notation in the proof of this proposition:

Why $k[X,Y]/(F)=k[x,y]$ with small letters? What's the difference between $k[X,Y]$ and $k[x,y]$ in this context?

Thanks

Answer 1

$k[x,y]$ just refers to a $k$-algebra generated by $2$ elements $x, y$, which need not be algebraically independent over $k$ (as $X, Y$ are). Indeed, $x, y$ are the images $X + (F), Y + (F)$ in the quotient ring $k[X,Y]/(F)$.