I'm reading through the chapters in Algebraic Geometry, Harris, about Blow-ups and Resolving Singularities and he keeps mentioning fibers (for instance, in one example he saysthe fiber over that point [the origin in ] is a copy of corresponding to the lines through that point
), however, he doesn't seem to ever actually define what a fiber is, only something called the fiber product (which I assume is also something I will need).
So if someone could tell me what he means that would be much appreciated (I also noticed he never actually explains what a homogeneous equation is, which I know is a really basic concept (and we have subsequently covered it in lectures so it doesn't matter), but still!)
Thank you very much!
. The fibre over
in topology. However, in AG you need a bit more but that requires you to know about schemes (you have
, and the fibre of
A homogeneous equation is exactly what you think it means --- the total degree of every term appearing in the equation is the same.
So you blow up
at the origin, getting
are the coordinates for
are the coordinates of
. The fibre over any point of
is still a point, but the fibre over (0,0) is the entire
since there isn't any condition on the