Chapter 1
Warming up to
Enumerative Geometry
Enumerative geometry is an old subject that has been revisited exten-
sively over the past 150 years. Enumerative geometry was an active
field in the
19th
century. Much progress was made, and there was
much excitement about what was to come. Indeed, enumerative ge-
ometry was the subject of Hilbert’s 15th problem. Hilbert’s famous
problem list was delivered at the beginning of the 20th century and
was quite influential in shaping mathematics. A good overview of the
influence of the Hilbert problems, including Hilbert’s 15th, appears in
[Hilbert].
Unfortunately, many fundamental enumerative problems eluded
the best mathematicians for most of the 20th century. Progress came
from a seemingly unlikely source: string theory in physics. In this
book we will learn many fundamentals of enumerative geometry and
begin to appreciate some of the connections of these ideas to physics.
The basic question of enumerative geometry in its most general
form is simply stated as
“How many geometric structures of a given type satisfy a given col-
lection of geometric conditions?”
A trivial example would be the question:
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