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:

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:

1