2 1. Warming up to Enumerative Geometry

“How many points in the plane lie on each of two given lines?”

However, even this simple example reveals several potential diﬃ-

culties of a type which we will have to frequently address:

• The answer depends on the choice of lines:

(1) If the lines are chosen generally, there is exactly one

such point, the point of intersection of the two lines.

(2) The point can “disappear to inﬁnity” if the lines are

moved into a parallel position, in a precise way to be

explained shortly.

(3) The answer can be inﬁnite (if the two lines coincide

with each other).

• Note as well that the question can be translated into a ques-

tion of algebra: how many solutions (x, y) are there to the

system of equations

ax + by = c,

dx + ey = f?

This somewhat cumbersome answer to a very simple question is

unpleasant. In enumerative geometry, this is dealt with by changing

the question slightly, sacriﬁcing the simplicity of the question in favor

of simplicity of the answer, in this case an unequivocal “1”.

Let’s shift from the context of geometry to the context of algebra,

as we just saw we can do in this

example.1

Let’s stay in the realm of

the elementary and ask the enumerative (algebraic) question: “how

many roots does a polynomial of degree n in one variable have?”

For this problem, there is again a range of subcases, revealing a

range of potential diﬃculties. Let’s consider a polynomial of degree d

f(x) =

a0xd

+

a1xd−1

+ ··· + ad−1x + ad,

for the moment with real coeﬃcients ai, and solve f(x) = 0. Let’s

ﬁrst consider d = 1. The solution is x = −a1/a0, and there is at ﬁrst

glance exactly one solution. There is a diﬃculty, which we will see is

more than a simple issue of semantics: suppose we agree to consider

f(x) = 0 · x + a1 as a degenerate degree 1 polynomial; after all, f(x)

1This

ability to move freely from geometry to algebra and back is at the core of

the subject called algebraic geometry.

“How many points in the plane lie on each of two given lines?”

However, even this simple example reveals several potential diﬃ-

culties of a type which we will have to frequently address:

• The answer depends on the choice of lines:

(1) If the lines are chosen generally, there is exactly one

such point, the point of intersection of the two lines.

(2) The point can “disappear to inﬁnity” if the lines are

moved into a parallel position, in a precise way to be

explained shortly.

(3) The answer can be inﬁnite (if the two lines coincide

with each other).

• Note as well that the question can be translated into a ques-

tion of algebra: how many solutions (x, y) are there to the

system of equations

ax + by = c,

dx + ey = f?

This somewhat cumbersome answer to a very simple question is

unpleasant. In enumerative geometry, this is dealt with by changing

the question slightly, sacriﬁcing the simplicity of the question in favor

of simplicity of the answer, in this case an unequivocal “1”.

Let’s shift from the context of geometry to the context of algebra,

as we just saw we can do in this

example.1

Let’s stay in the realm of

the elementary and ask the enumerative (algebraic) question: “how

many roots does a polynomial of degree n in one variable have?”

For this problem, there is again a range of subcases, revealing a

range of potential diﬃculties. Let’s consider a polynomial of degree d

f(x) =

a0xd

+

a1xd−1

+ ··· + ad−1x + ad,

for the moment with real coeﬃcients ai, and solve f(x) = 0. Let’s

ﬁrst consider d = 1. The solution is x = −a1/a0, and there is at ﬁrst

glance exactly one solution. There is a diﬃculty, which we will see is

more than a simple issue of semantics: suppose we agree to consider

f(x) = 0 · x + a1 as a degenerate degree 1 polynomial; after all, f(x)

1This

ability to move freely from geometry to algebra and back is at the core of

the subject called algebraic geometry.