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.
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