1. Warming up to Enumerative Geometry 9

any of its factors. Sometimes we will want to consider multiplicities

and then we have to put extra powers back in.

In particular we can state an easy converse of Theorem 1.2 as say-

ing that any set Z ⊂

P1

consisting of d points (including multiplic-

ity) is the zero locus of a degree d homogeneous polynomial f(x0, x1),

unique up to a scalar multiple. If we have points (ai, bi) ∈

P1

with

multiplicities ri and

∑

i

ri = d, we can take f to be the degree d

polynomial

f(x0, x1) =

i

(bix0 −

aix1)ri

.

Just as for

P1,

we have a notion of the dehomogenization of

F (x0, . . . , xn):

(5) f(x1, . . . , xn) = F (ψ0(x1, . . . , xn)) = F (1, x1, . . . , xn)

and a notion of the homogenization of f(x1, . . . , xn):

(6) F (x0, . . . , xn) = x0f

d

x1

x0

, . . . ,

xn

x0

,

where d is the degree of f(x1, . . . , xn), the maximum total degree of

any term in f.

Compare with [Hulek, Sections 2.1, 2.2], a text containing an in-

troduction to algebraic geometry with minimal prerequisites. [Reid]

and [Fulton3] are other algebraic geometry texts readily accessible

to undergraduates. For graduate level algebraic geometry, see, e.g.,

[Miranda] for complex algebraic geometry in dimension 1, [GH] for

complex algebraic geometry more generally, or see either [Harris] or

[Hartshorne] for algebraic geometry over arbitrary ﬁelds.

By a line in

P2,

we mean the hypersurface deﬁned by a homoge-

neous polynomial of degree 1 in (x0, x1, x2) (degree 1 hypersurfaces in

Pn

are more generally called hyperplanes). In other words, the poly-

nomial ax + by = c in

C2

can be homogenized to get ax1 + bx2 = cx0,

and there is a similar homogenization for dx+ey = f (note that (x, y)

has been replaced by (x1, x2) here).

Now lines that were parallel in the plane meet in

P2!

Consider

for instance the lines x + y = 1 and x + y = 2. Their homogenizations

are x1 + x2 = x0 and x1 + x2 = 2x0. Subtracting these equations,

any of its factors. Sometimes we will want to consider multiplicities

and then we have to put extra powers back in.

In particular we can state an easy converse of Theorem 1.2 as say-

ing that any set Z ⊂

P1

consisting of d points (including multiplic-

ity) is the zero locus of a degree d homogeneous polynomial f(x0, x1),

unique up to a scalar multiple. If we have points (ai, bi) ∈

P1

with

multiplicities ri and

∑

i

ri = d, we can take f to be the degree d

polynomial

f(x0, x1) =

i

(bix0 −

aix1)ri

.

Just as for

P1,

we have a notion of the dehomogenization of

F (x0, . . . , xn):

(5) f(x1, . . . , xn) = F (ψ0(x1, . . . , xn)) = F (1, x1, . . . , xn)

and a notion of the homogenization of f(x1, . . . , xn):

(6) F (x0, . . . , xn) = x0f

d

x1

x0

, . . . ,

xn

x0

,

where d is the degree of f(x1, . . . , xn), the maximum total degree of

any term in f.

Compare with [Hulek, Sections 2.1, 2.2], a text containing an in-

troduction to algebraic geometry with minimal prerequisites. [Reid]

and [Fulton3] are other algebraic geometry texts readily accessible

to undergraduates. For graduate level algebraic geometry, see, e.g.,

[Miranda] for complex algebraic geometry in dimension 1, [GH] for

complex algebraic geometry more generally, or see either [Harris] or

[Hartshorne] for algebraic geometry over arbitrary ﬁelds.

By a line in

P2,

we mean the hypersurface deﬁned by a homoge-

neous polynomial of degree 1 in (x0, x1, x2) (degree 1 hypersurfaces in

Pn

are more generally called hyperplanes). In other words, the poly-

nomial ax + by = c in

C2

can be homogenized to get ax1 + bx2 = cx0,

and there is a similar homogenization for dx+ey = f (note that (x, y)

has been replaced by (x1, x2) here).

Now lines that were parallel in the plane meet in

P2!

Consider

for instance the lines x + y = 1 and x + y = 2. Their homogenizations

are x1 + x2 = x0 and x1 + x2 = 2x0. Subtracting these equations,