Affine Algebraic Varieties
Undergraduate Seminars: Toric Varieties
Lena Ji
February 3, 2016
Contents
1. Algebraic Sets
2. The Zariski Topology
3. Morphisms of Affine Algebraic Sets
4. Dimension
References
1
3
5
6
6
1. Algebraic Sets
Let k be an algebraically closed field, for instance the complex numbers C. We
define affine n-space over k to be the set of n-tuples of elements in k. That is,
An = {(a1 , . . . , an ) | ai 2 k}.
We denote by k[x1 , . . . , xn ] the polynomial ring in n variables with coefficients in
k. Polynomials f 2 k[x1 , . . . , xn ] can be viewed as maps from An to A by evaluating
f at each point, and so we can consider the set of zeroes of a polynomial. More
generally, for a collection of polynomials {fi }i2I , we define their zero set to be
V ({fi }i2I ) = {(a1 , . . . , an ) 2 An | fi (a1 , . . . , an ) = 0 8i 2 I}.
Definition 1.1. A subset of An of the form V ({fi }i2I ) is called an affine algebraic
set.
Note that these are referred to as affine algebraic varieties in Smith, et al.
However, we will follow Fulton and adopt the following definition.
Definition 1.2. An algebraic set V ✓ An is irreducible if, for any expression
V = V1 [ V2 where Vi are algebraic sets in An , V1 = V or V2 = V .
Definition 1.3. An affine algebraic variety is an irreducible affine algebraic set.
Example 1.4.
(1) V (xy) ✓ C2 is an affine algebraic set, but it is not irreducible. (Figure 1)
(2) V (y 2 x2 x3 ) ✓ C2 is an affine algebraic variety. (Figure 2)
Here are many more examples of affine algebraic sets. Due to artistic limitations,
pictures are over R.
Example 1.5. A point (a1 , . . . , an ) 2 An is an affine algebraic variety because
V (x1 a1 , . . . , xn an ) = {(a1 , . . . , an )}.
1
/(, , }
2
}, , )\
Figure 2. V (y 2
Figure 1. V (xy)
Figure 3. V (x2 + y 2
f
z2)
Figure 4. V (y 2
x2
x(x2
x3 )
1))
Example 1.6. A hypersurface in An is the zero set of a single nonconstant polynomial, for example the quadratic cone V (x2 + y 2 z 2 ) ✓ C2 . A hypersurface in
A2 is called an affine plane curve. The affine variety given in Figure 2 is an affine
plane curve, as is the elliptic curve V (y 2 x(x2 1)).
Example 1.7. This very nice heart is a hypersurface given by the solutions of the
9 2 3
equation (x2 + 94 y 2 + z 2 1)3 x2 z 3 80
y z = 0.
Example 1.8. The Whitney umbrella is defined by the equation x2
y 2 z = 0.
Affine Algebraic Varieties
3
Example 1.9. The torus with major radius R and minor radius r is defined by
the equation (x2 + y 2 + z 2 + R2 r2 )2 = R2 (x2 + y 2 ).
Example 1.10. The special linear group SL(n, C) = {A 2 Mn (C) | det(A) = 1}
2
is a hypersurface in Mn (C) ⇠
= Cn . This follows from the fact that the determinant
is a polynomial in n2 variables; for example when n = 3, then
0
1
a b c
det @d e f A = aei + bf g + cdh ceg bdi af h.
g h i
Example 1.11. The unit sphere S n 1 ✓ Cn is an affine algebraic variety defined
by the equation x21 + . . . + x2n = 1. However, the unit open ball (in the Euclidean
topology) and defined as the set {(a1 , . . . , an ) 2 Cn | a21 + . . . + a2n < 1} is not; if a
polynomial vanishes on an open subset of Cn in the Euclidean topology, then it is
uniformly 0.
2. The Zariski Topology
Recall that a collection T of subsets of a space X defines a topology on X if
(1) X and ; are in T ;
(2) the union of any subcollection of elements of T is contained in T ;
(3) the intersection of any finite subcollection of elements of T is in T .
We would like to use affine algebraic sets to define the closed sets of a topology
on An , and so we must check that
F
(1 ) An and ; are affine algebraic sets;
F
(2 ) the arbitrary intersection of affine algebraic sets is itself an affine algebraic
set;
F
(3 ) the finite union of affine algebraic sets is itself an affine algebraic set.
Let’s verify these! If a 2 A is nonzero, then the polynomial equation a = 0 has
no solutions, and so V (a) = ;. However, the equation 0 = 0 is satisfied by every
F
point in An , and so V (0) = An . So condition (1 ) is satisfied.
F
For (2 ), let {V↵ }↵2A be a collection
T of affine algebraic sets, where each V↵ =
V ({fi↵ }i↵ 2I↵ ). Then the intersection ↵2A V↵ is the common zero set of {fi↵ }i↵ 2I↵
over all ↵ 2 A, i.e.
!
\
[
V↵ = V
{fi↵ }i↵ 2I↵ .
↵2A
↵2A
The twisted cubic curve in Figure 5 illustrates this, as it is given by the intersection
of two surfaces: V (x2 y) \ V (x3 z) = V (x2 y, x3 z).
F
So it remains to show (3 ). By induction, it is enough to check the union of
two affine algebraic sets.
/(, , }
4
Figure 5. An intersection
f
}, , )\
Figure 6. A union
Proposition 2.1 ([3] Exercise 1.2.1). The union of two affine algebraic sets in An
is an affine algebraic set.
Proof. Let V ({fi }i2I ) and V ({gj }j2J ) be affine algebraic sets. We claim that
V ({fi }i2I ) [ V ({gj }j2J ) = V ({fi gj }(i,j)2I⇥J ).
Certainly ✓ holds, since if p = (a1 , . . . , an ) 2 V ({fi }i2I )[V ({gj }j2J ), then fi (p) =
0 for all i or gj (p) = 0 for all j. In either case, then fi gj (p) = 0 for all i and all j,
and so p 2 V ({fi gj }(i,j)2I⇥J ).
Now let q 2 V ({fi gj }(i,j)2I⇥J ) and suppose that q 62 V ({fi }i2I ) [ V ({gj }j2J ).
Then there exist i and j with fi (q) 6= 0 and gj (q) 6= 0. But this implies fi gj (q) 6= 0,
a contradiction, so q must be in V ({fi }i2I ) [ V ({gj }j2J ) and we have shown ◆. ⇤
An example of this is the union of the x-axis and yz-plane in Figure 6: V (x) [
V (y, z) = V (xy, xz).
Definition 2.2. The topology on An where the closed sets are of the form V ({fi }i2I )
is called the Zariski topology.
The Zariski topology on An is very di↵erent from the Euclidean topology. Open
subsets in this topology are very big; in fact they are dense and quasi-compact.
Additionally, two non-empty open sets will always intersect, and so the Zariski
topology is not Hausdor↵ on An for n > 0.
Example 2.3. Let k = C. Then the Zariski topology on A1 is the cofinite topology
on C — closed sets are ;, C, and finite sets — since polynomials in one variable
have finitely many roots.
Example 2.4 ([3] Exercise 1.2.2). The Zariski topology on A2 is not the product
topology on A1 ⇥ A1 . Recall that the product topology on X ⇥ X is generated by
open sets of the form U1 ⇥ U2 , where U1 , U2 are open subsets of X. So if A1 ⇥ A1 is
equipped with the product topology, where each A1 has the Zariski topology, open
sets are of the form A2 {finitely many horizontal lines [ vertical lines [ points}
The diagonal of A1 ⇥ A1 , defined A1 ⇥A1 = {(a1 , a2 ) 2 A1 ⇥ A1 | a1 = a2 },
is not closed in the product topology, where each A1 is endowed with the Zariski
topology, since A1 is not Hausdor↵. However, it is the zero set of the polynomial
x y 2 A[x, y], so A1 ⇥A1 = V (x y) ✓ A2 is closed in the Zariski topology on A2 .
Affine Algebraic Varieties
5
If V ✓ An is an affine algebraic set, then we can endow V with the subspace
topology induced by the Zariski topology on An . Then closed subsets of V are of
the form V \ W , where W ✓ An is an affine algebraic set.
3. Morphisms of Affine Algebraic Sets
Definition 3.1. Let V ✓ An and W ✓ Am be affine algebraic varieties. A morphism of algebraic varieties is a map F : V ! W given by the restriction of
a “polynomial map” An ! Am (meaning that each of the m components is given
by a polynomial in k[x1 , . . . , xn ]).
So when we compose F with the inclusion i : W ,! Am , the resulting map
will be of the form i F = (F1 , . . . , Fm ) where each Fi is the restriction to V of a
(non-unique) polynomial in k[x1 , . . . , xn ].
Definition 3.2. A morphism F : V ! W is an isomorphism if it has an inverse
morphism. In this case we say that V and W are isomorphic.
Example 3.3. Let C be the plane parabola given by the equation y
Then C is isomorphic to A1 via the maps
'|C , where ' : A2 ! A1
x2 = 0.
A1 ! C
t 7! (t, t2 ).
(x, y) 7! x
Example 3.4 ([3] Exercise 1.3.2). The twisted cubic V = V (x2
Figure 5 is isomorphic to A1 . Since
y, x3
z) in
V = {(t, t2 , t3 ) 2 A3 | t 2 A},
we can define a morphism A1 ! V by t 7! (t, t2 , t3 ). The restriction to V of the
projection A3 ! A1 onto the first factor, defined (x, y, z) 7! x, gives an inverse
morphism.
Proposition 3.5 ([3] Exercise 1.3.1). If F : V ! W is a morphism of affine
algebraic sets, then F is continuous in the Zariski topology.
Proof. Any closed subset of W is of the form V ({fi }i2I )\W where fi 2 k[x1 , . . . , xm ].
F
1
(V ({fi }i2I ) \ W ) = F
1
(V ({fi }i2I )) \ F
1
(W ) = V ({fi F̃ }i2I ) \ V
is a closed subset of V , where F̃ : An ! Am is any “polynomial map” that restricts
to F , so F is continuous.
⇤
6
/(, , }
f
}, , )\
4. Dimension
Definition 4.1. The dimension of an affine algebraic set V is the length of the
longest chain of distinct nonempty affine closed subvarieties of V
sup{d | Vd ) Vd
1
) · · · ) V0 }.
Hence the dimension of an affine algebraic set is equal to the maximum of the
dimensions of its irreducible components (maximal irreducible subsets).
Example 4.2. The quadratic cone V (x2 + y 2
z 2 ) has dimension 2.
Example 4.3. The affine algebraic set V (xy, xz) has dimension 2 = max{1, 2}.
Definition 4.4. An affine algebraic set is equidimensional if all of its irreducible
components have the same dimension.
So in the earlier examples, V (xy, xz) is not equdimensional but V (x2 + y 2
and V (xy) (Figure 1) are.
z2)
Definition 4.5. The codimension of an affine algebraic set V ✓ An is defined
codim V = n dim V .
Example 4.6. Hypersurfaces in An are precisely the affine algebraic sets of codimension 1.
References
[1] William Fulton. Algebraic Curves, 2008.
[2] Herwig Hauser. Algebraic Surfaces Gallery. http://homepage.univie.ac.at/herwig.hauser/
bildergalerie/gallery.html.
[3] Karen Smith, Lauri Kahanpää, Pekka Kekäläinen, and William Traves. An Invitation to Algebraic Geometry. Springer, 2004.