ANDREAS LEOPOLD KNUTSEN
Part 1.
Additional material to [Hart, Chapter I]
1. Morphisms
Let X ⊆ An and Y ⊆ Am be affine varieties with coordinate rings A(X) = k[y1, . . . , yn]/I(X) and A(Y) = k[x1, . . . , xm]/I(Y).
Let
X ϕ //Y
P //
h(x1)(P), . . . , h(xm)(P) and
A(X)oo h A(Y)
f ◦ϕoo f
be the morphism and ring homomorphism, respectively, corresponding to each other, as given in the proof of [Hart, Prop. 3.5]. (The xis denote the classes of the xis.)
Let now Z ⊆X and W ⊆ Y be closed subsets, with corresponding ideals I(Z) ⊆ A(X) and I(W)⊆ A(Y). (We will as usual write mP =I(P) for the maximal ideal corresponding to a point.)
Lemma 1.1. We have (a) I(ϕ(Z)) = h−1I(Z).
(b) ϕ(Z)⊆W ⇔I(W)⊆h−1I(Z).
(c) If P ∈X is a point, then Q=ϕ(P)⇔h−1mP =mQ. (d) ϕis dominant (i.e., ϕ(X) =Y) ⇔h is injective.
Proof. We have a commutative diagram X ϕ //Y
Z?
OO
ϕ|Z
//W?
OO
Date: August 30th, 2012.
1
if and only if we have a commutative diagram A(X)
A(Y)
oo h
A(X)/I(Z) A(Z)oo A(W) A(Y)/I(W).
But the latter diagram holds if and only if h(I(W)) ⊆ I(Z), which means that I(W)⊆h−1I(Z). This proves (b).
In particular, we obtain that I(ϕ(Z))⊆h−1I(Z)
To prove the opposite inclusion, whence (a), we first claim that
(1) ϕ(Z)⊆Z(h−1I(Z))
Indeed, ify∈ϕ(Z), theny=ϕ(x) for somex∈Z. Iff ∈h−1I(Z), thenh(f)∈I(Z), so that
f(y) = f(ϕ(x)) = (f◦ϕ)(x) = h(f)(x) = 0, whencey ∈Z(h−1I(Z)), proving (1).
From (1) we get
ϕ(Z)⊆Z(h−1I(Z)) whence
I(ϕ(Z))⊇I(Z(h−1I(Z))) =h−1I(Z),
by [Hart, I, Prop. 1.2(d)], because h−1I(Z) is radical as I(Z) is. Therefore, (a) is proved.
Now (c) follows from (a) since
Q=ϕ(P)⇔Q=ϕ(P)⇔mQ =I(Q) =h−1I(P) =h−1mP. Finally, (d) follows from (a) since
ϕdominant ⇔ ϕ(X) =Y ⇔h−1I(X) =I(Y)
⇔ h−1((0)) = (0)⇔hinjective.
Similarly, we have the following local descriptionof morphisms:
Lemma 1.2. ([Hart, Exc. I.3.3]) Let ϕ:X −→Y be a morphism of varieties.
(a) For each P ∈X, ϕ induces a local homomorphism 1 of local rings ϕ∗P :Oϕ(P),Y −→ OP,X.
In the case of affine varieties, this is just the localization of h above:
Oϕ(P),Y 'A(Y)mϕ(P) −→A(X)mP ' OP,X.
(b) ϕ is an isomorphism ⇔ ϕ is a homeomorphism and ϕ∗P is an isomorphism for all P ∈X.
(c) If ϕ(X) is dense in Y, then ϕ∗P is injective for all P ∈X.
1If A and B are local rings with maximal ideals mA and mB respectively, a homomorphism f :A→B is called alocal homomorphismiff−1mB=mA.
Proof. (a) We define
Oϕ(P),Y ϕ
∗
P //OP,X
hU, fi //hϕ−1U, f ◦ϕi
Note thatϕ−1U is an open neighborhood of P inX asϕ is continuous, andf ◦ϕ is regular on ϕ−1U, since ϕis a morphism.
To check that the map is well-defined (that is, that it is compatible with the equivalence relations on the rings), note that if hU, fi = hV, gi in Oϕ(P),Y, then f = g on U ∩V, whence f ◦ϕ = g ◦ϕ on ϕ−1(U ∩V) = ϕ−1U ∩ϕ−1V, so that hϕ−1U, f ◦ϕi=hϕ−1V, g◦ϕi.
The map is a ring homomorphism because
ϕ∗P(hU, fi+hV, gi) = ϕ∗P(hU∩V, f +gi) =hϕ−1(U ∩V),(f+g)◦ϕi
= hϕ−1U, f ◦ϕi+hϕ−1V, g◦ϕi
= ϕ∗P(hU, fi) +ϕ∗P(hV, gi), and similarly for multiplication.
Next, we check that the homomorphism is local. By abuse of notation, denote the maximal ideals inOϕ(P),Y andOP,X bymϕ(P) andmP, respectively. We want to show that ϕ∗P−1mP =mϕ(P). This follows from
hU, fi ∈mϕ(P) ⇔ f(ϕ(P)) = 0⇔(f◦ϕ)(P) = 0
⇔ hϕ−1U, f ◦ϕi ∈mP ⇔ϕ∗P(hU, fi)∈mP.
Clearly, ϕ∗P is the localization of the homomorphism h : A(Y) −→ A(X) defined above, using [Hart, I, Thm. 3.2(c)] and Lemma 1.1(c).
(b) The direction “⇒” is clear since we see that (ϕ−1)∗P is an inverse homomorphism toϕ∗P.
Conversely, suppose that ϕis a homeomorphism andϕ∗P is an isomorphism for all P ∈ X. Let ψ be the inverse continuous map to ϕ. To show that ψ is a morphism, we have left to prove the regularity condition.
So let U ⊆ X be an open set and f : U → k a regular function. We must show that f ◦ψ :ψ−1U →k is regular as well.
Let Q ∈ ψ−1U = ϕ(U) and let P = ψ(Q) ∈ U ⊆ X. Since f is regular by assumption, we have that hU, fi ∈ OP,X. Since ϕ∗P : OQ,Y −→ OP,X is surjective, there exists a hV, gi ∈ OQ,Y such that
hϕ−1V, g◦ϕi=ϕ∗P(hV, gi) = hU, fi.
By the definition of the equivalence relation in OP,X, this means that g◦ϕ=f onϕ−1V ∩U.
Composing (on the right) with ϕ−1 =ψ we obtain
g =f ◦ψ onϕ(ϕ−1V ∩U) =V ∩ϕ(U).
Since g is regular, we obtain that f◦ψ is regular on the neighborhood V ∩ϕ(U) of Q. Hence f◦ψ is regular on ψ−1U, as we wanted to show.
(c) Quick proof: By [Hart, Prop. 4.3], any variety has a base for the Zariski topology consisting of open affine subsets (that is, open subsets that are isomorphic to affine varieties). Since the question is local around every P ∈X, we can cover X and Y by affine subsets and reduce to the case of affine varieties. Then the result follows from the last assertion in (a) and Lemma 1.1(d), since localization preserves the injectivity.
Direct proof: Let 06=hU, fi ∈ Oϕ(P),Y. The set S ={Q∈U |f(Q)6= 0} ⊆U
is open (since S = f−1(A1 − {0}), regarding f : U → k = A1, f is continuous, by [Hart, Lemma 3.1], and A1 − {0} is open in A1). We have S 6= ∅, since we assume that hU, fi 6= 0.
Since ϕ(X) is dense in Y, we also have that ϕ(X)∩U is dense in U. Therefore we must have ϕ(X)∩S 6= ∅. This yields that (f ◦ϕ)(Q) = f(ϕ(Q))6= 0 for some ϕ(Q)∈U. Hence f◦ϕ6= 0 in a neighborhood ofQ in ϕ−1U, whence
ϕ∗P(hU, fi) = hϕ−1U, f ◦ϕi 6= 0.
Hence ϕ∗P is injective.
2. Tangent spaces and nonsingular points
As an hors d’œuvres, consider an irreducible plane affine curve (= a one-dimensional variety in A2)
C =Z(f), with 06=f ∈k[x, y], with I(C) = (f). (In particular, f is an irreducible polynomial.)
LetP = (a, b)∈C and consider the equation
(2) ∂f
∂x(a, b)(x−a) + ∂f
∂y(a, b)(y−b) = 0.
Then, we have two possibilities:
(i) (2) defines a line. This is thetangent lineof C atP. In this case we say that P is a nonsingular point.
(ii) ∂f∂x(a, b) = ∂f∂y(a, b) = 0, in which case (2) defines the whole A2. In this case we say that P is a singular point.
If we define the linear subspace given by (2) to be the tangent space of C at P, denoted by TPC, we could summarize the two properties as:
P is a nonsingular point⇐⇒dimTPC= 1.
We now move to the case of an affine variety of arbitrary dimension.
First we will need the following definition:
Definition 2.1. Letf ∈A =k[x1, . . . , xn]. Thelinear part of f at P = (a1, . . . , an) is
fP(1) :=
n
X
i=1
∂f
∂xi(P)(xi−ai).
Let now Y ⊆An be an affine variety, with
I(Y) = (f1, . . . , ft), f1, . . . , ft∈A=k[x1, . . . , xn].
LetP = (a1, . . . , an)∈Y be a point.
Definition 2.2. The tangent space ofY at P is the linear subspace ofAn defined as TPY :=∩f∈I(Y)Z(fP(1)) = Z(f1,P(1), . . . , ft,P(1)).
(Note that P ∈Y.)
Clearly, considering TPY ⊆An as a subvectorspace with origin centered at P, we have
(3) dimTPY =n−rk∂fi
∂xj(P) .
The following lemma shows that this definition of tangent space is the natural one:
indeed, it is the space spanned by all lines that are tangent to Y atP. Lemma 2.3. Let L⊆An be a line through the point P. Then
L⊆TPY ⇔f|Lhas a multiple zero at P for allf ∈I(Y).
Proof. LetL be given by the system of equations xi =ai +bit, t∈A1, where (b1, . . . , bn)∈kn is a direction vector of the line.
Set
fL(t) :=f(a1+b1t, . . . , an+bnt), for any f ∈I(Y). SinceP ∈Y, we have
fL(0) =f(a1, . . . , an) =f(P) = 0 for allf ∈I(Y).
Then
fLhas a multiple zero at P ⇔ dfL
dt (0) = 0⇔
n
X
i=1
bi∂f
∂xi(P) = 0
⇔
n
X
i=1
∂f
∂xi(P)tbi = 0 for allt ∈k
⇔
n
X
i=1
∂f
∂xi(P)(xi−ai) = 0 for all (x1, . . . , xn)∈L.
⇔ L⊆Z(fP(1)).
Hence
fL has a multiple zero at P for all f ∈I(Y) ⇔ L⊆TPY,
which is what we wanted to prove.
Continuing with the notation above, let
aP := (x1 −a1, . . . , xn−an)⊆A=k[x1, . . . , xn] denote the maximal ideal ofP and
aP ⊆A(Y) = A/I(Y) its image in A(Y). Moreover, let
mP ⊂ OP =OP,Y be the maximal ideal in the local ring OP.
We will need the following lemma,
Lemma 2.4. (cf. proof of [Hart, I, Thm. 5.1])
aP/(a2P +I(Y))'aP/aP2 'mP/m2P. Proof. The left isomorphism is immediate.
From [Hart, I, Thm. 3.2] we have
A −→A(Y)' O(Y)⊆ OP 'A(Y)aP.
(Note that mP in [Hart, I, Thm. 3.2] is in fact the ideal aP in our notation.) Under the composition of these homomorphisms, we see that aP is mapped to mP. In fact, the homomorphism from A(Y) to OP above, is nothing but the natural localization monomorphism
A(Y) //A(Y)aP
f // f1
and mP =aPA(Y)aP. From this, and considering A(Y) as a subring of A(Y)aP, we have aP =A(Y)∩mP. It therefore follows that we get a natural induced inclusion morphism
φ : aP/aP2 //mP/m2P To prove that φ is surjective, let
f
g ∈mP withf, g ∈A, f(P) = 0, g(P)6= 0.
Then
f g(P) −f
g =f 1 g(P) − 1
g
∈m2P, whence
φ f g(P)
= f
g modm2P.
The following result is basically the same as [Hart, I, Thm. 5.1] (including the proof), but we use the notion of tangent space defined above.
Theorem 2.5. Let Y ⊆ An be an affine variety and P ∈ Y a point. There is a natural isomorphism
TPY '
mP/m2P)∗ (:= Homk(mP/m2P), k)).
Proof. We use the same notation as above. In particular, P = (a1, . . . , an).
Let (kn)∗ := Homk(kn, k) and let
hx1−a1, . . . , xn−ani
be a basis for (kn)∗. (This just correponds to making a translation sendingP to the origin. We do this to treat TP(Y) as a subvector space ofAn, with origin at P.)
Define a k-linear map by
A θ // (kn)∗
f //fP(1) :=Pn i=1
∂f
∂xi(P)(xi−ai).
Since aP := (x1−a1, . . . , xn−an) and hx1−a1, . . . , xn−ani is a basis for (kn)∗, we have that
(4) aP θ|aP// //(kn)∗
is surjective. Moreover, we have that
(5) kerθ|aP =a2P.
The inclusion TPY ⊆An =kn induces a surjection
(6) (kn)∗ ////(TPY)∗
(which is nothing but the restriction of k-linear maps). Combining (4) and (6) we obtain a surjective k-linear map
(7) aP Θ // //(TPY)∗.
We now claim that
(8) ker Θ =a2P +I(Y).
Indeed, recalling that I(Y) = (f1, . . . , ft), we have g ∈ker Θ ⇔ gP(1)|T
PY = 0⇔gP(1) ∈I(TP(Y))
⇔ gP(1) =
n
X
i=1
aifi,P(1) ⇔gP(1)−
n
X
i=1
aifi,P(1) = 0
⇔ g−
n
X
i=1
aifi ∈kerθ|aP ⇔g ∈kerθ|aP +I(Y).
This, combined with (5), proves (8).
From this we get that
aP/(a2P +I(Y))'(TPY)∗,
and the theorem now follows by dualising and using Lemma 2.4.
Now recall that for any noetherian local ringAwith maximal idealmwe have that dimkm/m2 ≥dimA. If equality occurs, we say thatA is a regular local ring.
Also recall that byNakayama’s lemma,m1, . . . , mngenerate the idealmif and only if their images generatem/m2 as ak-vector space. Therefore, for any noetherian local ringA, its maximal idealmcannot be generated by fewer than dimAelements, andA is a regular local ring precisely whenm can be generated by exactly dimA elements.
These elements are then called a regular system of parameters for A.
Also note that dimOP,Y = dimY for any variety Y (by [Hart, I, Exc. 3.12].
Hence, combining (3) with Theorem 2.5 we obtain the following:
Theorem-Definition 2.6. Let Y ⊆ An be an affine variety and P ∈ Y a point.
Then dimTPY ≥dimY and the following conditions are equivalent (i) dimTPY = dimY;
(ii) OP is a regular local ring;
(iii) rk ∂fi
∂xj(P)
=n−dimY.
If these conditions are satisfied, we say that P is a nonsingular point of Y. Oth- erwise, P is a singularpoint of Y.
To extend the notion of singular and nonsingular point to any variety, we first note that condition (ii) makes sense for any variety (and does not depend on the isomorphis class, that is, it is intrinsic).
Using Theorem 2.5, we can also make the notion of tangent space intrinsic:
Definition 2.7. ([Hart, I, Exc. 5.10(b)]) Let X be a variety and P ∈X. Let OP,X be the local ring of P on X with maximal idealmP.
We define theZariski tangent spaceTP(X)ofX atP to be the dualk-vector space of mP/m2P, that is
TPX := Homk(mP/m2P, k).
Note that the Zariski tangent space and the tangent space are denoted by the same symbols, but this does not matter, by Theorem 2.5.
TakingTPY as the Zariski tangent space in Theorem-Definition 2.6, also condition (i) becomes intrinsic. We therefore have:
Theorem-Definition 2.8. LetY be any variety andP ∈Y a point. ThendimTPY ≥ dimY and the following conditions are equivalent
(i) dimTPY = dimY;
(ii) OP is a regular local ring;
If these conditions are satisfied, we say that P is a nonsingular point of Y. Oth- erwise, P is a singularpoint of Y.
(Also see [Hart, Exc. I.5.10(a)].)
As mentioned above, if P is a nonsingular point, then mP ⊆ OP is generated by dimY elements x1, . . . , xdimY, called a regular system of parameters. These corre- spond to local coordinates around P.
Finally, we note that for any morphism between varieties, there is a natural induced map of tangent spaces, as we are used to in differential geometry:
Lemma 2.9. ([Hart, Exc. I.5.10(b)]) Let ϕ:X −→Y be a morphism of varieties.
Then there is a natural induced k-linear map
dϕP :TPX −→Tϕ(P)Y.
Proof. Since
ϕ∗P :Oϕ(P),Y −→ OP,X
is a local homomorphism of local rings by Lemma 1.2(a) (that is,ϕ∗P−1mP =mϕ(P)), we have an induced k-linear map
(9) mϕ(P)/m2ϕ(P) −→mP/m2P
Dualizing, that is, applying the contravariant functor Homk(−, k) to the last map, we obtain an induced k-linear map
TP(X) := Homk(mP/m2P, k)−→Homk(mϕ(P)/m2ϕ(P)) =:Tϕ(P)(Y).
Definition 2.10. The mapdϕP, sometimes denoted bydPϕ, is called thedifferential of ϕat P.
We conclude with an example:
Example 2.11. (The cubic cuspidal curve) Let Y ⊂ A2 be the (irreducible) curve given by y2 =x3. Let f(x, y) =x3−y2. Then
∂f
∂x(a, b) = 3a2 and ∂f
∂y(a, b) = −2b.
The tangent space at the pointP = (a, b) of Y is defined by the single equation 3a2(x−a)−2b(y−b) = 0
(where 3a = 0 if chark= 3 and 2b = 0 if chark = 2) which has dimension one (and is a line) if and only if (a, b)6= (0,0). Hence
(10) P ∈C is nonsingular⇐⇒P 6= (0,0), by the criterion (i) (or (iii)) in Theorem-Definition 2.6.
Let us now prove (10) by checking condition (ii) in Theorem-Definition 2.6.
We have
A(Y) = k[x, y]/(x3−y2) andaP = (x−a, y−b).
Moreover,
a2P = ((x−a)2,(x−a)(y−b),(y−b)2).
Using Lemma 2.4, we have
m(0,0)/m2(0,0) 'a(0,0)/(a2(0,0)+I(Y)) = (x, y)/(x2, xy, y2, x3−y2) = (x, y),
wherexandyare the residue classes ofxandyrespectively. Hence, dimkm(0,0)/m2(0,0) = 2, showing that (0,0) is a singular point.
In the case P = (a, b) 6= (0,0) we can perform a linear change of coordinates sending P to the origin, to obtain
m(a,b)/m2(a,b) ' a(a,b)/(a2(a,b)+I(Y)) = (x, y)/(x2, xy, y2,(x−a)3−(y−b)2)
= (x, y)/(x2, xy, y2,−3a2x+ 2by−a3−b2)
=
(x) = (y) if chark 6= 2,3;
(x) if chark = 3;
(y) if chark = 2;
showing that dimkm(a,b)/m2(a,b) = 1. So this proves again (10).
Consider now the map
A1
ϕ //Y ⊆A2
t //(t2, t3).
This is a morphism by [Hart, Lemma 3.6]. The corresponding ring homomorphism on the rings of regular functions is given by
A(Y) =k[x, y]/(x3−y2) h // k[t] =A(A1)
x //t2
y // t3,
wherexandyare the residue classes ofxandyrespectively (from the proof of [Hart, Prop. 3.5] or the beginning of Section 1).
LetQ= 0 ∈A1 (whence ϕ(Q) = (0,0)). We now prove that the differential (11) dϕQ :A1 'TQA1 −→Tϕ(Q)Y 'A2
in Lemma 2.9 is the zero map. Indeed, by the proof of Lemma 2.9, this map is induced by the homomorphism of local rings
ϕ∗Q :Oϕ(Q),Y −→ OQ,X in Lemma 1.2, which is nothing but the localization
A(Y)(x,y)=
k[x, y]/(x3−y2)
(x,y) −→k[t](t) =A(A1)(t)
of the homomorphism h. Using the isomorphisms in Lemma 2.4, we get that (11) is the dual of the natural map
(12) (x, y)/
(x, y)2+I(Y)
−→(t)/(t)2,
induced by h. But this sends the residue classes ofx andy tot2 and t3, respectively, which are zero in (t)/(t)2. Hence (12) is the zero map, whence (11) is also the zero map.
3. Products of varieties
3.1. Products of affine varieties. If p= (a1, . . . , an)∈An and q = (b1, . . . , bm)∈ Am are points we can define the point
(p, q) := (a1, . . . , an, b1, . . . , bm)∈An+m,
so that we can identify An+m as the set of pairs (p, q) with p∈An and q∈Am. Let now X ⊆ An and Y ⊆ Am be algebraic sets. Let A(An) = k[x1, . . . , xn] and A(Am) = k[y1, . . . , ym] and let I(X) = (f1, . . . , fs) with fi ∈ k[x1, . . . , xn] for i= 1, . . . , s and I(Y) = (g1, . . . , gt) with gj ∈k[y1, . . . , ym] forj = 1, . . . , t.
We define the product of X and Y to be
X×Y :={(p, q)∈An+m |p∈X andq ∈Y} ⊆An+m. This is again an algebraic set: Indeed, it is easy to see that
X×Y =Z(f1, . . . , fs, g1, . . . , gt),
where we consider thefi’s andgj’s as polynomials inA[An+m] =k[x1, . . . , xn, y1, . . . , ym].
In other words,
I(X×Y) = (f1, . . . , fs, g1, . . . , gt) = I(X)k[y1, . . . , ym] +I(Y)k[x1, . . . , xn].
Proposition 3.1. ([Hart, Exc. I.3.15]) LetX ⊆An andY ⊆Am be affine varieties.
Then
(a) X×Y ⊆An+m (with its induced topology) is also an affine variety.
(b) A(X×Y)'A(X)⊗kA(Y)
(c) X×Y is a product in the category of varieties.
(d) dim(X×Y) = dimX+ dimY.
Remark 3.2. Note that (c) means that (i) the natural projections πX :X×Y −→
X and πY ×Y −→ Y are morphisms, and (ii) given a variety Z with morphisms α : Z −→ X and β : Z −→ Y, there exists a unique morphism γ : Z −→ X ×Y making a commutative diagram:
(13) Z
α
γ
β 5555555555555555
X×Y
πX
{{wwwwwwwww
πY
##G
GG GG GG GG
X Y.
(Often, the morphism γ is denoted by α×β.)
Proof. (a) Assume that X × Y = Z1 ∪ Z2, a union of two closed subsets. Let Zi =Z(hi1, . . . , hiui) with each hij ∈ k[x1, . . . , xn, y1, . . . , ym] for i = 1,2, and u1, u2 positive integers. Define
Xi :={x∈X |x×Y ⊆Zi} ⊆X, for i= 1,2.
We claim that X =X1 ∪X2. Indeed, assume to get a contradiction that there is a pointp0 = (a1, . . . , an)∈X−(X1∪X2). Asp0×Y ⊆X×Y we must have
Y =Y1∪Y2, whereYi :={y∈Y |(p0, y)∈Zi}.
Defining the polynomials
hij,p0 :=hij(a1, . . . , an, y1, . . . ym)∈k[y1, . . . , ym] we see that
Yi =Z({hij,p0}),
so that each Yi is closed. The irreducibility of Y implies that Y =Y1 or Y =Y2, say Y =Y1. Hence, for all y ∈ Y, we have (p0, y)∈ Z1, but this means that p0 ∈ X1, a contradiction.
So we have proved thatX =X1∪X2. Moreover, defining for eachy= (b1, . . . , bm)∈ Y, the polynomials
hy,ij :=hij(x1, . . . , xn, b1, . . . , bm)∈k[x1, . . . , xn] we see that
Xi =Z({hy,ij}y∈Y)
so that each Xi is closed. Therefore, X =X1 orX =X2 by the irreducibility of X, so that X×Y =Z1 or Z2. Therefore X×Y is also an affine variety.
(b) Define a homomorphism
A(X)⊗kA(Y) ϕ //A(X×Y) = k[x(f1,...,xn,y1,...,ym]
1,...,fs,g1,...,gt)
by the condition
ϕ X
i
Fi⊗Gi
(x, y) = X
i
Fi(x)Gi(y).
The map is clearly surjective, as the residue classes of the coordinate functions xi: k[x1, . . . , xn] //A(X) = k[x1, . . . , xn]/(f1, . . . , fs)
xi //xi
lie in the image ofϕ, and similarly also the residue classes of the coordinate functions yj, and these functions generate A(X×Y).
The map is also injective. Indeed, without loss of generality assume that the {Gi} are linearly independent, and that
ϕ X
i
Fi⊗Gi
(x, y) =X
i
Fi(x)Gi(y) = 0.
Then, for any fixed (a1, . . . , an)∈X ⊆An, we would have X
i
Fi(a1, . . . , an)Gi(y) = 0
showing that Fi(a1, . . . , an) = 0 for all i, whence Fi = 0 in A(X).
(c) Consider the homomorphism of k-algebras
A(X)id⊗k1//A(X)⊗kA(Y).
By [Hart, I, Prop. 3.5] and (b) this corresponds to a morphism of varieties X×Y // X,
(x, y) //x
which is the projectionπX. HenceπX andπY in the remark are morphisms. Denoting by hα and hβ the homomorphisms of k-algebras corresponding to α and β given by [Hart, I, Prop. 3.5], the commutative diagram corresponding to (13) is
O(Z)
A(X)⊗kA(Y).
hγ
OO
A(X)
hα
??
id⊗k1
77o
oo oo oo oo oo
A(Y),
hβ
__??????
??????
???????
1⊗kid
ggOOOOOOOOOOO
where we have to prove the existence and uniqueness of hγ. But this now follows from the corresponding universal property of the tensor product.
(d) Assume that dimX = q and dimY = r. The chains of irreducible, closed subsets of X, resp. Y, in the definition of dimension can be extended by [Hart, I, Thm. 1.8] to chains of irreducible, closed subsets of An, resp. Am:
X0 $X1 $· · ·$Xq =X $Xq+1 $· · ·$Xn =An, Y0 $Y1 $· · ·$Yr =Y $Yq+1 $· · ·$Ym =Am. We therefore obtain a chain of irreducible, closed subsets
X0×Y0 $X0×Y1 $· · ·$X0×Yr $X1×Yr $· · ·$Xq×Yr
$Xq×Yr+1 $· · ·$Xq×Ym
$Xq+1×Ym $· · ·$Xn×Ym =An×Am =An+m.
This chain is maximal, since dimAn+m = n+m. But then also the first line of the chain must be maximal, so that dim(X×Y) =q+r= dimX+ dimY.
It is also easy to see that ifX ⊆Anand Y ⊆Am are quasi-affinevarieties, then so is alsoX×Y ⊆An+m. Indeed, writeX =X1\X0 andY =Y1\Y0, withX0, X1 ⊆An and Y0, Y1 ⊆Am closed. Then one easily sees that
X×Y = (X1×Y1)\
(X1×Y0)∪(X0×Y1) .
3.2. Products of quasi-projective varieties and the Segre embedding. The definition of the product of affine varieties (and quasi-affine varieties) in the previous section was very natural. For quasi-projective varieties the situation is a bit more complicated. The problem is of course that, whereas there is a natural identification An×Am = An+m as sets, which allows us to give An×Am a natural structure of affine variety, there is no identification ofPn×Pm with Pn+m due to the equivalence relations defining projective spaces. The trick is to identify the set Pn×Pm with a subset in a projective space bigger than Pn+m and show that this subset is in fact a projective variety.
To study this situation, we consider the map Pn×Pm
ψ //PN, N =nm+n+m
(a0 :· · ·:an)×(b0 :· · ·:bm) //(· · ·:aibj :· · ·) (in lexicographic order) called the Segre embedding, after the Italian mathematician Beniamino Segre 2.
We will also need the homomorphism of k-algebras given by T =k[{zij}] θ //S =k[x0, . . . , xn, y0, . . . , ym]
zij //xiyj.
Lemma 3.3. (cf. [Hart, Exc. I.2.14]) (a) ψ is well-defined and injective.
(b) The image of ψ is a (closed) subvariety of PN; more precisely,
imψ =Z(kerθ) = Z({zijzkl−zkjzil}; 0≤i, k≤n and0≤j, l ≤m)⊆PN. (c) The Segre embedding has the followinglocal property: for any pointsx∈Pnand y ∈ Pm there exist open affine neighborhoods U and V of x and y respectively, such thatψ(U×V)is open inψ(Pn×Pm)andψ gives an isomorphism between the product U ×V as affine varieties defined above to the subvariety ψ(U ×V)⊆ψ(Pn×Pm).
Proof. (a) ψ is obviously independent of multiplying the coordinates of the points in Pn and Pm by a common scalar, so the map is well-defined. Moreover, if
ψ((a0 :· · ·:an)×(b0 :· · ·:bm)) = ψ((a00 :· · ·:a0n)×(b00 :· · ·:b0m)),
2Beniamino Segre (1903-1977) was an Italian mathematician who is remembered today as a major contributor to algebraic geometry and one of the founders of combinatorial geometry. He was born and studied in Turin. His main contributions to algebraic geometry concerned birational invariants of algebraic varieties, and the investigation of singularities. (From Wikipedia)
then, picking any bj0 6= 0, we get ai =a0i· b0j
0
bj0 for all i= 0, . . . , n,
so that (a0 : · · · : an) = (a00 : · · · : a0n). Similarly, (b0 : · · · : bm) = (b00 : · · · : b0m), so that ψ is injective.
(b) First note that if f ∈kerθ, then
f(ψ((a0 :· · ·:an)×(b0 :· · ·:bm)) = f(· · ·:aibj :· · ·) = 0
= θ(f)((a0 :· · ·:an)×(b0 :· · ·:bm)).
In particular, imψ ⊆Z(kerθ).
For any 0≤i, k ≤n and 0≤j, l ≤m,
θ(zijzkl−zkjzil) =xiyjxkyl−xkyjxiyl= 0, so that
({zijzkl−zkjzil})⊆kerθ ⊆T =k[{zij}], whence
Z(kerθ)⊆Z({zijzkl−zkjzil}).
What we have left to prove of (b) is therefore that (14) Z({zijzkl−zkjzil})⊆imψ
Let P ∈ Z({zijzkl −zkjzil}) and write P = (. . . : tij : . . .). At least one of the homogeneous coordinates of P is not zero, say tk0l0 6= 0 for some 0 ≤ k0 ≤ n and 0≤l0 ≤m. Define ai := ttil0
k0l0
for every 0≤i≤n and bj = ttk0j
k0l0
for every 0≤j ≤m.
Then, as P ∈Z({zijzkl−zkjzil}), we have
tijtk0l0 −tk0jtil0 = 0, which gives
tij
tk0l0 = til0 tk0l0 · tk0j
tk0l0 =aibj. But this means that
P =ψ((a0 :· · ·:an)×(b0 :· · ·:bm)), proving (14).
(c) Let the xi’s, the yj’s and the zij’s denote the homogeneous coordinates of Pn, Pm and PN, respectively. Assume for simplicity that x ∈ An0 := D(x0) and y∈Am0 :=D(y0). Define
AN00 :={(· · ·:cij :· · ·)∈PN |c00 6= 0} ⊆PN which is open. Then clearly
ψ(An0 ×Am0 ) = imψ∩AN00,
which is open in imψ =ψ(Pn×Pm). Moreover,ψ restricted to An0 ×Am0 is nothing but
An0 ×Am0 //AN00⊆PN,
(1, a1, . . . , an)×(1, b1, . . . , bm) //(a1, . . . , an, b1, . . . , bm, . . . , aibj, . . .), with i, j >0, so thatψ(An0 ×Am0 )'An×Am, the product of affine varieties defined
above.
We can now use the Segre embedding and identifyPn×Pm with its imageψ(Pn× Pm)⊆PN, whence give it a structure of projective variety. Moreover, for any subsets X ⊂Pn and Y ⊂Pm we identify X×Y with ψ(X×Y). Then we have
Proposition 3.4. ([Hart, Exc. I.3.16]) Assume that X ⊂ Pn and Y ⊂ Pm are quasi-projective varieties and consider X×Y ⊆Pn×Pm. Then
(a) X×Y is a quasi-projective variety.
(b) X×Y is a projective variety if both X and Y are projective.
(c) X×Y is a product in the category of varieties.
(d) dim(X×Y) = dimX+ dimY.
Proof. All of these properties can be checked locally, for instance by taking the open covering of ψ(Pn×Pm)⊆PN given by the various
ANij :={(· · ·:cij :· · ·)∈PN |cij 6= 0} ⊆PN, as in the proof of Lemma 3.3(c).
Then (a)-(d) follow from Proposition 3.1 and Lemma 3.3(c).
Remark 3.5. There is a subtle point in proving (c), though: one must convince oneself that all the morphismsγ as in Remark 3.2 constructed locally (and uniquely) on each open set glue together on the open sets in a compatible way and glue to a morphism. The fact that they glue toether in a compatible way (that is, coincide on all intersections) follows from the uniqueness property shown in Proposition 3.1(c) (by definition) and then the properties of a morphism (continuity and preservation of regular functions) can be checked locally. The precise concept of glueing morphisms is given in Step 3 in the proof of [Hart, II, Thm. 3.3].
We have now given meaning to the affine variety An×Am and to the projective varietyPn×Pm. Similarly, considering an embeddingAm $ Pm as one of the standard affine open subsets, we have also defined the product Pn×Am ⊂ Pn×Pm, which is a quasi-affine variety by Proposition 3.4(a).
Finally, we would like to determine explicitly which are the closed subsets ofPn×Pm and Pn×Am. As above we let x0, . . . , xn denote the homogeneous coordinates ofPn and y0, . . . , ym (respectively, y1, . . . , ym) denote the homogeneous coordinates of Pm (respectively of Am).
Proposition 3.6. A subset X ⊆ Pn×Pm (respectively, X ⊆ Pn×Am) is closed if and only if it is given by a system of polynomial equations
gk(x0, . . . , xn, y0, . . . , ym) = 0, fork = 1, . . . , t, that is bihomogeneous in the xi’s and yj’s (respectively,
gk(x0, . . . , xn, y1, . . . , ym) = 0, fork = 1, . . . , t, that is homogeneous in the xi’s).
Proof. We only prove the part concerning X ⊆Pn×Pm.
A closed subset V ⊆PN is defined by homogeneous polynomial equations fk(z00, . . . , znm) = 0,
denoting the homogeneous coordinates ofPN by zij as before. Making the substitu- tion zij =xiyj we can write these in the coordinates xi and yj as equations
gk(x0, . . . , xn, y0, . . . , ym) = 0,
that are bihomogeneous in the xi’s and yj’s, that is, homogeneous in each set of the variables x0, . . . , xn and y0, . . . , ym, and of the same degree in both.
Conversely, it is easy to see that a polynomial with these two homogeneity proper- ties can be written as a polynomial in the productsxiyj, thus defining a closed set in Pn×Pm (by writing the corrisponding polynomials in the zij’s). However, equations that are bihomogeneous in thexi’s andyj’s always define a closed subset ofPn×Pm even if the degrees of homogeneity in the two sets of variables are different. Indeed, if g(x0, . . . , xn, y0, . . . , ym) has degree r in the xi’s and s in the yj’s, say with r > s, then g = 0 is equivalent to the system of equations
yr−si g = 0 fori= 0, . . . m,
and we have just proved that such equations define a closed subset of Pn×Pm. A polynomial f ∈k[x0, . . . , xn, y0, . . . , ym] is said to be bihomogeneous of bidegree (r, s) if it is homogeneous of degree r in the variables x0, . . . , xn and homogeneous of degree s in the variables y0, . . . , ym. As above one can prove that the regular functions on a subvariety X ⊆ Pn×Pm are given by functions that can locally be written as quotients of polynomials that arebihomogeneous of the same bidegreeand such that the denominator does not vanish.
The definition of products of varieties can easily be extended to a product of any number of varieties.
We conclude this section with a classical example:
Example 3.7. The quadric surface in P3 ([Hart, Exc. I.2.15])
Let w, x, y, z be the homogeneous coordinates of P3. By Lemma 3.3, the Segre embedding Q:=ψ(P1×P1)⊆P3 is given by the equation xy−zw= 0.
Let nowt= (a0 :a1)∈P1. Then one sees thatt×P1 ⊆Qis given by the equations a1w=a0yand a1x=a0z,
so it is a line in P3.
Therefore,Q contains the family of lines
{Lt:=t×P1}t∈P1, and likewise the family of lines
{Mu :=P1×u}u∈P1
Clearly, two distinct lines in the same family do not intersect, whereas Lt∩Mu = (t, u), one point.
This surface is the simplest example of a ruled surface ([Hart, V, §2]). In fact it has two rulings given by each of the two families of lines above.
Since the surfaceQ contains other curves besides these lines, like for instance the curves defined by
xy−zw=x−y = 0,
we see that the induced topology on Q ' P1 ×P1 via the Segre embedding in P3 is not the product topology.
4. The Veronese (or d-uple) embedding
Letnanddbe positive integers. The homogeneous polynomials of degreedinn+1 variables x0, . . . , xn form a vector space of dimension n+dn
. This space is generated by the monomials
Mi0,...,in :=xi00· · ·xinn,
where i0, . . . , in are nonnegative integers such that i0+· · ·+in =d.
Consider the map
Pn
ρn,d //PN, N = n+dn
−1
a= (a0 :· · ·:an) // (· · ·:Mi0,...,in(a) :· · ·)
called the Veronese embedding, after the Italian mathematician Giuseppe Veronese
3. The map is easily seen to be well-defined and injective.
Denote the homogeneous coordinates of PN by vi0,...,in and of Pn by x0, . . . , xn as usual.
We will need the homomorphism of k-algebras given by T =k[{vi0,...,in}] θ //S =k[x0, . . . , xn]
vi0,...,in //Mi0,...,in :=xi00· · ·xinn. Note that kerθ is a prime ideal, as θ induces a monomorphism
T /kerθ //S
3Giuseppe Veronese (1854 - 1917) was an Italian mathematician. He was born in Chioggia, near Venice. His most famous monograph isFondamenti di geometria a pi`u dimensioni e a pi`u specie di unit`a rettilinee esposti in forma elementare(1891).
and S in an integral domain. It is also easy to see that if θ maps a polynomial to zero, then it must map every homogeneous component of the polynomial to zero. So kerθ is a homogeneous prime ideal, whence
Z(kerθ)⊆PN is a projective variety.
Proposition 4.1. ([Hart, Exc. I.2.12, I.3.4]) ρn,d is an isomorphism from Pn onto the projective variety Z(kerθ)⊆PN
Proof. We will first prove that
(15) imρn,d=Z(kerθ).
Iff ∈kerθ is homogeneous and P = (· · ·:Mi0,...,in(a) :· · ·)∈imρn,d, then f(P) =f(· · ·:Mi0,...,in(a) :· · ·) = 0.
This proves that imρn,d⊆Z(kerθ).
To prove the opposite inclusion, first note that θ
(vi0,...,in)d
=
xi00· · ·xinnd
=xdi00· · ·xdinn
= θ
(vd,0,...,0)i0 ·(v0,d,...,0)i1· · ·(v0,...,0,d)in , whence
vdi0,...,in −(vd,0,...,0)i0 ·(v0,d,...,0)i1· · ·(v0,...,0,d)in ∈kerθ.
Therefore, if
P = (. . .:bi0,...,in :. . .)∈Z(kerθ), we have for each homogeneous coordinate
(16) bdi
0,...,in = (bd,0,...,0)i0 ·(b0,d,...,0)i1· · ·(b0,...,0,d)in, so that at least one of the b0,...,d,...,0 is nonzero.
Assume now for simplicity that bd,0,...,0 6= 0. Then one computes that ρn,d(bd,0,...,0 :bd−1,1,...,0 :· · ·:bd−1,0,...,0,1) =P.
Indeed, this follows from the fact that θ
(vd,0,...,0)i0 ·(vd−1,1,...,0)i1 · · · (vd−1,0,...,0,1)in
= (xd0)i0 ·(xd−10 x1)i1· · ·(xd−10 xn)in
= xd(i0 0+···+in)−i1−···−inxi11· · ·xinn
= xi00 ·xi11· · ·xinn ·xd(d−1)0
= θ
vi0,...,in·(vd,0,...,0)d−1 , so that
(vd,0,...,0)i0 ·(vd−1,1,...,0)i1· · ·(vd−1,0,...,0,1)in −vi0,...,in ·(vd,0,...,0)d−1 ∈kerθ,
