Table Of ContentTANGENTS AND SECANTS OF ALGEBRAIC VARIETIES
F. L. Zak
Central Economics Mathematical Institute
of the Russian Academy of Sciences
CONTENTS
Index of Notations iii
Introduction 1
Chapter I. Theorem on Tangencies and Gauss Maps 14
1 Theorem on tangencies and its applications 15
2 Gauss maps of projective varieties 21
3 Subvarieties of complex tori 28
Chapter II. Projections of Algebraic Varieties 35
1 A criterion for existence of good projections 36
2 Hartshorne’s conjecture on linear normality and its relative
analogues 41
ChapterIII. VarietiesofSmallCodimensionCorrespon-
ding to Orbits of Algebraic Groups 47
1 Orbits of algebraic groups, null-forms and secant varieties 48
2 HV-varieties of small codimension 54
3 HV-varieties as birational images of projective spaces 64
Chapter IV. Severi Varieties 69
1 Reduction to the nonsingular case 70
2 Quadrics on Severi varieties 73
3 Dimension of Severi varieties 79
4 Classification theorems 84
5 Varieties of codegree three 90
Chapter V. Linear Systems of Hyperplane Sections on
Varieties of Small Codimension 101
1 Higher secant varieties 102
2 Maximal embeddings of varieties of small codimension 109
Chapter VI. Scorza Varieties 116
1 Properties of Scorza varieties 117
2 Scorza varieties with δ =1 122
3 Scorza varieties with δ =2 126
4 Scorza varieties with δ =4 131
5 End of classification of Scorza varieties 145
Bibliography 149
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ii
CHAPTER I
THEOREM ON TANGENCIES AND GAUSS MAPS
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14
1. THEOREM ON TANGENCIES AND ITS APPLICATIONS 15
1. Theorem on tangencies and its applications
Let Xn ⊂ PN be an irreducible nondegenerate (i.e. not contained in a hyper-
plane)n-dimensionalprojectivevarietyoveranalgebraicallyclosedfieldK,andlet
Yr ⊂PN be a non-empty irreducible r-dimensional variety. We set
(cid:169) (cid:175) (cid:170)
∆ =(Y ×X)∩∆ = (y,x)∈Y ×X (cid:175)x=y ,
Y X
where ∆ is the diagonal in X×X,
X
(cid:169) (cid:175) (cid:170)
S0 ⊂(Y ×X\∆ )×PN, S0 = (y,x,z)(cid:175)z ∈(cid:104)x,y(cid:105) ,
Y,X Y Y,X
where (cid:104)x,y(cid:105) denotes the chord joining x with y. We denote by S the closure of
Y,X
S0 inY ×X×PN,bypY theprojectionofS ontotheithfactorofY ×X×PN
Y,X i Y,X
(i=1,2), and by ϕY:S →PN the projection onto the third factor, and put
Y,X
pY =pY ×pY:S →Y ×X, S(Y,X)=ϕY(S ),
12 1 2 Y,X Y,X
(cid:161) (cid:162) (cid:175) (cid:161) (cid:162)
T(cid:48) = pY −1(∆ ), ψY =ϕY(cid:175) , T(cid:48)(Y,X)=ψY T(cid:48) .
Y,X 12 Y T(cid:48) Y,X
Y,X
1.1. Definition. The variety S(Y,X) is called the join of Y and X or, if
Y ⊂X, the secant variety of X with respect to Y.
We observe that in the case when Y = X the above definition reduces to the
usual definition of secant variety of X; we shall denote S(X,X) simply by SX.
In what follows we shall assume that Yr ⊂Xn is a subvariety of X.
1.2. Definition. ThevarietyT(cid:48)(Y,X)iscalledthevarietyof(relative)tangent
stars of X with respect to the subvariety Y.
We observe that T(cid:48)(X,X) = T(cid:48)X is the usual variety of tangent stars (cf. [45;
97]).
(cid:179)(cid:161) (cid:162) (cid:180)
1.3. Definition. The cone T(cid:48) = ψY pY −1(y×y) is called the (pro-
Y,X,y 12
jective) tangent star to X with respect to Y ⊂X at a point y ∈Y.
From this definition it is evident that T(cid:48) is a union of limits of chords
Y,X,y
(cid:104)y(cid:48),x(cid:48)(cid:105), y(cid:48) ∈Y, x(cid:48) ∈X, y(cid:48),x(cid:48) →y.
ItisalsoclearthatT(cid:48) ⊂T(cid:48) ⊂T ,whereT(cid:48) =T(cid:48) isthe(projective)
Y,X,y X,y X,y X,y X,X,y
tangent star to X at y (cf. [45; 97]) and T is the (embedded) tangent space to
X,y
X at y. On the other hand, T(cid:48) ⊃T(cid:48) , where T(cid:48) =T(cid:48) is the (projective)
Y,X,y y,X y,X y,X,y
tangent cone to X at the point y.
By definition, T(cid:48)(Y,X) = ∪ T(cid:48) . If X is nonsingular along Y, i.e. Y ∩
Y,X,y
y∈Y
SingX = ∅ and Y ⊂ SmX = X \SingX, then T(cid:48)(Y,X) = T(Y,X) = ∪ T is
X,y
y∈Y
the usual variety of tangents.
16 I. THEOREM ON TANGENCIES AND GAUSS MAPS
1.4. Theorem. An arbitrary irreducible subvariety Yr⊂ Xn, r ≥ 0 satisfies
one of the following two conditions:
a) dimT(cid:48)(Y,X)=r+n, dimS(Y,X)=r+n+1;
b) T(cid:48)(Y,X)=S(Y,X).
Proof. Lett=dimT(cid:48)(Y,X). Itisclearthatt≤r+n. Inthecasewhent=r+n
the theorem is obvious since S(Y,X) is an irreducible variety, S(Y,X)⊃T(cid:48)(Y,X)
and dimS(Y,X)≤r+n+1.
Suppose that t<r+n, and let LN−t−1 be a linear subspace of PN such that
L∩T(cid:48)(Y,X)=∅. (1.4.1)
We denote by π:PN \L→Pt the projection with center at L and put X(cid:48) =π(X),
(cid:175)
Y(cid:48) = π(Y). Since π(cid:175) is a finite morphism, we have dim(Y(cid:48)×X(cid:48)) = r+n > t,
X
and from the connectedness theorem of Fulton and Hansen (cf. [26] and [27, 3.1])
it follows that
(cid:161) (cid:175) (cid:175) (cid:162)
Y× X = π(cid:175) ×π(cid:175) −1(∆ )
X(cid:48) Y X Pt
is a connected scheme.
I claim that
Supp(Y× X)=∆ . (1.4.2)
X(cid:48) Y
Infact,supposethatthisisnotso. Thenbydefinitionforall(y,x)∈(Y× X)\∆
X(cid:48) Y
we have (cid:179)(cid:161) (cid:162) (cid:180)
ϕY pY −1(y,x) ∩L(cid:54)=∅,
12
and therefore for each point (y,y)∈∆ ∩((Y× X)\∆ )
Y X(cid:48) Y
(cid:179)(cid:161) (cid:162) (cid:180)
T(cid:48)(Y,X)∩L⊃T(cid:48) ∩L=ϕY pY −1(y,y) ∩L(cid:54)=∅
Y,X,y 12
contrary to (1.4.1). This proves (1.4.2).
From (1.4.2) it follows that L∩S(Y,X)=∅. Hence
t≤dimS(Y,X)≤N −dimL−1=t,
i.e. condition b) holds. (cid:164)
1.5. Corollary. codim T(cid:48)(Y,X)≤1.
S(Y,X)
1.6. Definition. Let L ⊂ PN be a linear subspace. We say that L is tangent
to a variety X ⊂ PN along a subvariety Y ⊂ X (resp. L is J-tangent to X along
Y, resp. L is J-tangent to X with respect to Y) if L⊃T (resp. L⊃T(cid:48) , resp.
X,y X,y
L⊃T(cid:48) ) for all points y ∈Y.
Y,X,y
It is clear that if L is tangent to X along Y, then L is J-tangent to X along Y
and if L is J-tangent to X along Y, then L is J-tangent to X with respect to Y.
If X is nonsingular along Y, then all the three notions are identical.
1. THEOREM ON TANGENCIES AND ITS APPLICATIONS 17
1.7. Theorem. Let Yr ⊂ Xn and Zb ⊂ Yr be closed subvarieties, and let
Lm ⊂PN,n≤m≤N−1bealinearsubspacewhichisJ-tangenttoX withrespect
toY alongY \Z (i.e.L⊃T(cid:48) forallpointsy ∈Y \Z). Then r ≤m−n+b+1.
Y,X,y
Proof. It is clear that Theorem 1.7 is true (and meaningless) for r ≤ b + 1.
Suppose that r > b + 1. Without loss of generality we may assume that Y is
irreducible. Let M be a general linear subspace of codimension b+1 in PN. Put
X(cid:48) =X∩M, Y(cid:48) =Y ∩M, L(cid:48) =L∩M.
It is clear that
n(cid:48) =dimX(cid:48) =n−b−1,
r(cid:48) =dimY(cid:48) =r−b−1, (1.7.1)
m(cid:48) =dimL(cid:48) =m−b−1
and L(cid:48) is J-tangent to X(cid:48) with respect to Y(cid:48) along Y(cid:48). In other words,
T(cid:48)(Y(cid:48),X(cid:48))⊂L(cid:48). (1.7.2)
In particular, from (1.7.2) it follows that
dimT(cid:48)(Y(cid:48),X(cid:48))≤m(cid:48). (1.7.3)
Sincen>r >b+1,fromtheBertinitheoremitfollowsthatthevarietiesX(cid:48) andY(cid:48)
are irreducible. By [58, Lema 1, Corolario 1], the variety X(cid:48) is nondegenerate, and
so the relative secant variety S(Y(cid:48),X(cid:48)) containing X(cid:48) does not lie in the subspace
L(cid:48). From (1.7.2) it follows that
S(Y(cid:48),X(cid:48))(cid:54)=T(cid:48)(Y(cid:48),X(cid:48)). (1.7.4)
In view of (1.7.4) Theorem 1.4 yields
dimT(cid:48)(Y(cid:48),X(cid:48))=r(cid:48)+n(cid:48). (1.7.5)
Combining (1.7.3) and (1.7.5) we see that r(cid:48) + n(cid:48) ≤ m(cid:48), and in view of (1.7.1)
r ≤m−n+b+1. (cid:164)
1.8. Corollary (Theorem on tangencies). If a linear subspace Lm ⊂PN is
tangent to a nondegenerate variety Xn ⊂ PN along a closed subvariety Yr ⊂ Xn,
then r ≤m−n.
1.9. Remark. It is clear that if Z does not contain components of Y, then in
the statement of Theorem 1.7 we may assume that Z ⊂Y ∩SingX.
We give an example showing that the bound in Theorem 1.7 is sharp.
1.10. Example. Let Xn ⊂PN , N =2n−b−2 be a cone with vertex Pb over
the Segre variety P1×Pn−b−2 ⊂P2n−2b−3, n>b+2. Then
X∗ =(X(cid:48))∗ =P1×Pn−b−2 ⊂(Pb)∗ =P2n−2b−3,
18 I. THEOREM ON TANGENCIES AND GAUSS MAPS
and a subspace Lm ⊂ PN, n ≤ m ≤ N −1 is tangent to X at a point x ∈ SmX
(andallpointsofthe(b+1)-dimensionalaffinelinearspace(cid:104)x,Pb(cid:105)\Pb)ifandonly
if the (N−m−1)-dimensional linear subspace L∗ is contained in the (N−n−1)-
dimensional linear subspace T∗ ⊂X∗ (here and in what follows asterisk denotes
X,x
dual variety and (cid:104)A(cid:105) denotes the linear span of a subset A ⊂ PN). It is easy to
see that an arbitrary (n−b−3)-dimensional linear subspace lying in X∗ coincides
with T∗ for some x∈X.
X,x
Let
Pn−b−2 ⊂X∗ =P1×Pn−b−2
be a linear subspace, and let L∗ be an arbitrary (N −m−1)-dimensional linear
subspace of Pn−b−2. Then the m-dimensional linear subspace L=(L∗)∗ is tangent
to X at all points of Y \Pb, where
(cid:169) (cid:175) (cid:170)
Y =Pm−n+b+1 ⊃Pb, Y = x∈X (cid:175)L∗ ⊂T∗ ⊂Pn−b−2 .
X,x
Thus for the subspace L and the subvarieties Y = Pm−n+b+1 ⊂ Pn−1 ⊂ X and
Z =SingX =Pb the inequality in Theorem 1.7 turns into equality.
1.11. Proposition. Let Xn ⊂ PN be a nondegenerate variety satisfying
condition R (cf. [30, Chapter IV , (5.8.2)]) (in other words, X is regular in
k 2
codimension k, i.e. b = dim(SingX) < n−k), and let L be an m-dimensional
linear subspace of PN. Put X(cid:48) = X · L, and let b(cid:48) = dim(SingX(cid:48)). Then
b(cid:48) ≤2N−m−n+b−1=b+c+ε−1, i.e.X(cid:48) satisfiesconditionR , where
k−c−2ε+1
c=codim X =N −n, ε=codim L=N −m.
PN PN
Proof. For an arbitrary point λ of the (ε−1)-dimensional linear subspace L∗ ⊂
PN∗ weputX =X·λ∗, whereλ∗ isthehyperplanecorrespondingtoλ. Itisclear
λ
that X(cid:48) = ∩ X . Let Y = SingX(cid:48), Y = SingX , λ ∈ L∗. It is easy to see that
λ λ λ
λ∈L∗
Y ⊂ ∪ Y , so that
λ
λ∈L∗
b(cid:48) =dimY ≤ maxb +ε−1, (1.11.1)
λ
λ∈L∗
where b =dimY . It is clear that the hyperplane λ∗ is tangent to X at all points
λ λ
of Y \SingX. Hence from Theorem 1.7 it follows that
λ
b ≤b+c. (1.11.2)
λ
Combining (1.11.1) and (1.11.2) we obtain the desired bound for b(cid:48). (cid:164)
ThefollowingsimpleexampleshowsthattheboundinProposition1.11issharp.
1.12. Example. Let XN−1 ⊂ PN be a quadratic cone with vertex Pb, and
(cid:163) (cid:164)
let N+b +1 ≤ m ≤ N −1 (here and in what follows [a] is the largest integer
2
not exceeding a given number a ∈ R). Then X∗ is a nonsingular quadric in the
(N −b−1)-dimensional linear subspace (Pb)∗ ⊂ PN∗. It is well known (cf. [28,
Volume II, Chapter 6; 37, Chapter XIII]) that X∗ contains a linear subspace of
(cid:163) (cid:164)
dimension N−b−2 . Let L∗ be its linear subspace of dimension N −m−1. Put
2
L=(L∗)∗, X(cid:48) =X·L. Then dimL=m, and it is easy to see that Y =SingX(cid:48) is
an (N −m+b)-dimensional linear subspace.
1. THEOREM ON TANGENCIES AND ITS APPLICATIONS 19
1.13. Corollary. Suppose that a variety Xn ⊂PN satisfies conditions S =
ε+1
S and R = R , and let Lm ⊂ PN be a linear subspace for
N−m+1 c+2ε−1 3N−2m−n−1
which dim(X·L) = n−ε = m+n−N. Then the scheme X ·L is reduced. In
particular, if X is nonsingular, N < 2(m+n+1), and dimX·L = m+n−N,
3
then X·L is a reduced scheme.
Proof. From Proposition 1.11 it follows that in the conditions of Corollary 1.13
X(cid:48) = X ·L satisfies condition R . Since dimX(cid:48) = n−ε, X(cid:48) satisfies condition S
0 1
(cf. [61,§17]). Hence to prove Corollary 1.13 it suffices to apply Proposition 5.8.5
from [30, Chapter IV ]. (cid:164)
2
1.14. Corollary. IfXn ⊂PN satisfiesconditionsS =S andR =
ε+2 N−m+2 c+2ε
R and Lm ⊂ PN is a linear subspace such that dim(Xn·Lm) = n−ε =
3N−2m−n
m+n−N,thentheschemeX·Lisnormal(andthereforeirreducibleandreduced).
In particular, if X is nonsingular, N ≤ 2(m+n) and dim(X·L) = m+n−N,
3
then X·L is a normal scheme.
Proof. From Proposition 1.11 it follows that in the conditions of Corollary 1.14
X(cid:48) = X ·L satisfies condition R . Since dimX(cid:48) = n−ε, X(cid:48) satisfies condition S
1 2
(cf. [61,§17]). Hence to prove Corollary 1.14 it suffices to apply Serre’s normality
criterion (cf. [30, Chapter IV , (5.8.6)]). (cid:164)
2
Of special importance to applications is the case when L is a hyperplane. We
formulate our results in this case.
1.15. Corollary. a) If a variety Xn ⊂ PN is nondegenerate and normal and
N ≤ 2n−b−1, where b = dim(SingX), then all hyperplane section of
X are reduced. In particular, if X is nonsingular and N < 2n, then all
hyperplane sections of X are reduced.
b) If a nondegenerate variety Xn ⊂ PN has properties S and R (the
3 N−n+2
last assumption means that N <2n−b−2), then all hyperplane sections
of X are normal (and therefore irreducible and reduced). In particular, if
X is nonsingular and N < 2n−1, then all hyperplane sections of X are
normal.
1.16. Remark. Corollary 1.15 gives a much more precise information than
Bertini type theorems describing properties of generic hyperplane sections (cf. e.g.
[80]),but,asshownbyExamples1.18and1.19below,theassumptionsinitsstate-
ment cannot be weakened.
1.17. Remark. If K = C and b = −1, then in the assumptions of Corol-
lary1.15b)irreducibilityofhyperplanesectionsfollowsfromtheBarth-Larsenthe-
orem according to which for N <2n−1 the Picard group PicX (cid:39)Z is generated
by the class of hyperplane section of X (cf. [54; 60; 65]).
We give examples showing that the bounds in Corollary 1.15 are sharp.
1.18. Example. Let X = P1 × Pn−b−1 ⊂ P2n−2b−1, n > b + 1, and let
0
Y = x × Pn−b−2 ⊂ X be a linear subspace. We denote by X(cid:48) ⊂ P2(n−b−1)
0 0
the section of X by a general hyperplane passing through Y . It is easy to see
0 0
that X(cid:48) is a nonsingular projectively normal variety (cf. e.g. [73]). Let Xn ⊂ PN
20 I. THEOREM ON TANGENCIES AND GAUSS MAPS
, N = 2n−b−1 be the projective cone with vertex Pb over X(cid:48). It is clear that
X is a normal variety and dim(SingX)=b, so that X satisfies conditions S and
2
R =R . However X has a non-reduced hyperplane section corresponding
n−b−1 N−n
to the hyperplane in P2n−2b−1 which is tangent to X along Y (cf. Example 1.10).
0 0
1.19. Example. Let X = Pn−b−2 ⊂ P2n−2b−3, n > b+2, and let X be the
0
projective cone with vertex Pb over X . Then Xn ⊂ PN, N = 2n−b−2 is a
0
Cohen-Macaulay variety (cf. e.g. [47; 73]) and dim(SingX)=b, so that X satisfies
conditions S and R = R . However for each hyperplane L such that
3 n−b−1 N−n+1
L∗ ∈X∗ =X∗ L·X is a reducible and therefore non-normal variety, viz. L·X =
0
H ∪H , where H =Pn−1 and H is the cone with vertex Pb over P1×Pn−b−3, is
1 2 1 2
a reducible and therefore non-normal variety, and Sing(L·X) = H ∩H = Pn−2
1 2
(cf. Example 1.10).
2. GAUSS MAPS OF PROJECTIVE VARIETIES 21
2. Gauss maps of projective varieties
Let Xn ⊂ PN be an irreducible nondegenerate variety. For n ≤ m ≤ N −1 we
put
(cid:169) (cid:175) (cid:170)
P = (x,α)∈SmX×G(N,m)(cid:175)L ⊃T ,
m α X,x
where G(N,m) is the Grassmann variety of m-dimensional linear subspaces in PN,
L isthelinearsubspacecorrespondingtoapointα∈G(N,m),andthebardenotes
α
closure in X ×G(N,m). We denote by p : P → X (resp. γ : P → G(N,m))
m m m m
the projection map to the first (resp. second) factor.
2.1. Definition. The map γ is called the mth Gauss map, and its image
m
X∗ = γ (P ) is called the variety of m-dimensional tangent subspaces to the
m m m
variety X.
2.2. Remark. Of special interest are the two extreme cases, viz.m = n and
m = N −1. For m = n we get the ordinary Gauss map γ: X (cid:57)(cid:57)(cid:75) G(N,n), and
for m = N −1 we see that X∗ = X∗ ⊂ PN is the dual variety and if X is
(cid:161) N−1 (cid:162)
nonsingular, then PN−1 = P N (−1) , where N is the normal bundle
N−1 PN/Xn PN/Xn
to X in PN (cf. [16, Expos´e XVII]).
2.3. Theorem. Let dim(SingX)=b≥−1. Then
(cid:161) (cid:162)
a) for each point α∈γ p−1(SmX) , dimγ−1(α)≤m−n+b+1;
m m m
a(cid:48)) dimX∗ ≥(m−n)(N −m−2)+(m−b−1);
m (cid:169) (cid:170)
b) for a general point α∈X∗, dimγ−1(α)≤max b+1, m+n−N −1 ;
(cid:169) m m (cid:170)
b(cid:48)) dimX∗ ≥min (m−n)(N −m)+n−b−1, (m−n+1)(N −m)+1 ;
m
c) if charK = 0 and γm = νm◦γ˜m is the Stein factorization of the mor-
phismγ , thenν isabirationalisomorphismandthegenericfiberofthe
m m
morphism γ (and γ˜ ) is a linear subspace of PN of dimension dimP −
m m m
dimX∗.
m
Proof. a) immediately follows from Theorem 1.7, and since
dimP =dimX+dimG(N −n−1,m−n−1)
m
=n+(m−n)(N −m), (2.3.1)
a(cid:48)) follows from a).
b) Suppose first that m = N −1. It is clear that dimγ−1 (α) ≤ n−1, and it
N−1
sufficestoverifythatifn−1≥b+2,i.e.n≥b+3,thenforageneralpointα∈X∗
we have dimγ−1 (α) (cid:54)= n−1. Suppose that this is not so, and let x be a general
N−1
point of X. Since n−1 > b+1, from Theorem 1.7 it follows that the system of
divisors
(cid:161) (cid:162)
Y =p γ−1 (α) , α∈T∗
α N−1 N−1 X,x
isnotfixed,andthereforeX =∪Y ,whereαrunsthroughthesetofgeneralpoints
α
α
of T∗ . Hence for a general point y ∈X there exists a hyperplane Λ ⊂T∗ such
X,x y X,x
that for a general point β ∈Λ we have L ⊃T . But then
y β X,y
(cid:104)T ,T (cid:105)⊂(Λ )∗ =Pn+1,
X,x X,y y
