TANGENTS AND SECANTS OF ALGEBRAIC VARIETIES. F. L. Zak Central Economics Mathematical Institute of the Russian Academy of Sciences

TANGENTS AND SECANTS OF ALGEBRAIC VARIETIES F. L. Zak Central Economics Mathematical Institute of the Russian Academy of Sciences

CONTENTS Index of Notations Introduction 1 Chapter I. Theorem on Tangencies and Gauss Maps 14 1 Theorem on tangencies and its applications 15 Gauss maps of projective varieties 1 3 Subvarieties of complex tori 8 Chapter II. Projections of Algebraic Varieties 35 1 A criterion for existence of good projections 36 Hartshorne s conjecture on linear normality and its relative analogues 41 Chapter III. Varieties of Small Codimension Corresponding to Orbits of Algebraic Groups 47 1 Orbits of algebraic groups, null-forms and secant varieties 48 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 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 10 Maximal embeddings of varieties of small codimension 109 Chapter VI. Scorza Varieties 116 1 Properties of Scorza varieties 117 Scorza varieties with δ = 1 1 3 Scorza varieties with δ = 16 4 Scorza varieties with δ = 4 131 5 End of classification of Scorza varieties 145 Bibliography 149 iii ii Typeset by AMS-TEX

CHAPTER I THEOREM ON TANGENCIES AND GAUSS MAPS Typeset by AMS-TEX 14

1. THEOREM ON TANGENCIES AND ITS APPLICATIONS 15 1. Theorem on tangencies and its applications Let X n P N be an irreducible nondegenerate (i.e. not contained in a hyperplane) n-dimensional projective variety over an algebraically closed field K, and let Y r P N be a non-empty irreducible r-dimensional variety. We set Y = (Y X) X = { (y, x) Y X x = y }, where X is the diagonal in X X, S 0 Y,X (Y X \ Y ) P N, S 0 Y,X = { (y, x, z) z x, y }, where x, y denotes the chord joining x with y. We denote by S Y,X the closure of SY,X 0 in Y X PN, by p Y i the projection of S Y,X onto the ith factor of Y X P N (i = 1, ), and by ϕ Y : S Y,X P N the projection onto the third factor, and put p Y 1 = p Y 1 p Y : S Y,X Y X, S(Y, X) = ϕ Y (S Y,X ), T Y,X = ( p Y 1 1) ( Y ), ψ Y = ϕ Y T, Y,X T (Y, X) = ψ Y ( T 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 Y r X n is a subvariety of X. 1.. Definition. The variety T (Y, X) is called the variety of (relative) tangent stars of X with respect to the subvariety Y. We observe that T (X, X) = T X is the usual variety of tangent stars (cf. [45; 97]). 1.3. Definition. The cone T Y,X,y = ψy ( (p Y 1 ) 1 (y y) ) is called the (projective) tangent star to X with respect to Y X at a point y Y. From this definition it is evident that T Y,X,y is a union of limits of chords y, x, y Y, x X, y, x y. It is also clear that T Y,X,y T X,y T X,y, where T X,y = T X,X,y is the (projective) tangent star to X at y (cf. [45; 97]) and T X,y is the (embedded) tangent space to X at y. On the other hand, T Y,X,y T y,x, where T y,x = T y,x,y is the (projective) tangent cone to X at the point y. By definition, T (Y, X) = T Y,X,y. If X is nonsingular along Y, i.e. Y y Y Sing X = and Y Sm X = X \ Sing X, then T (Y, X) = T (Y, X) = T X,y is y Y the usual variety of tangents.

16 I. THEOREM ON TANGENCIES AND GAUSS MAPS 1.4. Theorem. An arbitrary irreducible subvariety Y r X n, r 0 satisfies one of the following two conditions: a) dim T (Y, X) = r + n, dim S(Y, X) = r + n + 1; b) T (Y, X) = S(Y, X). Proof. Let t = dim T (Y, X). It is clear that t r+n. In the case when t = r+n the theorem is obvious since S(Y, X) is an irreducible variety, S(Y, X) T (Y, X) and dim S(Y, X) r + n + 1. Suppose that t < r + n, and let L N t 1 be a linear subspace of P N such that L T (Y, X) =. (1.4.1) We denote by π: P N \ L P t the projection with center at L and put X = π(x), Y = π(y ). Since π X is a finite morphism, we have dim (Y X ) = r + n > t, and from the connectedness theorem of Fulton and Hansen (cf. [6] and [7, 3.1]) it follows that Y X X = ( π Y π X ) 1 ( P t) is a connected scheme. I claim that Supp (Y X X) = Y. (1.4.) In fact, suppose that this is not so. Then by definition for all (y, x) (Y X X)\ Y we have ϕ Y ( ( p Y 1 ) 1(y, x) ) L, and therefore for each point (y, y) Y ((Y X X) \ Y ) T (Y, X) L T Y,X,y L = ϕ Y ( ( p Y 1 ) 1(y, y) ) L contrary to (1.4.1). This proves (1.4.). From (1.4.) it follows that L S(Y, X) =. Hence i.e. condition b) holds. t dim S(Y, X) N dim L 1 = t, 1.5. Corollary. codim S(Y,X) T (Y, X) 1. 1.6. Definition. Let L P N be a linear subspace. We say that L is tangent to a variety X P N 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 X,y (resp. L T X,y, resp. L T Y,X,y ) for all points y 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 Y r X n and Z b Y r be closed subvarieties, and let L m P N, n m N 1 be a linear subspace which is J-tangent to X with respect to Y along Y \ Z (i.e. L T Y,X,y for all points y Y \ Z). Then r m n + b + 1. 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 P N. Put X = X M, Y = Y M, L = L M. It is clear that n = dim X = n b 1, r = dim Y = r b 1, m = dim L = m b 1 (1.7.1) and L is J-tangent to X with respect to Y along Y. In other words, In particular, from (1.7.) it follows that T (Y, X ) L. (1.7.) dim T (Y, X ) m. (1.7.3) Since n > r > b+1, from the Bertini theorem it follows that the varieties X and Y are irreducible. By [58, Lema 1, Corolario 1], the variety X is nondegenerate, and so the relative secant variety S(Y, X ) containing X does not lie in the subspace L. From (1.7.) it follows that In view of (1.7.4) Theorem 1.4 yields S(Y, X ) T (Y, X ). (1.7.4) dim T (Y, X ) = r + n. (1.7.5) Combining (1.7.3) and (1.7.5) we see that r + n m, and in view of (1.7.1) r m n + b + 1. 1.8. Corollary (Theorem on tangencies). If a linear subspace L m P N is tangent to a nondegenerate variety X n P N along a closed subvariety Y r X n, 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 Sing X. We give an example showing that the bound in Theorem 1.7 is sharp. 1.10. Example. Let X n P N, N = n b be a cone with vertex P b over the Segre variety P 1 P n b P n b 3, n > b +. Then X = (X ) = P 1 P n b (P b ) = P n b 3,

18 I. THEOREM ON TANGENCIES AND GAUSS MAPS and a subspace L m P N, n m N 1 is tangent to X at a point x Sm X (and all points of the (b + 1)-dimensional affine linear space x, P b \ P b ) if and only if the (N m 1)-dimensional linear subspace L is contained in the (N n 1)- dimensional linear subspace TX,x X (here and in what follows asterisk denotes dual variety and A denotes the linear span of a subset A P N ). It is easy to see that an arbitrary (n b 3)-dimensional linear subspace lying in X coincides with TX,x for some x X. Let P n b X = P 1 P n b be a linear subspace, and let L be an arbitrary (N m 1)-dimensional linear subspace of P n b. Then the m-dimensional linear subspace L = (L ) is tangent to X at all points of Y \ P b, where Y = P m n+b+1 P b, Y = { x X L T X,x P n b }. Thus for the subspace L and the subvarieties Y = P m n+b+1 P n 1 X and Z = Sing X = P b the inequality in Theorem 1.7 turns into equality. 1.11. Proposition. Let X n P N be a nondegenerate variety satisfying condition R k (cf. [30, Chapter IV, (5.8.)]) (in other words, X is regular in codimension k, i.e. b = dim (Sing X) < n k), and let L be an m-dimensional linear subspace of P N. Put X = X L, and let b = dim (Sing X ). Then b N m n + b 1 = b + c + ε 1, i.e. X satisfies condition R k c ε+1, where c = codim P N X = N n, ε = codim P N L = N m. Proof. For an arbitrary point λ of the (ε 1)-dimensional linear subspace L P N we put X λ = X λ, where λ is the hyperplane corresponding to λ. It is clear that X = X λ. Let Y = Sing X, Y λ L λ = Sing X λ, λ L. It is easy to see that Y Y λ, so that λ L b = dim Y max b λ + ε 1, (1.11.1) λ L where b λ = dim Y λ. It is clear that the hyperplane λ is tangent to X at all points of Y λ \ Sing X. Hence from Theorem 1.7 it follows that b λ b + c. (1.11.) Combining (1.11.1) and (1.11.) we obtain the desired bound for b. The following simple example shows that the bound in Proposition 1.11 is sharp. 1.1. Example. Let X N 1 P N be a quadratic cone with vertex P b, and let [ ] N+b + 1 m N 1 (here and in what follows [a] is the largest integer not exceeding a given number a R). Then X is a nonsingular quadric in the (N b 1)-dimensional linear subspace (P b ) P N. It is well known (cf. [8, Volume II, Chapter 6; 37, Chapter XIII]) that X contains a linear subspace of dimension [ ] N b. Let L be its linear subspace of dimension N m 1. Put L = (L ), X = X L. Then dim L = m, and it is easy to see that Y = Sing X is an (N m + b)-dimensional linear subspace.

1. THEOREM ON TANGENCIES AND ITS APPLICATIONS 19 1.13. Corollary. Suppose that a variety X n P N satisfies conditions S ε+1 = S N m+1 and R c+ε 1 = R 3N m n 1, and let L m P N be a linear subspace for which dim (X L) = n ε = m + n N. Then the scheme X L is reduced. In particular, if X is nonsingular, N < 3 (m + n + 1), and dim X L = m + n N, then X L is a reduced scheme. Proof. From Proposition 1.11 it follows that in the conditions of Corollary 1.13 X = X L satisfies condition R 0. Since dim X = n ε, X satisfies condition S 1 (cf. [61, 17]). Hence to prove Corollary 1.13 it suffices to apply Proposition 5.8.5 from [30, Chapter IV ]. 1.14. Corollary. If X n P N satisfies conditions S ε+ = S N m+ and R c+ε = R 3N m n and L m P N is a linear subspace such that dim (X n L m ) = n ε = m+n N, then the scheme X L is normal (and therefore irreducible and reduced). In particular, if X is nonsingular, N 3 (m + n) and dim (X L) = m + n N, then X L is a normal scheme. Proof. From Proposition 1.11 it follows that in the conditions of Corollary 1.14 X = X L satisfies condition R 1. Since dim X = n ε, X satisfies condition S (cf. [61, 17]). Hence to prove Corollary 1.14 it suffices to apply Serre s normality criterion (cf. [30, Chapter IV, (5.8.6)]). 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 X n P N is nondegenerate and normal and N n b 1, where b = dim (Sing X), then all hyperplane section of X are reduced. In particular, if X is nonsingular and N < n, then all hyperplane sections of X are reduced. b) If a nondegenerate variety X n P N has properties S 3 and R N n+ (the last assumption means that N < n b ), then all hyperplane sections of X are normal (and therefore irreducible and reduced). In particular, if X is nonsingular and N < n 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, as shown by Examples 1.18 and 1.19 below, the assumptions in its statement cannot be weakened. 1.17. Remark. If K = C and b = 1, then in the assumptions of Corollary 1.15 b) irreducibility of hyperplane sections follows from the Barth-Larsen theorem according to which for N < n 1 the Picard group Pic X 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 0 = P 1 P n b 1 P n b 1, n > b + 1, and let Y 0 = x P n b X 0 be a linear subspace. We denote by X P (n b 1) the section of X 0 by a general hyperplane passing through Y 0. It is easy to see that X is a nonsingular projectively normal variety (cf. e.g. [73]). Let X n P N

0 I. THEOREM ON TANGENCIES AND GAUSS MAPS, N = n b 1 be the projective cone with vertex P b over X. It is clear that X is a normal variety and dim (Sing X) = b, so that X satisfies conditions S and R n b 1 = R N n. However X has a non-reduced hyperplane section corresponding to the hyperplane in P n b 1 which is tangent to X 0 along Y 0 (cf. Example 1.10). 1.19. Example. Let X 0 = P n b P n b 3, n > b +, and let X be the projective cone with vertex P b over X 0. Then X n P N, N = n b is a Cohen-Macaulay variety (cf. e.g. [47; 73]) and dim (Sing X) = b, so that X satisfies conditions S 3 and R n b 1 = R N n+1. However for each hyperplane L such that L X = X 0 L X is a reducible and therefore non-normal variety, viz. L X = H 1 H, where H 1 = P n 1 and H is the cone with vertex P b over P 1 P n b 3, is a reducible and therefore non-normal variety, and Sing (L X) = H 1 H = P n (cf. Example 1.10).

. GAUSS MAPS OF PROJECTIVE VARIETIES 1. Gauss maps of projective varieties Let X n P N be an irreducible nondegenerate variety. For n m N 1 we put P m = { (x, α) Sm X G(N, m) L α T X,x }, where G(N, m) is the Grassmann variety of m-dimensional linear subspaces in P N, L α is the linear subspace corresponding to a point α G(N, m), and the bar denotes closure in X G(N, m). We denote by p m : P m X (resp. γ m : P m G(N, m)) the projection map to the first (resp. second) factor..1. Definition. The map γ m is called the mth Gauss map, and its image Xm = γ m (P m ) is called the variety of m-dimensional tangent subspaces to the variety X... 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 G(N, n), and for m = N 1 we see that XN 1 = X P N is the dual variety and if X is nonsingular, then P N 1 N 1 = P ( N P N /X n( 1)), where N P N /X n is the normal bundle to X in P N (cf. [16, Exposé XVII])..3. Theorem. Let dim (Sing X) = b 1. Then ( a ) for each point α γ m p 1 m (Sm X) ), dim γm 1 (α) m n + b + 1; a ) dim Xm (m n)(n m ) + (m b 1); b ) for a general point α Xm, dim γm 1 (α) max { b + 1, m + n N 1 } ; b ) dim Xm min { (m n)(n m) + n b 1, (m n + 1)(N m) + 1 } ; c ) if char K = 0 and γ m = ν m γ m is the Stein factorization of the morphism γ m, then ν m is a birational isomorphism and the generic fiber of the morphism γ m (and γ m ) is a linear subspace of P N of dimension dim P m dim Xm. Proof. a) immediately follows from Theorem 1.7, and since dim P m = dim X + dim G(N n 1, m n 1) = n + (m n)(n m), (.3.1) a ) follows from a). b) Suppose first that m = N 1. It is clear that dim γ 1 N 1 (α) n 1, and it suffices to verify that if n 1 b +, i.e. n b + 3, then for a general point α X we have dim γ 1 N 1 (α) n 1. Suppose that this is not so, and let x be a general point of X. Since n 1 > b + 1, from Theorem 1.7 it follows that the system of divisors ( Y α = p N 1 γ 1 N 1 (α)), α TX,x is not fixed, and therefore X = α Y α, where α runs through the set of general points of T X,x. Hence for a general point y X there exists a hyperplane Λ y T X,x such that for a general point β Λ y we have L β T X,y. But then T X,x, T X,y (Λ y ) = P n+1,

I. THEOREM ON TANGENCIES AND GAUSS MAPS i.e. for a general pair of points x, y X we have dim (T X,x T X,y ) = n 1. From this it follows that either all n-dimensional linear subspaces from γ n (X) are contained in an (n + 1)-dimensional linear subspace P n+1 P N or they all pass through an (n 1)-dimensional subspace P n 1 P N. But in the first case X is a hypersurface and by Theorem 1.7 dim Y α = n 1 b + 1, contrary to our assumption, and in the second case the intersection of X with a general linear subspace P N n+1 P N is a nonsingular strange curve (we recall that a projective curve of degree is called strange if all its tangent lines pass through a fixed point). It is well known (cf. [59; 34, ChapterIV; 39 or 75]) that the only nonsingular strange curves are conics in characteristic. Therefore in the second case X is a quadric, and we again come to a contradiction. Thus assertion b) holds for m = N 1 (if char K = 0, then one can simplify the proof using the reflexivity theorem according to which (X ) = X (cf. [96])). Next we prove assertion b) for m = k under the assumption that it holds for m = k + 1. It is clear that for general points α k Xk, α k+1 Xk+1 we have dim Y αk dim Y αk+1. (.3.) If b + 1 k + n N, then from the induction hypothesis it follows that dim Y αk dim Y αk+1 b + 1. Suppose that dim Y αk+1 k + n N > b + 1. (.3.3) If dim Y αk < dim Y αk+1, then assertion b) for m = k immediately follows from (.3.3). Otherwise from (.3.) and (.3.3) it follows that for a general point x X and a general point α k+1 Xk+1 for which Y α k+1 x each hyperplane in L αk+1 containing T X,x is tangent to X at all points of a (dim Y αk+1 )-dimensional component of Y αk+1 that are nonsingular on X, and by Theorem 1.7 dim Y αk+1 b + 1. But then dim Y αk = dim Y αk+1 b + 1, so that inequality b) holds also in this case. Assertion b) is proved. b ) immediately follows from b) in view of (.3.1). c) Let α m be a general point of X m. The linear subspace L m α m P N is tangent to X at all points of the subvariety Y αm Sm X, Y αm = p m ( γ 1 m (α m ) ), and it is easy to see that Y αm Sm X = ( Yα Sm X ), (.3.4) L α L α m

. GAUSS MAPS OF PROJECTIVE VARIETIES 3 where α runs through the set of points of X for which L α L αm. From the reflexivity theorem (cf. e.g. [49]) it follows that if char K = 0, then for a general point α X we have Y α = p N 1 ( γ 1 N 1 (α)) = (T X,α) (.3.5) is a linear subspace of P N of dimension N dim X 1. From (.3.4) and (.3.5) it follows that Y αm = Y αm Sm X = (T X,α) L α L α m is also a linear subspace of P N. Since char K = 0, the morphism γ m is separable and therefore smooth at a general point. Hence ν m is a birational isomorphism. This completes the proof of assertion c) and Theorem.3. We observe that if char K = p > 0, then assertion c) of Theorem.3 is no longer true. As an example, it suffices to consider the hypersurface in P n+1 defined by equation n+1 x p+1 = 0 (in this case γ is the Frobenius map). The case of positive i=0 i characteristic is treated in [50]..4. Corollary. If char K = 0, X n P N is a nonsingular variety, and N n + 1 m N 1, then a general m-dimensional tangent subspace is tangent to X along a linear subspace of dimension at most m + n N 1 (for N n this bound is better than the one given in Theorem 1.7). For n m N n + 1 a general m-dimensional tangent subspace is tangent to X at a single point..5. Corollary. Let X n P N, X n P n, n = dim X, b = dim (Sing X). Then n n b 1. In particular, for a nonsingular variety n n. If n b + 3, then n N n + 1 (this bound is better than the preceding one if N n b 1). The following example shows that both bounds in Corollary.5 are sharp..6. Example. Let X 0 = P 1 P n b P n b 3, n > b +, and let X be a projective cone with vertex P b and base X 0. Then X n P N, N = n b, dim (Sing X) = b, X = X 0 X 0, and n = n b 1 = N n + 1..7. Remark. In the case when char K = 0 and b = 1, the inequality n N n + 1 was independently proved by Landman (cf. [50]). Another proof was earlier given by the author (cf. [96, Proposition 1] for n = ; the general case is quite similar)..8. Corollary. Let X n P N, X n P n, b = dim (Sing X). Then dim γ(x) n b 1. In particular, for a nonsingular variety, dim γ(x) = dim X and γ is a finite morphism. If in addition char K = 0, then γ is a birational isomorphism (i.e. γ is the normalization morphism)..9. Remark. In the case when K = C and b = 1, Griffiths and Harris [9] proved that dim γ n (X) = dim X. Different proofs of finiteness of γ n in this case were later given by Ein [18] and Ran [68]. In our first proof of Corollary.8 (and Theorem 1.7) we used methods of formal geometry. Since related techniques is used in 3, we give this proof here.

4 I. THEOREM ON TANGENCIES AND GAUSS MAPS As in the proof of Theorem 1.7, considering the intersection of X with a general (N b 1)-dimensional linear subspace of P N we reduce everything to the case when b = 1. Suppose that the n-dimensional linear subspace L corresponding to a point α L G(N, n) is tangent to X along an irreducible subvariety Y, dim Y > 0, i.e. Y γ 1 (α L ). Let X = X /Y be the completion of X along Y, and let G = γ(x) /αl be the formal neighborhood of the point α L in the variety γ(x) G(N, n). Since X n P n, dim γ(x) > 0. Hence H 0 (G, O G ) and H 0 (X, O X ) H 0 (G, O G ) are infinite-dimensional vector spaces over the field K. On the other hand, let M P N be a linear subspace, dim M = N n 1, M L =, and let π : X P n be the projection with center at M. Then π /Y : X P n /π(y ) is an isomorphism of formal spaces, and therefore H 0 (X, O X ) H 0 (L, O L ), (.9.1) where L = L /Y P n /π(y ) is the completion of L along Y. But by the well-known theorem on formal functions (cf. [31, Chapter V; 36]), H 0 (L, O L ) = K which is impossible since H 0 (X, O X ) is infinite-dimensional in view of (.9.1). The above contradiction shows that dim Y = 0, i.e. γ is a finite morphism. Although, as we have already seen, the bounds in Theorem.3 are sharp, one can still prove stronger results for certain special classes of projective varieties. An important example is given by complete intersections..10. Proposition. Let X n P N be a nondegenerate nonsingular complete intersection. Then all Gauss maps γ m, n m N 1 are finite and dim X m = dim P m = n+(m n)(n m). If in addition char K = 0, then all γ m, n m N 1 are birational isomorphisms. Proof. Let α m Xm, α X be points for which there is an inclusion of the corresponding linear subspaces L αm L α. Then it is clear that γm 1 (α m ) γ 1 N 1 (α). Hence it suffices to prove Proposition.10 in the case when m = N 1. We recall that P N 1 = P ( N P N /X n( 1)) (cf. Remark.). Furthermore, the morphism γ N 1 : P N 1 XN 1 is defined by a linear subsystem without fixed points of the complete linear system O PN 1 (1), where O PN 1 (1) is the tautological sheaf on P ( N P N /X n( 1)) (cf. [16, Exposé XVII]). In view of [30, Chapter II, 6.6.3] and [31, Chapter III], to show that γ N 1 is finite it suffices to verify that N P N /Xn( 1) is an ample vector bundle. But if X is complete intersection of hypersurfaces F i, deg F i = a i, i = 1,..., N 1, then N P N /X N n n( 1) = O X (a i 1), and by [31, Chapter III] N P N /Xn( 1) is an ample bundle. The remaining assertions of Proposition.10 follow from (.3.1) and assertion c) of Theorem.3..11. Remark. The above proof of Proposition.10 can also be interpreted in elementary terms; cf. [4]. i=1

. GAUSS MAPS OF PROJECTIVE VARIETIES 5 The Gauss map γ : X G(N, n), where X n P N, X n P n is a nonsingular variety, can also be interpreted in another way. To begin with, γ is the map corresponding to the vector bundle N P N /Xn( 1) with a distinguished (N + 1)- dimensional vector subspace of sections corresponding to points of K N+1 (where P N = (K N+1 \ 0)/K ; cf. [8]). Furthermore, let L P N, dim L = N n 1 be a general linear subspace, and let π L : X P n be the projection with center in L. We denote by R L the ramification divisor of the finite covering π L, R L = { x X T X,x L }. The Gauss map γ is defined by the linear system R L generated by the divisors R L, L G(N, N n 1). This linear system does not have fundamental points, and ramification divisors R L corresponding to various linear subspaces L N n 1 P N are preimages of Schubert divisors on G(N, n) (cf. [8, Chapter 1; 37, Chapter XIV, 8])..1. Proposition. The linear system R L is ample. Proof. Proposition.1 immediately follows from Corollary.8 in view of [30, Chapter II, 6.6.3]..13. Remark. In the case when char K = 0 Ein [18] proved that ramification divisor is ample for an arbitrary nonsingular finite covering of P n of degree greater than one. Let X n P N, X n P n be a nonsingular variety. The exact sequences 0 T X O N+1 X N ( 1) 0, 0 O X ( 1) T X Θ X ( 1) 0, where Θ X is the tangent bundle to X and T X = γ (S) is the preimage of the standard vector subbundle S of rank n + 1 on G(N, n) (so that projectivizations of fibres of T X naturally correspond to projective tangent spaces to X), show that γ ( O G(N,n) (1) ) det T X K X (n + 1) = K X O X (n + 1), where K X is the canonical line bundle on X (cf. [64, 6.19]; we denote by the same symbol a bundle and the corresponding sheaf of sections). We remark that the property that a section of the line bundle K X (n + 1) vanishes along a divisor from R L lies in the basis of the classical definition of canonical class. An immediate consequence of Proposition.1 is the following.14. Corollary. Let X n P N, X n P n be a nonsingular variety. Then K X (n + 1) is an ample line bundle..15. Remark. It is worthwhile to compare Corollary.14 with some known results on the index of Fano varieties [51]. In general the role of very ampleness versus ampleness in such type of results is still to be investigated. However in the conditions of Corollary.14 the bundle K X (n +1) is actually very ample, at least if

6 I. THEOREM ON TANGENCIES AND GAUSS MAPS char K = 0 (cf. [18]). This is easily shown by induction on n using the fact that X has sufficiently many nonsingular hyperplane sections, and by Kodaira s vanishing theorem, for such a section H n 1 X n the complete linear system K H + nh = K X + (n + 1)H H is cut by the linear system K X + (n + 1)H (here K H is the canonical class of H; we denote by the same symbol the canonical divisor class and the canonical line bundle)..16. Proposition. Let X n P N be a nondegenerate variety, and let Y r X n be a subvariety of X for which m n = codim L Y < codim P N X = N n, where L m = Y is the linear span of Y. Then r min { n 1, [ ]} N+b, where b = dim (Sing X). Proof. Without loss of generality we may assume that Y Sing X. From our assumption it follows that for an arbitrary point y Y Hence dim (T X,y L) dim T Y,y r. (.16.1) γ(y ) = γ(y Sm X) { α G(N, n) dim L α L r } = S(L, r) G(N, n), where S(L, r) is the corresponding Schubert cell and γ : X G(N, n) is the Gauss map. Since by our assumption m r < N n, i.e. n + m r < N, from (.16.1) it follows that for each point y Y Sm X there exists a hyperplane M containing L which is tangent to X at y. Put S(M, L, r) = { α G(N, n) L α M, dim L α L r }. Then S(M, L, r) S(L, r) and dim S(M, L, r) = (r + 1)(m r) + (n r)(n n 1), dim S(L, r) = (r + 1)(m r) + (n r)n n), codim S(L,r) S(M, L, r) = n r = codim X Y. { Replacing if necessary r by min dim (TX,y L) } we may assume that y Y Then γ(y ) S(M, L, r) Sm (S(L, r)). dim (γ(y ) S(M, L, r)) dim γ(y ) codim S(L,r) S(M, L, r) = (r f) (n r) = r n f, (.16.) where f is the dimension of general fiber of γ Y. On the other hand γ(y ) S(M, L, r) = γ ({ y Y Sm X T X,y M }), (.16.3)

. GAUSS MAPS OF PROJECTIVE VARIETIES 7 and from Theorem 1.7 it follows that dim (γ(y ) S(M, L, r)) N n + b f. (.16.4) Combining (.16.3) and (.16.4), we conclude that r n f N n + b f, i.e. r [ ] N+b. Proposition.16 is proved. We observe that [ ] N+b < n 1 for N < n b..17. Remark. For K = C, b = 1 Proposition.16 can be also deduced from the Barth-Larsen theorem on the structure of integral cohomology of X (cf. [54])..18. Remark. It is worthwhile to compare Proposition.16 with the known classical result the first rigorous proof of which was probably given by Lluis (cf. [58, Lema 1, Corolario 1]) in which r is arbitrary, but L is a general linear subspace..19. Example. Let X0 n b 1, n b + 5, n + b 1 (mod ) be a general linear projection of the Grassmann variety G( n b+1, 1) in P n b 5, and let X n P N, N = n b 4 be a cone with vertex P b and base X 0. Then X 0 G( n b+1, 1) (cf. [33; 38]) and dim (Sing X) = b. Furthermore, X n Y r, where Y r, b < r < n, r n (mod ) is the cone with vertex P b over Y0 r b 1, and Y 0 is the projection of a Grassmann subvariety G( r b+1, 1) G( n b+1, 1). Then m = b+1+(r b 1) 3 = r b 4, and m r = r b 4 < N n = n b 4. On the other hand, for r = n we have an equality in Proposition.16, viz. r = [ ] N+b = n 4..0. Corollary. If X n P n, then X does not contain linear subspaces of dimension greater than [ ] N+b. If X is not a hypersurface (i.e. N > n + 1), then X does not contain projective hypersurfaces of dimension greater than [ ] N+b. The following examples show that the bound in Corollary.0 is sharp..1. Example. a 1 ) Let X0 n b 1, n b + be a nonsingular quadric, and let X n P N, N = n+1 be a cone with vertex P b and base X 0. Then dim (Sing X) = b, ] = [ N+b and X contains a linear subspace Y r = P r, where r = b + 1 + [ n b 1 (cf. [8, Volume, Chapter 6; 37]). a ) Let X 0 = P 1 P n b, b b + 3 be a Segre variety, and let X n P N, N = n b be a cone with vertex P b and base X 0. Then dim (Sing X) = b, and X contains a linear subspace Y n 1 = P n 1. In this case r = n 1 = N+b. b) In the assumptions of Example.19, let n = b + 7. Then X n P n+3 contains the quadratic cone Y n with vertex P b whose base is a nonsingular fourdimensional quadric G(3, 1). Here n = n+b+3 = N+b. Apparently, it is hard to construct examples of multi-dimensional varieties containing a hypersurface of dimension [ ] N 1. ]

8 I. THEOREM ON TANGENCIES AND GAUSS MAPS 3. Subvarieties of complex tori Besides subvarieties of projective space there is another important class of varieties for which it is natural to introduce Gauss maps, viz. subvarieties of complex tori. Let A N be an n-dimensional complex torus, and let X n A N be an analytic subset. Let C N be the universal covering of A N, and let C N A N be the corresponding homomorphism of abelian groups. Using shifts, one can identify the tangent space to A N at an arbitrary point z A N with C N, and the tangent space to X at a point x X can be identified with a vector subspace Θ X,x C N. 3.1. Definition. Let A be a complex torus, and let Y A be a connected analytic subset. The smallest subtorus of A containing all the differences y y, y, y Y (in the sense of group structure on A) is called the toroidal hull of Y and is denoted by Y. We observe that for an arbitrary point y Y we have Y y + Y. 3.. Lemma. Let Y A N be a connected compact analytic subset whose tangent subspaces at smooth points are contained in a vector subspace C m C N. Then dim Y m. Proof. It is easy to see that there exist an N-dimensional torus ÃN and an m- dimensional subtorus T m ÃN, T m Y such that à is locally isomorphic to A in a neighborhood of Y. It is clear that in a suitable neighborhood of T in à and therefore in sufficiently small neighborhoods of Y in à and Y in A there exist N m analytically independent holomorphic functions. On the other hand, from [5] and [36] it follows that in a small neighborhood of Y in A there exist exactly dim A dim Y analytically independent holomorphic functions. Hence dim A dim Y N m, i.e. dim Y m. 3.3. Definition. Let X n A N be an n-dimensional analytic subset of an N- dimensional torus A, and let Y r be an r-dimensional analytic subset of X. We say that a vector subspace C m C N is tangent to X along an analytic subset Y X if C m Θ X,y for all y Y. 3.4. Lemma. Let X n A N be an analytic subset, and let C m C N be a vector subspace which is tangent to X along a connected compact analytic subset Y r X n. Then there exist an N-dimensional complex torus ÃN, an m-dimensional complex subtorus T m ÃN, T m Y r, neighborhoods U A N, U Y, U + Y A = U, Ũ ÃN, Ũ Y, Ũ + Y à = Ũ, and an analytic subset X T Ũ such that Ũ U, Y A Y à = Y T, and X X U, and the mappings U Ũ, T à and X U T Ũ are compatible with the action of Y. Proof. The tori à and T are constructed as in Lemma 3.. To construct X it suffices to take the preimage of X in C N and to project it to the universal cover C m of the torus T m. Considering the quotient tori, it is easy to verify that this can be done equivariantly.

3. SUBVARIETIES OF COMPLEX TORI 9 3.5. Theorem. Let X n A n be an analytic subset of a complex torus, and let C m C N be a vector subspace which is tangent to X along a connected compact analytic subset Y r X n. Then for some neighborhood Y U A we have X U X U, where X U is a product of the torus Y A, dim Y = k and a (local) analytic subset of an (m k)-dimensional complex torus B m k, and there is a natural isomorphism C m C k C m k, where C k C N is the universal cover of the torus Y and C m k is the universal cover of the torus B. Proof. In the notations of Lemma 3.4 we consider the canonical holomorphic mappings From Lemma 3.4 it follows that where π : A A/ Y A, π : Ã Ã/ Y Ã, π T : T T/ Y T. X U = π(x U) π( X) π T ( X), π(y ) = y X U X = π(x). In particular, the neighborhood X U of the point y in X embeds as an analytic subset in the (m k)-dimensional torus B = T/ Y T. We put X = π 1 (X ) A, XU = X U = π 1 (X U ). Then X U is the desired analytic subset of U, and for an arbitrary point z X U the tangent space to X U at z has dimension not exceeding m and is tangent to X along the analytic subset X π 1 (π(z)) = X (z + Y A ). 3.6. Corollary (Theorem on tangencies for subvarieties of complex tori). Let X n A N be an analytic subset of a complex torus, and let C m C N be a vector subspace which is tangent to X along a compact analytic subset Y r X n. Then r k m, where k m is the maximal dimension of complex subtorus C A such that dim (X + C) m. 3.7. Remark. In contrast to the case of subvarieties of projective spaces (cf. Corollary 1.8), in Corollary 3.6 we do not assume that X is nondegenerate (an analytic subset X A is called nondegenerate if X = A). However if dim X m, then k dim X n and Corollary 3.6 is trivial. Let X n A N be an analytic subset of a complex torus, let n m N 1, and let P = { (x, α) Sm X Gras (N, m) L α Θ X,x }, where Gras (N, m) G(N 1, m 1) is the Grassmann variety of m-dimensional vector subspaces in C N, L m α C N is the vector subspace corresponding to a point α Gras (N, m), and the bar denotes closure in X Gras (N, m). We denote by p m : P m X (resp. γ m : P m Gras (N, m)) the projection map to the first (resp. second) factor.

30 I. THEOREM ON TANGENCIES AND GAUSS MAPS 3.8. Definition. The mapping γ m is called the mth Gauss map, and its image Xm = γ m (P m ) Gras (N, m) is called the variety of tangent m-spaces to the variety X. In particular, for m = n we obtain the usual Gauss map γ : X Gras (N, n), and for m = N 1 we get a map γ N 1 : P N 1 N 1 PN 1. 3.9. Proposition. Let X n A N be an irreducible compact analytic subset. Then there exists an analytic subtorus C k A N such that (i) X + C = C; (ii) γ = γ π X, where π : A B, B = A/C is the canonical holomorphic map and γ : X Gras (N, n) and γ : X Gras (N k, n k), X = π(x) B are the Gauss maps; (iii) the map γ : X γ (X ) Gras (N k, n k) is generically finite. Proof. Arguing by induction, we assume that Proposition 3.9 is already verified for N < N and prove it in the case dim A = N. If the map γ is generically finite, then it suffices to put C = 0, X = X. Suppose that for a general point x X we have dim γ 1 (γ(x)) > 0, and let Y be a positive-dimensional component of γ 1 (γ(x)). By Lemma 3. 0 < k = dim Y n. Since a continuous family of complex analytic subtori of A is constant, we conclude that if x is another general point of X and Ỹ is a positive-dimensional component of the fiber γ 1 (γ( x)), then Ỹ = Y. We put C = Y, X = π(x) B, B = A/C, x = π(y ) = π(x). Since the tangent space to X is constant along Y Sm X and the kernel of the differential d x ( π X ) coincides with Θπ 1 (x ),x, we see that Y lies in a fiber of the Gauss map for the subvariety π 1 (x ) C. But Y spans C and dim C n < N (otherwise Y = X = C = A and Proposition 3.9 is obvious), so that from the induction hypothesis it follows that Y = C. Thus a general and therefore each fiber of the map π X coincides with the corresponding fiber of the map π : A B; moreover, X + C = C and X is a locally trivial analytic fiber bundle over X with fiber C. Furthermore, Sing X = π 1 (Sing X ), Θ X,x = Θ X,x Ck, where C k C N is the universal covering of C and γ = γ π X. 3.10. Corollary. Let X n A N be a compact complex submanifold. Then the Gauss map γ : X Gras (N, n) can be represented in the form γ = γ π, where π : X X is a locally trivial analytic fiber bundle whose fiber is a complex subtorus C k A N, X is a compact complex subvariety of the torus B = A/C, and the Gauss map γ : X Gras (N k, n k) is finite. In particular, if X does

3. SUBVARIETIES OF COMPLEX TORI 31 not contain complex subtori (e.g. if A is a simple torus), then the Gauss map γ is finite. Proof. Corollary 3.10 is an immediate consequence of Theorem 3.5 and Proposition 3.9. Our results also allow to describe the structure of Gauss maps γ m for arbitrary n m N 1. 3.11. Theorem. Let X n A N be a compact analytic submanifold, n m N 1. Then a) there exist finitely many subtori C 1,..., C l A such that if X i = X + C i, i = 1,..., l, α Xm, L α is the m-dimensional vector subspace ( of C N corresponding to α, and Y is a connected component of p m γ 1 m (α) ), then for some 1 i l we have Y = C i, L α is tangent to X i along a torus y + C i, y Y ( so that in particular α ( ) Xi m = γ m( Pm ( X i ) )), and Y is a connected component of the analytic subset X (y + C i ); b) the components of general fibers of the Gauss maps are ( the same. More precisely, if n m, m N 1 and x X, α m γ m p 1 m (x) ), α m ( γ m p 1 m (x)) are general points, then in a neighborhood of x we have p m ( γ 1 m (α m ) ) = p m ( γ 1 m (α m )). If C X is the maximal analytic subtorus of A for which X + C = X, then a general subspace L α C N, α X m is tangent to X along a union of tori of the form x + C, x X. Proof. Theorem 3.11 is an immediate consequence of Theorem 3.5 and Corollary 3.10. 3.1. Corollary. If X does not contain complex subtori, then for an arbitrary n m N 1 the mth Gauss map γ m is generically finite. In particular, dim X m = dim P m = n + dim ( Gras (N n, m n) ) = n + (m n)(n m) (compare with (.3.1)) and X N 1 = PN 1. If A is a simple torus (i.e. A does not contain proper analytic subtori), then all Gauss maps γ m : P m X m are finite. 3.13. Let X n A N be an analytic submanifold. Then the tangent bundle Θ X naturally embeds in the restriction of the tangent bundle Θ A on X (which is a trivial bundle on X with fiber C N ), and we can consider the normal bundle N A/X = ( )/ Θ X A ΘX. It is clear that if S (resp. Q) is the canonical vector sub- (resp. quotient-) bundle on Gras (N, n) and γ : X Gras (N, n) is the Gauss map, then Θ X = γ (S) and N A/X = γ (Q). In other words, the Gauss map γ is induced by the normal bundle N A/X and the linear map Γ(A, Θ A ) Γ(X, N A/X ) of the corresponding vector spaces of sections (cf. [8, Volume 1, Chapter I, 5]). Similarly, the map γ N 1 : P ( N A/X ) P N 1 is induced by the invertible sheaf O N (1) on P ( N A/X ) = PN 1.

3 I. THEOREM ON TANGENCIES AND GAUSS MAPS The exact sequence shows that 0 Θ X Θ A X N A/X 0 det N A/X = det Θ X = K X, where K X is the canonical line bundle on X. Since for the Plücker embedding we have det Q = O Gras (N,n) (1), the map γ is also defined by a (base point free) linear subsystem of the canonical linear system K X, viz. by the linear system spanned by the ramification divisors R L = { x X dim (Θ X,x L) > 0 }, where L runs through the set of general (N n)-dimensional vector subspaces of C N (compare with Section ). 3.14. Proposition. Let X n A N be an analytic submanifold. a) The following conditions are equivalent: (i) The bundle N A/X is ample; (ii) The mappings γ m, n m N 1 are finite; (iii) XN 1 = PN 1 and γ N 1 : P ( ) N A/X P N 1 is a finite covering. b) Suppose that condition ( ) (iii) from a) holds. Then either n = N 1 or deg γ N 1 = c n Ω 1 X = ( 1) n c n (X) = e(x) N 1, where e(x) is the (topological) Euler-Poincaré characteristic of X and Ω 1 X is the sheaf of differential forms of rank one. Proof. a) (i) (iii) in view of the definition of ampleness of vector bundle (cf. [31, Chapter III]), Corollary 6.6.3 from [31, Chapter II] and Proposition.6. from [30, Chapter III 1 ], (ii) (iii) is obvious, and (iii) (ii) follows from the fact that for m < N 1 the fibers of γ m (or, more precisely, their projections to X) are contained in the fibers of γ N 1. b) From the description of the map γ N 1 given in 3.13 it immediately follows that deg γ N 1 = c n (Θ X) = c n ( Ω 1 X ) = ( 1) n c n (X) = e(x). In [55, 3.1] it is shown that if Y is a complex manifold and π : Y P k is a finite covering of degree k 1, then Pic Y = Z. To verify b) it suffices to put k = N 1, Y = P ( N A/X ) and to observe that for N n 1 > 0 we have rk ( Pic ( P ( N A/X ))). We observe that in view of Corollary 3.1 assertions (i) (iii) hold in the case when A is a simple torus. 3.15. Proposition. Let X n A N be an analytic submanifold. Then the canonical linear system K X is base point free, and its suitable multiple defines a holomorphic mapping π : X X making X a locally trivial analytic fiber bundle over a complex manifold X ; the fiber of π is the maximal analytic subtorus C A for which X + C = X (where X embeds isomorphically in B = A/C). Proof. In view of the above description of Gauss map (cf. 3.13), Proposition 3.15 immediately follows from Corollary 3.10 and Corollary 6.6.3 from [30, Chapter II].

3. SUBVARIETIES OF COMPLEX TORI 33 3.16. Corollary. Let X n A N be a nondegenerate complex submanifold (i.e. X = A). Then there exists an analytic subtorus C A such that if π : A B = A/C is the projection map, then (i) π X : X X B is a locally trivial analytic fiber bundle with fiber C (so that X = π 1 (X )); (ii) the mapping π X is equivalent to the mapping defined by a sufficiently high multiple of the canonical class K X ; (iii) the canonical class K X is ample; (iv) B = X is an abelian variety. 3.17. Corollary. An analytic submanifold X n A N is a variety of general type (i.e. the canonical dimension of X coincides with its dimension) if and only if the canonical class K X is ample. 3.18. Remark. From Corollary.8 it follows that for a nonsingular variety X n P n over an algebraically closed field of characteristic zero the Gauss map γ is birational, and according to Remark.15, the map defined by the complete linear system K X + (n + 1)H, where H is a hyperplane section of X, is an isomorphism. However for submanifolds of complex tori the map γ and the canonical map defined by the complete linear system of canonical divisors can be finite maps of degree greater than one. As an example, it suffices to consider a hyperelliptic curve X of genus g > 1 embedded in its Jacobian variety J X. In this case the Gauss map coincides with the canonical map which clearly has degree two (it is clear that the normal bundle N JX /X is ample, and all the Gauss maps γ m, 1 m g 1 are finite; cf. Proposition 3.14). In [83] it is shown that in the conditions of Proposition 3.14 deg γ X e(x) N n. 3.19. Remark. The study of submanifolds of complex tori was begun by Hartshorne [3] and continued by Sommese [84] who revealed the role of complex subtori using the notion of k-ampleness. At the same time Ueno [93, 10] undertook a thorough investigation of properties of the canonical dimension of submanifolds of complex tori (his results easily follow from ours, but are stated in different terms) and announced in [9] our Corollary 3.17, but his proof turned out to be erroneous (cf. [93, 10.13]). Griffiths and Harris [9, 4 b)] showed that the map γ from our Corollary 3.10 is generically finite, and basing on their result Ran [68] gave a different proof of Corollary 3.17 and of Proposition 3.0 below in the case c = 0. The following two results are analogs of Proposition.16 for submanifolds of complex tori. 3.0. Proposition. Let X n A N be a complex ] submanifold, and let Y r X n be a complex subtorus. Then r, where c is the maximum of [ n(n n)+c N n+1 dimensions of complex subtori C A such that X+C = X (this bound is nontrivial for c < n N 1). In particular, if X is a hypersurface (i.e. n = N 1) containing a complex subtorus Y r of dimension r > n, then X is a locally trivial analytic bundle whose fiber is a complex torus and whose base is a hypersurface in a complex torus of smaller dimension. Proof. It is clear that for an arbitrary point y Y we have Θ X,y Θ Y,y = C r,

34 I. THEOREM ON TANGENCIES AND GAUSS MAPS where C r C N is the universal covering of the torus Y. Hence and by Corollary 3.10 γ X (Y ) S Y = { α Gras (N, n) L α C r} r c dim γ X (Y ) dim S Y = dim (Gras (N r, n r)) = (n r)(n n) which implies the assertion of the proposition. 3.1. Proposition. Let X n A N be a complex submanifold, and let Y r X n be an analytic subset for which dim Y = m, where m r = codim Y Y < codim A X = N n. Denote by d the maximal dimension of complex subtori D A for which X + D A. Then r [ ] n+d. In particular, if A is a simple torus, then r [ ] n. Proof. Proposition 3.1 can be proved in essentially the same way as Proposition.16. In the notations corresponding to those of.16 we have dim ( γ(y ) S(M, L, r) ) dim γ(y ) codim S(L,r) S(M, L, r) = (r f) (n r) = r n f, (3.1.1) where f is the dimension of general fiber of γ Y other hand, from Corollary 3.6 it follows that (compare with (.16.)). On the dim (γ(y ) S(M, L, r)) d f. (3.1.) Combining (3.1.1) and (3.1.) we get r n f d f, i.e. r [ ] n+d as required. We observe that [ ] n+d < n 1 for d < n. 3.. Corollary. If X A, then X does not contain complex subtori of dimension greater than [ ] n+d. If X is not a hypersurface (i.e. N > n + 1), then X does not contain hypersurfaces (in complex tori) of dimension greater than [ ] n+d. 3.3. Remark. In contrast to the case of subvarieties of projective spaces (cf. Proposition.16), in Proposition 3.1 we do not assume that X is nondegenerate. However if X = A, then d dim X n, so that in the degenerate case our results are trivial.

CHAPTER II PROJECTIONS OF ALGEBRAIC VARIETIES Typeset by AMS-TEX 35

36 II. PROJECTIONS OF ALGEBRAIC VARIETIES 1. An existence criterion for good projections Let Y r X n be a nonempty irreducible r-dimensional subvariety of an irreducible n-dimensional variety X defined over an algebraically closed field K, and let Y Y Y Y X be the diagonal. Denote by I Y the Ideal of Y in Y X and put Θ Y,X = Spec ( j=0 I j/ I j+1 ), Θ Y,X,y = Θ Y,X K(y), y Y. 1.1. Definition. We call Θ Y,X,y Y X at the point y Y. It is easy to see that the (affine) tangent star to X with respect to Θ y,x Θ Y,X,y Θ X,y Θ X,y, where Θ y,x = Θ y,x,y is the (affine) tangent cone to X at the point y, Θ X,y = Θ X,X,y is the (affine) tangent star to X at y, and Θ X,y is the Zariski tangent space to X at y. Furthermore, if X n P N and the bar denotes projective closure, then in the notations of Section 1 of Chapter 1 we have Θ y,x = T y,x, Θ X,y = T X,y, Θ Y,X,y = T Y,X,y, Θ X,y = T X,y (cf. [45]). 1.. Definition. Let f : X X be a morphism of algebraic varieties. We say that f is unramified in the sense of Johnson (J-unramified) with respect to Y X at a point y Y if the morphism d y f Θ is quasifinite. If f is J-unramified with Y,X,y respect to Y at all points y Y, then we say that f is J-unramified with respect to Y. If moreover Y = X, then the morphism f is called J-unramified. 1.3. Definition. In the notations of Definition 1. we say that f is an embedding in the sense of Johnson (J-embedding) with respect to Y X if f is J- unramified with respect to Y and is one-to-one on f 1 (f(y )). If moreover Y = X, then the morphism f is called J-embedding. 1.4. Remark. If X is nonsingular along Y, i.e. Y Sing X = and Y Sm X, then f is unramified with respect to Y if and only if f is unramified at all points y Y ; f is a J-embedding with respect to Y if and only if f is a closed embedding in some neighborhood of Y in X. 1.5. Proposition. Let X n P N be a projective algebraic variety, let Y r X n be a nonempty irreducible subvariety, let L N m 1 P N, L X = be a linear subspace, and let π : X P m be the projection with center in L. a) The following conditions are equivalent: (i) The morphism π is J-unramified with respect to Y ; (ii) L T Y, X) =.

1. AN EXISTENCE CRITERION FOR GOOD PROJECTIONS 37 b) The following conditions are equivalent: (i) The morphism π is unramified at the points of Y ; (ii) L T (Y, X) =. c) The following conditions are equivalent: (i) The morphism π is a J-embedding with respect to Y ; (ii) L S(Y, X) =. d) The following conditions are equivalent: (i) The morphism π is an isomorphic embedding; (ii) L S(Y, X) = L T (Y, X) =. Proof. Most of the assertions of the proposition are obvious. To verify a) it suffices to use the fact that π Θ is quasifinite iff π Y,X,y T is finite or equivalently Y,X,y L T Y,X,y = (we recall that T Y,X,y is a projective cone with vertex y). 1.6. Proposition. a) In the conditions of Proposition 1.5 suppose that the morphism π : X n P m is J-unramified with respect to an irreducible subvariety Y r X n, where m < r + n (i.e. dim L N n r). Then π is a J-embedding with respect to Y. b) In the conditions of Proposition 1.5 suppose that the morphism π : X n P m is unramified at all points y Y r, where Y r X n is an irreducible subvariety and m < r+n (i.e. dim L N n r). Then π is an isomorphism in a neighborhood of Y. Proof. In view of Proposition 1.5 a), our condition means that Therefore L T (Y, X) =. (1.6.1) dim T (Y, X) < codim P N L r + n. (1.6.) In view of Theorem 1.4 of Chapter I, from (1.6.) it follows that S(Y, X) = T (Y, X). (1.6.3) In view of Proposition 1.5 c), assertion a) of Proposition 1.6 now follows from (1.6.1) and (1.6.3). b) According to Proposition 1.5 b), our condition means that Therefore L T (Y, X) =. (1.6.4) dim T (Y, X) dim T (Y, X) < codim P N L r + n. (1.6.5) By Theorem 1.4 of Chapter I, from (1.6.5) it follows that S(Y, X) = T (Y, X). (1.6.6) In view of Proposition 1.5 d), our assertion now follows from (1.6.4), (1.6.6), and the obvious inclusion T (Y, X) T (Y, X).