manuscripta math. 94,427-435 (1997) manuscripta mathematica t Sprmgcr-VcrLag 1997

Variet ies w i t h low d i m e n s i o n a l dual variety

R o b e r t o Muf ioz *

Fac. Matemgtticas UCM. Dpto. r~lgebra. Ciudad Universitaria. 28040-Madrid. Spain.

Received July 3, 1996; in revised form July 31, 1997

In this note we classify the complex projective ,>folds X C IP N with dimen- sion equal to the dimension of its dual X* mimls one, ~t < 2 /3N and positive defect. We show that such n-folds are hyperplane sections of the n + 1-folds with dim X = dim X* (classified by Ein) and some scrolls over curves. The natural generalization is to consider the set Sv of positive defect n-folds with thin(X) = d im(X*) - ( p - 1) (p a fixed positive integer). We show that the set of possible values of the pair (N, 7~) with 7t _< 2 /3N corresponding to nondegenerate it-tblds in @ which m'e not scrolls is finite.

i I n t r o d u c t i o n

Let X be a complex projective ~L-fbld in the projective space 11 ~'N. Tile dual variety of X, denoted by X*, is the variety of hyperplanes tangent to X (i.e., that contain the embedded tangent space in a point). For most n-folds their dual variety is a hypersurface. We will call the difference N - 1 - dim(X*) the defect of X. The goal of this paper is to classify n-folds X with positive defect under some additional hypothesis on the difference between the dimension of X and the dimension of its dual.

By Zak's ' theorem of tangencies [15] d i m ( X ) <_ dim(X*). Then the defect will be nmximum when the equality holds. The nonlinear positive detect varieties ,,,itl~ di ' , , (X) = d'i,,~(X*) and ,~ < ~ have been dassmed by L. Ein [11 and they are the following (we call them Ein varieties):

(a) the Segre embedding of 1P 1 x lP ''~-t in IP '2'z-1. (b) the Pliicker embedding in D 9 of the grassmannian of lines in IF 4, G(2, 5). (c) the 10-dimensional spinor variety Slo, which parametrizes tile II~4's ill a

smooth quadric hylmrsurface of IP 9, embedded in II ~'ts. Ill this paper, tbllowing a suggestion of F.L. Zak, we extend this result of

Ein and classify the nonlinear positive defect n-f(dds with the property that d i m ( X ' ) = d i m ( X ) + 1 and n <_ ~ - . We conclude that these 'l>folds aro:

(a) I[ ~u- ~-bundles over nonsingular ctlrves sllch that tile fihers m'e embedded linearly with N = 2m

(b) hyl)erplmm sections of Ein 's~ + 1-folds.

• Part. supported by ;I research project of the I)C, CICYT, ref. PB93-0440-C03-0I and by a grant of the (hominid" d Autdnoma (le MadrM.

428 Roberto bhlfioz

We can remark tha t the complete intersections have defect equal to zero. Then the above lists wilt be tile complete lists of -n-folds with the conditions requested if the Har tshorne 's conjecture on coml)lete intersections is true.

These classifications of varieties with d i m ( X ) = d im(X*) and witll d i m ( X ) = d i m ( X * ) - 1 suggest us the s tudy of the general case di re(X) = d i m ( X * ) - ( p - 1) (p a fixed positive integer). Let S t, be the set of complex nondegenerate n-folds in IP 'N with positive defect and d i m ( X ) = d i m ( X * ) - (p - 1). Let us call the I?"-bundles over a m-fold Y such tha t the iibers are emlm(hled linearly scrolls over" Y. \Ve show tha t the set of pairs (N,'u) corresponding to eleme.nts in S r which are not scrolls is finite.

AcknowledgemenL 1 would like to thank my advisor Raquel Mallavibarreni~ and tnofessor I;.L. Zak for many helpful discussions.

2 P r e l i m i n a r y R e s u l t s

Let X be a complex projective irreducible algebraic variety. The cono'rmal variety of X, denoted by TO.\-, is tile following wtriety

T°x - = {(~:, H) : x e S i n ( X ) , H E (~..N)*, T.v,.~ C H ) .

where SIn(X) is the set of nonsingular points of X. Tile dual va.riety of X is the ' image of tile projection of TO.\- on (pN). . From

now on X will be a complex nonlinear 'u-fold in I, ~''N. We also assume tha t X is nondegenerate (the degenerate case is easy to study ([1], Prop. 1.1)).

Let H be a general tangent hyl)erplane to X. The biduality theorem (TOx = TOx* ) implies that the locus of points z such that the embedded tangent space on z is contained in H is a linear space of dimension N - 1 - d i m ( X * ) (= c le f (X) ) . This linear space is known ~ the tin, tact locus L of X and H.

T h e o r e m 2.1. (Zak's T h e o r e m o f t a n g e n e i e s ) [15] Let X be a nondegen- erate eornplex p'rojective n-fold in ~,N and H a l-plane (l >_ n). Let Y be the set of lmirds in X sueh. that its embedded tan..qc'nt space is contained in H. Then. d iu t (Y ) <_ l - 'u.

We can apply Zak's Theorem to a general tangent hyperl)hme H and so d im(X*) > d i m ( X ) . Thus, as it was said in the introduction, the defect will be maximum when equali ty holds ( that will be the case when X* is smooth). These maximal defect n-f¢flds with the addit ional hypothesis n < a N have been classified by Ein [1] and they are only the ones pointed out in the previous section.

In order to face the problem of classification of n-fohls iu the next step (i.e. with d i m ( X * ) = d i m ( X ) + 1) we need to recall some results of [1]. They are, mainly, some propert ies of the conormal bundle of the contact locus of a general tangent hylmrplane H in X.

T h e o r e m 2.2. [1] Let X be a n-fold as before with positiiJe defect k, H a gener'al tant.leut hylterplane and L the contact locus of X arzd H. Let us denote b?.t K x the eano'nicat bu'u,tle of X . It follows that:

(,,) HZI.\.(0 ~_ J%.l.\

Varietie.s with low dimensional duM variety 429

(b) K. \IL = o ( ~ ) . (c) I f , ,~:~-2 < 0 and t ( \ .

t w i s t s of.A/'[l[\, is."

H°(AfLl .x . (a ) ) = 0 H~(N/I . \ . (a)) = 0 H ~ ( N L I . \ . ( a ) ) = 0

H'(H/ I .~ (a ) ) = o

= O(b) (b an in, teger) th, en the cohomolog 9 of the

f o r all a <_ 0

f o r all a >_ - k

f o r all a > ,~3k 0 < i < k

f o r al la<_ ~ O < i < k

And, si~ce - , - k - 2 has to be an integer, n =- k (mod 2) 2

Scrolls with some conditions on the dimension of the fibers are natural examples of positive defect varieties. There are results tha t part ial ly characterize them, in the tbllowing sense:

T h e o r e m 2.3. [2] In the same conditions as above we have: (a) k = n - ' 2 "if and only i f X is a[P .... l-bundle over a smooth curve. (b) k = t t - 4 i f a'nd oaly i f X is a ~ .... 2-bundle ove'r a smooth, su'@u:e Ot >_ 7). (c) U'Ic >_ ~ then X is a scroll.

Remark . Grassmannians of lines in IP '-9", G(2, 2r + 1), with r a positive integer are other examples of n-folds with positive defect (k = 2).

We will use the following well known fact,:

L e m m a 2.4. (first bound) L e t X be a nondegenerate complex projective n-fold i'a ~,N 'with positive defect k, then k < 'n - 2.

PwoJ': Take H a general hyperplane tangent to X. Then k is the dimension of the singular locus of the section X fq H, so k _< n - 1. By the congruence n - k (mod 2) we have the lemnm.

Since 2.2 gives iMbrmation about the cohomology of the tensor products of Aft:ix with the powers of the twisting bundle OL(1), we can use the Beilinson spectral sequence (BSS) [131 of.N'Ll x to s tudy it. Then, denoting this sequence by E~? '~j, by definition

El'" : H"(N~:I.\-(p)) ® n - ~ ( - l , )

and converges to

other case

Each term of the BSS must have elements different Dora zero in some of its cohnnns because tile normal bundle cmmot be zero. This is tile key fact, to prove the following lemma:

L e m m a 2.5. ( s e c o n d b o u n d ) [1] Let X be a co'topic:z:, l,rojective a~,d 'non- defle~terat~e n-Jbld in ~.N with K\ - = O(b) (b an i'aleger) and positive ,lefl'x:t k, th.ea k < "77 .2 .

In most s i tuatkms 2.2 implies tha t tile first, t, erm of the BSS of.M~.fx has elements ditferent from zero oaly in some central cohmms. The next. lmmmt allows us to extract (:onsequcnccs of this fact.

430 Roberto Mufioz

L e m m a 2 .6 . Let F be a vecto'c bmdle over [P"~ a'n.d P its llilbert pol'yr~omi,.l, tl, at is P(m) = 2'a~=o(-l)Jdim ( H J ( F ( m ) ) ) . The'~l it follows that

','k(F) = Z5~=o(-1)"+J ( ' : ) P ( - , l + j) .

Pwof : Let us recall tha t the rank of a coherent sheaf is tile rank of the tibet' in [ p P , q a general point. By the convergence x~,~+l = E~ 'j) of the BSS of F it Mlows

p n p n ( l ~ P + q ~ . l . f P P , q ~ Since each t e l ' l i t that the rk(F) must I)e equal to ~ - , , = 0 ~ , l = 0 t - * / • ~ , ~ + l J . of the BSS is cons t ruc ted from the previous one by taking cohomology, we have:

Z2,,=nZ;'=o(-1)"+"rk( EJ"~'t) = 2"p=oZ;'=o(-1),'+q,.l~.( E~ ''')

ti)r all l)ositive integer j . \\re can conclude apply ing recursively this equal i ty from j = 1 to n + 1 be(:ause in the I - t e r m of the BSS ca(:h e lement E{ '''1 has rank

( ' , l _ p ) d i , n ( H " ( F ( p ) ) ) .

Another l)roof of this l emma is based on taking consecut ive restr ict ions of F(m) to the different l inear spaces in IP 'n, t ha t is, IP ' ' ' - l , It ~'>2 ... lP '° and COml)uting the Hill)ert po lynomia l by means of the corresponding exact sequences:

0 -~ F ( ( t - 1)Iv- -, F(a)le~ -+ F ( a ) l ~ , - t -~ 0.

3 V a r i e t i e s w i t h d i m ( X ) = d i m ( X ' ) -- 1.

Let X be a nondegenera te nonl inear complex project ive n-fold ill IP N wi th ti le p roper ty tha t dim(X) = dim(X*) - 1. "~Ve write, as before, k = def(X) . \Ve will show tha t the only posi t ive defect 7z-tblds satisfying these prol)ert ies and u <_ 2N/3 are ei ther scrolls over s m o o t h curves with dimension equal to th(.' codimension or general hyperp lane sect ions of Ein varieties.

By 2.4 ~l must be greater than or equal to N/2. If n = N/2 then k = Jl - 2. T h eo rem 2.3 inlplies tha t X is a scroll over a nonsingular curve C. \¥e can reM" to [7] for a s tudy of the cases of C ra t ional and elliptic. If C has gelltlS bigger than or equal to two no examples are known and it w~s conjectured in [14] tha t the genus o t C is bounded by one. Th is conjec ture has been l)roved in d imension two [8], three [15] and four [14] att(l it is, to t i le best of our knowledge, still open in the rest of dimensions.

Since u cannot be ~ (it would iml, ly k = n - 3) then 'n >_ 1 + @. Ba r th ' s Theorent and 2.2 imply tha t K\- = O ( - N / 2 ) and then N is necessarily even.

By the Second Bound Lemma, N - I~ - 2 _< ,~__~2 and so 7l > 2 / 3 ( N - 1). We ,)

assmnc tha t 'n < ~ N and we have 2 ( N - 1) /3 < n < 2N/3. Therefore we can dist inguish three groups of n-folds:

Firs t groul): N = {Jr - 2, 'lL = 4r -- 2, k = 2r -- 2. Second group: N = 6r + 2, 'n = 4'r + 1, k = 2r - 1. Th i rd group: N = 65 n = 4'5 k = 2r - 2.

wi th r a l)ositive integer (big enough to make k > 0).

Varieties with low dimensional dual variety 431

F i r s t g r o u p . ( N , n , k) = (67" - 2, 4r - 2, 2 r - 2) w i t h r all integer greater than or equaL to 2. Sillce n - 3 k - 2 2 - - r + 1 <_ 0, 2.2 shows tha t

H'(H~ i x ( a ) ) = 0 0 < i < k -'r > a or a >_ - r + 9 ~rt°(.A.r~1.\.(.)) = o . <_ o

H k ( N ~ I . \ . ( , , ) ) = 0 , _> - k

So the first t e rm E{ '''~ of the BSS of A/'[[.\. has only one cohmm (p = - r + 1)

with etenmnts ditfi~rent f rom zero. T h e n E l '''t = E~; q, i.e.,

H ; I . \ - = H " - ~ ( -NT,, , - ( t - 4 ) ~ e " - ' ( , . - 1).

(2, It implies tha t rk(N'~.l.\.) = 2r > \ ' r - 1 and then r _< 3.

If r - 2, N = 10, n -=- (i, k = 2 and /~ ' \ - = 0 ( - 5 ) . Since X is non degenera te then h°(X, (9(1)) = 11 + a wi th a a positive integer. T h e n X is the iso,norphic pro jec t ion of a Dcl Pezzo maniibld of degree 6+a , and it does not exist by Fuj i ta ' s classif ication [5,6].

If r -- 3, n = 10, k = 4, N = 16 and K.\- = O ( - 8 ) . \Ve have h°(O(1) ) = 17 + a like in the previous ca.se. T h e n X is the isomorphic pro jec t ion of a Mulmi var ie ty wit, h Picard group fl'ee of rank 1 generated by the hyperl) lane sect ion and sect iomd genus equals 8+a . It does not exist following the classifications of Mukai of these variet ies [11,12].

S e c o n d g r o u p . ( 6 r + 2, 4 r + 1 , 2 r - 1) w i t h r a posit ive integer.

Again '*- :~ -'-' =_, - - r + l < 0 s o

tt'(.M'[i.\.(a)) = 0 , 0 < ' / < k , - r + 2 < _ a o r a < - r - 1 .

T h e n the first te rm of the BSS of A/'~I x has only two cohmms (p = - r and

p = - r + 1) wi th e lements di tferent frolIl 0. It implies tha t E~ ~'q = E ~ q. Let P he the Hilber t po lynomia l of A/'~I.\., Le tmna 2.6 for A/'~I.\. shows tha t

rk(N'~.,x)= ( 2 " ( 1 ) l P ( - r ) - P ( - r + l)t.

Since ,t > ~ ( N - 1) X is l inearly normal 19].

I f r = 1, 'n = 5, N = 8, k = 1, K.v = O ( - 4 ) , X is a D e l P e z z o w~rie tyof degree 5, and by Fu j i t a ' s classification it is a hyl~erplane section of the Pliicker embeddi l ,g of G(2, 5) in 1t 1'9.

If r = 2, 'n = 9, N = 14, k = 3, K x = 0 ( - 7 ) , X is a Mukai wtriety of degree 12 and sect ional genus 7, so by Mukai 's cl&ssification X is a hyl)erl)hme section of the embedd ing of the spinor wtriety S'~0 in IP aS.

T h i r d g r o u p . (6r, 4r, 2r - 2) wi th r ml integer greater than or equal to 2.

432 Roberto Mufioz

Again n-:3k-22 = - r + 2 <_ 0 and so we have all t i le info lmat ion of 2.2 abou t ti le cohomology of A/'~lx(a ). T h e n 2.6 and the fact that P ( - r ) = P ( - r + 2) show

tha t

+ o__ (2 , - 2 ) > , / _ - - ,. \ , . - 1 ) r ( - , - + t ) )

We recall now the result of Nagura [10] which assures that tor m > 25 there ahwlys exists a pr ime n,md)er p such t lmt m < p < (l.2)m.. Now it. is easy to check tha t if r >_ 7 or r = 5 there is always a pr ime mnnber p such tha t r + 2 _< 19 _< 2'r - 2. But this is a cont radic t ion with the above equal i ty of ranks because such p divides bo th combina tor ia l numbers and does not divide 2 ( r + 1). For tim cease r = 6 we can see t lmt 3 makes the role of p and so r < 4.

I f r = 2, then 'n = 8, N = 12, k = 2 , K.x = 0 ( - 6 ) mid X is l inearly normal . So X is a Mukai var ie ty of degree 10 and sectional genus 6. It does not exist by Mukai 's classification.

If r = 3, then n = 12, N = 18, k = 4 and K x = 69(-9) . Since A/'[i.v ~- N'LIX(--1) their Chern polynomials are equal. By 2.2 and 2.6 we

have P(0) = 0 and 6 P ( - 2 ) - 8 P ( - a ) = 8 respectively. These facts and the R i e m m m Roch T h e o r e m lead to the following result about the Chern classes of N'~b \.:

cl = - 4 h e2 = ch 2 ca = (14 - 3c)h a ca = (91 - 14c + (1/2)c2)h 4

with h tile class of ti le hyperp lmm sect ion of L a n d e an integer. By the exac t sequence of t i le normal bundles

0 --+ .A/'~,-(1)IL -~ COg'" -~ A/'~,IX (1) ~ 0 (*)

Af~.I.\. (1) is spanned by global sect ions and so P(1) , whk:h is exact ly the d imen-

sion of tile global sections of A/'~I.\.(1 ), must be greater than ov equal to 9, i.e.,

c < 1 0 . If c = 10 then the d imens ion of the vector space H°(N'[I. \ .(1)) equals 10,

th~ls w~ Call COllStl'HCt t | l e e x e t e r s e q l l e l t c e

0 -~ F - ~ O ~ ' " -~ ~¢ ; i . \ . ( 1 ) - , o

where F is a bundle of rank 9 on L. Til ls is a contradic t ion with tile fact t ha t the inverse of the Chern polynomia l of N'~lX(1 ) has degree 4.

On the o ther band c. must be grea ter than or equal to 6 ([3], Col'. 2.12) and equal i ty holds when A/';.I. \- spl i ts as a sum of line bundles. Therefore c = 8.

Lvt us restr ic t Aft*l. x. to a p ro jec t ive plane in L. \\re denote this res t r ic t ion

I) 3' JV'~2 and tile d imension of tile cohomology vector spaces H ~ by Id. Using

the isoluOrl,hisnl A/'e*~ --~ A/'~._,(-1) and the R iemann-Roch Tlteormn we see tha t

h " ( A ( t ; ~ ( - i ) ) = 0 for i greater than (I, h- (YV' ; . , ( -y) ) = 0 I,,, j g,'eat,'r tha,, - " l , ) ( H e ~ ( _ 1 ) ) _9 ,m, t t h e ~ ' . . p l e " " t " * = (h (./~fF2),h (Jr'u,2)) ,:all ],,, ,Hie of (4,~), (3,1) or (2 ,0 ) .

We claim tha t h°(AfQ,*.~) < 3. Let, us ~ssmne tha t hU(Af~;.e) is bigger than or equal to three, then we can choose three l inearly imlependent global sect ions s L,

Vmicties with low dimensional dual variety 433

su and s3 of Aft*2. Since this bundle is a subbund le of a trivial one, these sections give the exact, sequence

where (2 is a bundle o[ rank 5 on IP '2. Thea in any line of this ]p,2 we have the seql le l lce

(f)(~'| ([14 0 -~ O ~ a ~ , ~ Cv~,~ ( - 1 ) -~ Q ] ~ ~ I)

o[ the rest, rictions. Since 5 9 ~ ( - 1 ) has no global sections then QI~ ~- C9~,~ ~) C9~14(-1). Thus Q is a ml i fonn vector bundle over ]t :'2 and its sl)litting type is (0,- 1,- 1,- 1,- 1). By the classification in [41, Th. 5.1, we can conclude tha t Q* ( - 1)

~03 either splits as a stun of line bundles or is cg~.j ~; Y272 (1). In both cases h 1 (Q) = (1

and so h I (Aft.)) = 0 which gives a contradict ion. Then we have concluded tha t the m, ir (h°(@~), hl(~V~*~))ml,st be (2,0).

Thus the first t e rm of the BSS of JV'~., has elements different fi'om zero only on the diagonal (p + q = 0) and it implies tha t

"1~ he. sequences

~.., _~ O~: q~ f;~2 (1) ¢ O~'_,-(-1).

and t i le BSS's of the different restrictions .h@ (s = 2, 3) show that t i le conormaI bundle N'Td.\. is isomorl)hic to /2L(1) (9 ~73(3) (where the notat ion /2}' means AJ'X?c).

We (:all make a blowing up of X along L. Following the nota t ions of Ein ([1], Lema 4.2) we call p : X --, X this blowing up and denote 02(a,b ) the line bundle p*Ox (a) ® (.92 (-bE) where E is the exceptional divisor.

Let us take the following exact sequence:

0 --~ 02(1 , 2) --* 0 2 ( 1 , 1) --~ COE(1,1) -~ 0

Since 1 is great(n" tll:-).l, ' i ' t - a ~ / 2 = [.) the l l ( I lL Lelllll 4.2 c) fI1(0.~.(1,2)) is (xttlat to (k "l'aking the cohomology sequence of the se(tllell('e al)ove we h~tve, the. inequal i ty h°((97;.(1, 1)) k h°(Ot , ' ( l , 1)) = h°(Af/ ix(1)) . This is at contradic t ion

because h°((92(1, 1)) = 14 and h°(JV'/Ix(1)) = 15. So r ( 'minor be 3.

If ' r = 4 the way of reasoning tile non existence of these varieties is completely mmlogous to the previous one, we only out l ine the proof.

Once again the is(mmrphisn~ .h/~l x ~_ AfLl.\.(-1 ) and 2.6 alh)w us to ('ore- Irate the Chern classes of tile conormal bundle Af[i x to obtain, with tile salllc nota t ions as Ileibre,

c! = - 5 h ("~ = c h ~ (':a = ( 3 0 - 4 c ) h 3 c4 = ( 5 7 - 9 c + lcU)hl

c..-, = ( - 2 9 7 + , l lc - c'-')h r' c, = ( - : j - c - 2370 - "2 + 'c:~)h[~

434 Roberto M u i'mz

5 ) 10 ([3}, Cor. 2.12) and the equali ty implies tha t \Ve also know tha t c > 2

A/'~I x splits as s tun of line bundles. On the other hand N'~Ix(1 ) is spanned 1oy

global sections and then c mus t be 12. Let us restrict to a IP '2. Making the same kind of a rgmnents as in the case

r = 3 we can conclude tha t

JV"~2 -~ 0~2 3 ~b °(D2fl 0 Ca

and tha t the restr ict ion to a IP 's is

It is easy to see tha t there is no vector bundle on lP -'x' with the required condi t ions and whose restr ict ion to a II ~s is the one of (**), q.e.d.

\ r e can suni t ip o l l r resu|ts in the fo l lowing theoreni.

T h e o r e m 3.1. Let X be a llo'nde.qe'ne'rate complex projective n-Jbld in n ~N with n < :aN,'2 positive defect k and such that d i m ( X ) = d i m ( X * ) - 1, th.en X must be one of the following n-folds:

(i) sc'mlls over smooth curves of d imension n in IP '2'~. (ii) hype'qdane sections of G(2, 5) or" $1o.

4 T h e G e n e r a l Case

Ein ' s m , I our classification suggest tile s tudy of positive defect, varieties in cases when d i m ( X ) = d i m ( X * ) - (I 7 - 1) with p a tixed positive integer. Let us call the set of complex nondegenera te and nonl inear n-folds in this s i tua t ion ,50. We show that , excluding scrolls, in each $~, there are only a finite set of possibilit ies for the couple (n, N)

Let us assume X is not a scroll. Theorem 2.3.c. implies tha t k <_ 0~/2) - 1 mid then i~ > 2 (N - p + 1)/3. Thus tbr N >_ 2 + 4p Bm'th 's Theorem implies tha t P i e ( X ) ~- Z. 'l'o apply 2.2.c we need tha t (n - 3k - 2) /2 _< 0, i.e., n < 3 ( N - 1 7 - (2 /3) ) /4 . By the ~ussumption of the inequal i ty n _< 2N/3 , this is t rue tin' N greater t han or equal to 9p+6 . Thus H~(N'~Ix (a)) = 0 for a >_ al = ( n , - 3 k ) / 2

and 0 < i < k and for a < a,2 = (k - 'n)/2. But the difference al - a2 is less than or equal t.o 217. This implies t ha t the first t e rm of the BSS of the conormal bundle o f ' a variety with d i m ( X ) = d i m ( X * ) - (17 - 1) and N ~ 917 + 6 has elements

I ;1t7 is dilt 'erent from zero only in the 2p central columns. But k - 21 ) >_ 5 N -

greater thau zero. Then , when N > 9p there are some cohmms in the BSS with no elements different from zero. So 2.6 gives equalit ies of the type

~ ' - I I k k / ~ + i n - k = ~ :c~ k e v e n x , C22 t = 0

' n - k = k+i :r~ k o d d . % E ' -

Varieties with low din:ensional dual variety 435

W e m u s t f ind a p r i m e n u m b e r b e t w e e n -~ + p + 1 (or ~k+: + p + 1) a n d k a n d such

t h a t it d o e s n o t d i v i d e ~ - : . B u t t h i s p r i m e n u m b e r ex i s t s by N a g u r a ' s r e su l t , n - k

b e c a u s e --7-. < § + p + 1 a n d b e c a u s e it is p o s s i b l e to t a k e N big e n o u g h to 1 m a k e k as big as u e e d e d (k > 5 N - p).

\,Ve have s h o w u t h a t :

T h e o r e m 4 .1 . Let p be a fixed positive integer and @) as before. Then the set of pairs (n , N) corresponding to n-folds X C ~,N of Sp with. n < '~N a'nd which are not sewlls is finite.

R e f e r e n c e s

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