Higher order dual varieties of projective surfaces

This article was downloaded by: [George Mason University]On: 11 May 2013, At: 02:27Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

Communications in AlgebraPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lagb20

Higher order dual varieties of projective surfacesAntonio Lanteri a & Raquel Mallavibarrena ba Departmentto Di Matwmatica, “F. Enrioues” Università Via, C.Salidini 50, Milano,1-20133, Italy E-mail:b Departmentyo De Algebra Facultad De Matemáticas, Universidad Complutense DeMadrid Ciudad Universitaria, Madrid, E-28040, Spain E-mail:Published online: 27 Jun 2007.

To cite this article: Antonio Lanteri & Raquel Mallavibarrena (1999): Higher order dual varieties of projective surfaces,Communications in Algebra, 27:10, 4827-4851

To link to this article: http://dx.doi.org/10.1080/00927879908826733

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any formto anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that thecontents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drugdoses should be independently verified with primary sources. The publisher shall not be liable for anyloss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arisingdirectly or indirectly in connection with or arising out of the use of this material.

http://www.tandfonline.com/loi/lagb20

http://dx.doi.org/10.1080/00927879908826733

http://www.tandfonline.com/page/terms-and-conditions

COMMUNICATIONS IN ALGEBRA, 27(10), 4827-4851 (1999)

HIGHER ORDER DUAL VARIETIES

OF PROJECTIVE SURFACES

E-mail address: lanteriOvmimat.mat.unirni.it

E-mail address: raquelmCQeucmax.sim.ucm.es

ABSTRACT. We investigate higher order dual varieties of projective manifa~1,ls whose osculatory behavior is the best possible. In particular, for a k-jet ample s,~.rface we prove the nondegeneratedness of the k-th dual variety and for 2-regular swfaces we investigate the degree of the second dual variety.

Key words and phrases. Surface (complex, projective); Je t bundles; k-jet ampleness; k- regularity; Duality.

oeveral papers in the literature discusg higher order dual varieties of special projective manifolds like scrolls, surfaces in etc., especially from t h ~ enumerative point of view IP2], [PSI, IMP], [Sh]. Though for different reasons, all such varieties appear quite special as to the behavior of their osculating hyperplanes.

In recent years the notions of k-spanned, k-very ample, and k-jet ample line bundle on a projective manifold have been introduced [BSl], [BS2], [BS3]. These notions, generalizing that of very ample line bundle, formalize the co~icept of higher- order embedding and have been extensively studied by several authors.

1991 Mathematics Subject Classification. Primary 14C20, 14560; secondarj 14525

Copyright Q 1999 by Marcel Dekker, Inc

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013

4828 LANTERI AND MALLAVIBARRENA

This paper grew out of the idea that, for a higher-order embedded smooth projective variety X , the behavior of its osculating hyperplanes should be the best possible, unlike the varieties mentioned above.

Furthermore k-jet ampleness implies another notion, k-regularity. By definition, k-regularity means that the k-th jet bundle J k L of the line bundle L giving the embedding of X is generated by the k-jets jks of the sections s E r ( X , L). If (X, L) is a k-regular smooth n-fold, then the k-th dual variety X: of X parametrizes all elements of the linear system 1LI having some singular point of multiplicity at least k + 1. It is reasonable to expect that in general X: is nondegenerate, that is, its dimension is as large as possible.

In fact, imitating the classical theory for the ordinary dual variety, we prove our first main result (Theorem (1.4)): if (X, L) is k-regular, then either %(JkL) = 0 and X: is degenerate, or %(JkL) = degnkdegx;, where nk is the morphism induced by the second projection of X x JLj on the projective bundle

If L is k-jet ample, then JkL is spanned by global sections, and so c,(JkL) 2 0. In fact, in the case of surfaces we obtain much more. Our second main result (Theorem (3.2)) says that for n = 2, the equality c2(JkL) = 0 occurs if and only if (X, L) = (p, O P ( ~ ) ) .

Combining the two results we get that, if a surface is embedded by a k-jet ample line bundle, then its k-th dual variety is nondegenerate except for the k-th Veronese surface. For k = 1 this fact was known classically [K, Theorem 21, p. 1961, [E, Proposition 3.1, (c)].

The proof of our second result relies heavily on two specific features of surface theory: the Enriques-Kodaira classification and some subtle adjunction theoretic properties of k-jet ample line bundles (Lemma (3.1)), which in turn depend on some adjunction theoretic results for k-very ample line bundles on surfaces proven in [BS3]. This is the reason why we focus on surfaces. -Moreover, for k = 2 and 3 we also improve the result, showing that the same characterization of the equality cz(JkL) = 0 holds in the more general setting of k-regular surfaces (Propositions (4.1) and (4.2)). We conjecture that this statement is true for any k.

Another point where jet-ampleness provides a useful tool is in investigating the birationality of the map ?rk. In fact, if L is (k + 1)-jet ample on X , then nk is birational (Proposition (2.4)). This certainly appears as a rather weak property compared with what is known [Z] about the birationality of the Gauss map nl. However, combined with some improvement we get for specific surfaces like Segre- Hirzebruch surfaces and elliptic conic bundles, this result applies in the study of the second dual variety of 2-regular projective surfaces.

In particular, in this setting we investigate the geography of 2-regular surfaces (S, L) in the plane (d,v), where d = L2 and v = cz(J2L), relating the extrinsic geometry of (S, L) to the intrinsic geometry of S. When 7 ~ 2 is birational we have v = degSi; hence our results (Propositions (4.3), (4.4), (4.5)) can be regarded

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013

HIGHER ORDER DUAL VARIETIES OF SURFACES 4829

as analogs of classical and recent results relating the degree and the class of a projective surface [Ll].

We consider only smooth complex projective varieties.

This work was done in the framework of the projects lOOB and III1997-0123 supported by MURST of the Italian Government and MEC of the Spanish Govern- ment. We are very grateful to Steve Kleiman for several useful remarks.

Let X C = P(V) be a smooth projective variety of dimension 71 2 2 (n-fold) not contained in any hyperplane. Sometimes we identify the embedded variety X with the pair (X, V) and we write IVI to denote the linear subsystern of IOx(l)l defined by V; equivalently, IV( = P(VV) = pV. (0.1) Let L be a line bundle on X. For every integer-k 2 1 we denote by J k L the k-th jet bundle of L. We recall the exact sequence of vector bundles on X holding for every k 2 I (0.1.1) o -+s~R: ,EIL-+ J ~ L - + J ~ - ~ L - + O ,

where R& stands for the cotangent bundle of X and Sk denotes the k-th symmetric power.

Let X be a surface S. Then (0.1.1) yields:

and

We will need the following lemma.

(0.2) Lemma. Let L be an ample line bundle on a surface S. T h m

if and only if (S, L) = (p, Ope (k)).

Proof. From (0.1.2) and the equality cl(JkL) = 0, we get L = ? ~ K S . Since L is ample we conclude that S is a Del Pezzo surface (see (0.5)); hence ~ ( 0 s ) = 1. Now from the condition c2(Jk L) = 0, by using Noether's formula and (,0.1.3), we get

-

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013


and taking into account the above expression of L this gives Ki = 9. Then the assertion follows from the classification of Del Pezzo surfaces [Dl. The converse is obvious. 0

(0.3) Let S be a surface. We say that S is ruled if it has Kodaira dimension s (S ) = -00. We frequently denote by q : S -t So a birational morphism from S to a minimal model So and by s ( 2 0) the number of blowing-ups q factors through. Recall that s does not depend on q; in fact s = K'& - Ki. We denote by E the effective divisor on S contracted by q to a finite set such that Ks = qWKs, + E. Note that E consists of s distinct irreducible components (each one being a smooth rational curve) with positive multiplicities. If S is a rational surface, but S # P, then we can replace the minimal model So in the above setting with a Segre- Hirzebruch surface Fe := P(Up @Up, (-e)), simply recalling that F1 is blown-up at a single point.

(0.4) For use several times below, we fix some notation and recall some basic results about P1-bundles p : S -t C over a smooth curve of genus g(C). Then the irregularity of S is q(S) = g(C). Moreover there exists a vector bundle E of rank 2 on C such that S = PC(&) and we can assume E to be normalized as in [Ha, p. 3731. Let e := - degE be the invariant of S ; we recall that e >_ -g(C). Let Co denote a section corresponding to the tautological line bundle of E and let f be a fibre. Recall that Pic(S) Z $ p8Pic(C), Z being generated by [Cob Moreover, since the numerical equivalence classes of Co and f generate Num(S), for every line bundle L on S we have L s [aCo + bf] for some integers a, b. In particular Ks r [-2Co + (2g(C) - 2 - e) f].

(0.5) Let L be an ample line bundle on a surface S. We say that (S, L) is a scroll if S is a P1-bundle over a smooth curve and Lf = Up (1) for every fibre f of S. We say that (S, L) is a conic fibration over a smooth curve B if S is a ruled surface over B and LF = U p (2) for the general fibre F of the ruling. We reserve the expression conic bundle for a conic fibration (S, L) whose fibres are all irreducible, i. e. when S is a !P1-bundle over 8. A Del Pezzo surface is a surface whose anticanonical bundle -Ks is ample. In this case we call (S, - K s ) a Del Pezzo pair. Note that according to this terminology the quadric surface (P' x P' , Op (1 , l ) ) is a scroll and the Del Pezzo pair (P' x P' , Up, x p ~ (2,2)) is a conic bundle.

(0.6) Let L be a very ample line bundle on a surface S and let d = L2 be the degree. We recall that the A-genus of (S, L) is defined as

We have the basic inequality A(S, L) 2 0; moreover if A(S, L) _< 1 then S is ruled with q(S) 5 1 (e. g. see (10, Propositions 2.3 and 2.41). If S is not ruled, then we have the stronger inequality

(e. g, see [Be, Lemma 1.41). If S is of general type, we can also obtain a little improvement of (0.6.1) by extending the argument-in [Be, Lemma 1.41 (see also [Ha, Ex. 6.2, p. 423)).

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013

HIGHER ORDER DUAL VARIETIES OF SURFACES 483 1

(0.7) Lemma. Let S C fl be a surface of general type of degree d, not contained in any hyperplane. Then d 2 2N - 1.

Proof. Let L be the hyperplane bundle on S and let C E ]LJ be a general element. Then C is a smooth curve and

(0.7.1) ~ O ( L ~ ) 2 h O ( ~ ) - 1 = N.

Since K(S) = 2, we have LKs > 0; hence deg LC < 2g(C) - 2. Then Clifford's theorem says that

1 (0.7.2) hO(Lc) 5 deg LC + 1.

Combining (0.7.1) and (0.7.2) gives

d = deg LC 2 2 h 0 ( ~ c ) - 2 2 2N - 2.

Now assume that d = 2N - 2. Then the equality holds in (0.7.2) aiid so, since L is very ample, the Clifford theorem gives us only the following possibilities: i) LC = Kc, or ii) C is hyperelliptic. But in the former case we would gc:t LKs = 0 by adjunction, which is impossible, while in the latter S would be a rxled surface by [SV, Main theorem], which is also impossible.

1. k-REGULARITY AND HIGHER ORDER DUAL VARIETIES

Let X C PN = P(V) be an n-fold and set L := Ox(l ) . For every i:nteger k 2 1, consider the bundle homomorphism

associating to every section s E V its k-th jet jk(s ) . Following [PSI and [P2] we make some basic definitions and results.

(1.1) 1) The fibres of j k determine the k-th osculating projective spaces to X . For x E X , let Ik(x) denote the image of the homomorphism j k , , . Then P(lk(x)) C

P(V) is the k-th osculating (projective) space to X at x. We denote it q Osck(X).

2) If the generic rank of jk is s(k) +,I, the k-th dual variety, . Y l , of X is the closure in pV of the set of hyperplanes containing an s(k)-dimensional k-th osculating space to X. Note that X: is only defined when

for the general x E X. This automatically rules out the Veronese maniffolds (XI L) = (P, U p (k)) since for such manifolds Osck(X) = p for all x E X. In fact, according to [FKPT], they are the only ones with this property.

3) (XI V) is said to be k-regular if the homomorphism in (1.0.1) is surjective; i. e. if JkL is generated by the k-jets of sections in V. We say that (XI 1;) is k-regular if this property holds for V = HO(L). In particular, that (XI L) is 1-regular simply

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013


means that L is very ample. More geometrically, that (X, V) is k-regular means that, if X is embedded in = P(V), then X has a k-th osculating space of the maximal dimension a t every point. We also say that (X, V) is generically k-regular when the generation property holds on a dense Zariski open subset U c X .

4) If (X, V) is generically k-regular we can determine the expected dimension of X z . Actually rephrasing the above definition we see that X l is the image via the second projection X x IVI -+ IVI of the closure of the projective bundle Pu (Kk), where Kk is the dual of the kernel of the evaluation map U x V -+ (JkL)u. So, if (X, V) is generically k-regular, then

In particular, for the usual dual variety (classical case), we have dimX;/ 5 N - 1, and for the second dual variety of a surface X C with N 2 6, we have d i m x i 5 N - 4.

Note also that the Veronese manifolds ( P , O p ( k ) ) are k-regular, but X i is not defined for them.

For (X, V) generically k-regular we say that X has a degenerate k-th dual variety if dim X: is less than the expected dimension n + N - (kin). In this case we define the k-th dual defect 6k to be the difference between the expected and the true dimension of X z :

We denote by .rrk : P(Kk) -t X: the restriction of the second projection of X x P ( V ) . If Jk = 0, then .rrk is generically finite. Note that a priori it could be that deg xk > 1.

We note the following fact. As will be apparent later, it extends a known property of k-very ample, or simply k-spanned, line bundles [BSl, (0.5.1)].

(1.2) Lemma. Let (X, V) be a k-regular n-fold with V 2 HO(L). Then for every irreducible reduced curve C c X , we have LC 2 k.

Proof. Let x E C,, be a smooth point of C and let 21,. . . , z, be local coordinates on X a t x such that zl is a local coordinate on C,,. The surjectivity of the homomorphism V + JkL implies that there is a section s E V whose k-th jet a t x is that of 2:. Thus the divisor D defined by s does not contain C and we have LC = D C > (DC), = k. 0

This simple fact has the following consequence.

(1.3) Proposition. If ( X , V ) is k-regular and k 2 - 2 , then X C P(V) cannot contain any lines.

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013


Our first main result generalizes what is known in the classical case ((BS4, p. 331, [K, Theorem 2, iv) a t p. 1861).

(1.4) Theorem. Let (X, V ) be a k-regular n-fold for some subspace V C HO(L).

(i) If X l is nondegenerate, then cn(Jk L ) = deg .rrk deg X l ;

(ii) X l is degenerate zf and only zf cn(JkL) = 0.

Proof. Set h := (k:n) - n and let A be a general linear @ in (VI. Then A is the projectivization of the linear span < s o , s l , . . . ,ah > of h + 1 general se r~ ions in V. Consider the locus

of the singular points of multiplicity 2 k + 1 occurring for the elements in the linear system A. It coincides with the degeneracy locus Dh(4) of the bundle homomorphism

@fh+l) + JkL 4 : Q x

defined by h

'$(fo, f l : . . . I fh) = fijk(si). i=O

Note that 4 is represented by a (k:n) x (h + 1) = ( h + n) x (h i- 1) matrix, hence codim(Dh($)) 5 (h + n - h)(h + 1 - h) = n. Moreover J k L is spanncd, since by assumption ( X , V) is k-regular, and in addition 4 is general, since so, . : sh are so. By the general Bertini theorem [Ot, Teorema 2.81 all these condition:; imply that Dh(+) has the expected codimension, which is n. Therefore its cohomo ogy class is given by

in view of the Porteous formula (e. g. see [Ot, p. 291).

Let q and p be the projections of X x IV\ onto the first and the se~csnd factors. Since q maps the scheme P(ICk) n p-'(A) isomorphically onto the scheme Dh(+), we deduce the equality of zero-cyles: q.((p(Kk)] . [p'n]) = [Dh(+)]. (>n the other hand, the projection formula yields this equation:

Moreover p,[B(Kk)] is either degzk [X;] or zero, according to whether X t has the expected codimension or not. Therefore, by taking degrees and reca!ling (1.4.1), we get that

d e g ~ k deg X l , if codim(Xl) = h;

otherwise.

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013

4834 LANTEFU AND MALLAVIBARRENA

We cannot assert that, if (X, V) is a k-regular n-fold with k 2 2, then X: is always nondegenerate. Moreover, if Xi is degenerate, then the morphism z k :

P(Kk) -t X: has positive dimensional fibres. This means that every k-osculating hyperplane to X (k 2 2) osculates along a positive dimensional subvariety of X. Of course osculating hyperplanes are tangent, so these tangent hyperplanes have positive dimensional contact loci. As is known the contact locus of a general tangent hyperplane is a linear space (by the theorem of biduality), of dimension &. Note that J1 > 0 cannot hold, otherwise, our X would contain lines, contradicting Proposition (1.3). So if X is k-regular and k 2 2, then X;/ is non degenerate. Moreover we conclude from this argument that if (X, V) is a k-regular n-fold with degenerate X:, then the positive dimensional contact loci of the k-th osculating hyperplanes are not linear spaces.

To have a n example of non linear "higher contact loci", consider ( X , L) = ( P , O p z ( k ) ) with k 2 3. Then, Xy is nondegenerate. Let 1 c P be a line. Then the hyperplane of cutting on X the divisor D = kl is k-osculating to X along the image of 1 embedded by 1 L 1, which is a rational normal curve of degree k.

In Sections 3 and 4 we will give a more precise result about the nondegerated- ness of higher order dual varieties of surfaces.

(2.1) In the last decade several notions of higher order embeddings have been introduced and studied by Beltrametti and Sommese [BSl], [BS2], [BS3]. The strongest one corresponds to the projective embeddings associated with k-jet ample line bundles. Let L be a line bundle on an n-fold X and let k be a nonnegative integer. According to [BS2, 2.21 we say that L is k-jet ample if for any T distinct points 21,. . . ,x, on X and any r positive integers kl, . . . , k , such that k -k 1 = EL1 kt the evaluation map

is surjective, where mi stands for the maximal ideal a t xi.

We also recall two weaker notions. L is said t o be k-very ample if for every 0-dimensional subscheme 2 of X of length k + 1 the restriction homomorphism

is surjective. Finally L is said to be k-spanned if the above condition is satisfied for every curvilinear 0-subscheme of length k + 1, i. e. a subscheme 2 contained in a smooth curve. These three notions are each one stronger than the following one [BS2, Proposition 2.2); moreover for k = 1 all three are equivalent to the notion of very ampleness and for k = 2 the last two coincide [BS2, (2.3)].

We will use the following basic fact very frequently. If L is a k-very ample line bundle on a smooth surface S, then LC 2 k for every irreducible curve C c S [BS3,

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013


(0.3.1)]. For k-jet ample line bundles this also follows from Lemma (l.i!:, combined with the following observation: if L is a k-jet ample line bundle on an n-fold X I then (XI L) is k-regular. Actually, taking r = 1 above, we see that the map

sending a section of L to the evaluation of its k-jet at x, is a surjection for any x E X. Here m stands for the maximal ideal of Ox,,. But the second term above is just the fibre of J k L at x, so we get the desired surjection in (1.0.1).

(2.2)Remark. Let IF, be the Segre-Hirzebruch surface of invariant e and let Co and f be as in (0.4). For any line bundle L E Pic(Fe) we have L = [do + bf] for a, b E Z. If (Fe, L) is k-regular, then, by Lemma (1.2), we get

k 5 Lf = a and k 5 LCo = b - ae.

These conditions are the same as those characterizing k-jet ample linf bundles on Pel (IBS2, Proposition 5.11). Recalling the observation in (2.1) we can thus complete ([BS2, Remark 5-11) with the following statement: (F,, L) is k-reguleu if and only if L is k-jet ample.

(2.3)Example. On the other hand, there are k-regular pairs (XI L) s ~ c h that L is not k-jet ample. Let C be a canonical curve of genus 4 and set L = KC; then IL( gives an embedding in p. Projecting C from a general point p E C into a plane we get a plane quintic of genus 4. Hence there are 2 trisecants to C pawing through p. Therefore L is not 2-jet ample.

On the other hand, if C is general in the sense of moduli, thtm C has an osculating plane at every point, so L is 2-regular. To see this, let Q(2 P1 x P1) C be a smooth quadric surface and let C E / U p ,PI (3,3)1 be a smooth cLrve. Then C is of genus 4 embedded by the hyperplane bundle O p ,p(1 ,1) in @ as a canonical curve of degree d = 6. Now assume that C is not 2-regular. Then there exists a point x E C at which the osculating space OSC&(Z) is a line 1 instead of a plane. But then 1 is contained in Q; hence it belongs to one bf the rulings of Q, and

But this cannot occur if C is general.

Indeed, let F be the cubic form cutting out C on Q. Think of Q as the image of the Segre embedding 4 : P x P1 r described in affine coordinates by (t, s) H (t, s , ts). Then C is the zero locus of h := 4'F. Note that h is a polynomial of degree 3 in each indeterminate s and t and that every such polynomial can be obtained as 4' of some cubic form. Moreover if the line 1 C Q is defined by s = so, then by (2.3.1) x is a point (to, so) on Q such that to is a triple root of h(t, so) = 0. In other words, at the point (to, so), we must have that

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013


But now it is clear that for F general, hence h general, there are no such points on Q. Therefore the general C E JOpr (3,3)) is 2-regular.

This fact can also be proved using the Weierstrass points of C. Indeed, if C has a point x satisfying (2.3.1), then the gap sequence of C at x [ACGH, p. 411 cannot be G, = {1,2,3,5). Of course 1 E G, and 2 E G, since C is not hyperelliptic, but 3 $ G,. In fact h0(3x) = 2, the linear series 13x1 being cut out on C by the ruling containing 1. Therefore x is a non-normal Weierstrass point (e. g. , see [GH, p. 2741 or [ACGH, p. 421). In fact, the gap sequence at x is G, = {1,2,4,5). Actually h0(6x) = 3, since 16x1 is cut out on C by (211; on the other hand, for rn > 7 we have hO(rnz) = m + 1 - 4 = ho((m - 1)x) + 1, since rnx is nonspecial. Now, the fact that C contains a non-normal Weierstrass point implies that C is non-general in the sense of moduli, according to [GH, p. 2771.

Let (X, L) be a k-regular n-fold and denote by IL - (k + l)xl the sublinear system cut out on X by the k-th osculating hyperplanes to X at x. This notation is appropriate because for a section s E HO(L) the condition jk(s)(x) = 0 exactly means that the divisor D = ( s ) ~ has a singular point of multiplicity at least k + 1 at x.

(2.4) Proposition. Let L be a (k + 1)-jet ample line bundle on an n-fold X . Then irk is birational, X: is nondegenerate, and c,(JkL) = deg X z .

Proof. The last assertion follows from Theorem (1.4) provided that i rk is birational. Assume that nk is not birational. Then there exists a point x such that the general element D E IL - (k + l)xl has, other than x, some singular point of multiplicity k + 1 somewhere on X. But, by the Bertini theorem, the singularities of the general D lie in the base locus of IL - (k + 1)xI. Since L is ( k + 1)-jet ample, the assertion follows from the following lemma.

(2.5) Lemma. Let L be a (k + 1)-jet ample line bundle on a projective manifold X, and let x E X. Then lL - (k + l ) x ( has no base points on X \ {x).

Proof. Since L is (k + 1)-jet ample, for every y E X , y # x, the homomorphism

is surjective. In particular this implies that there exists a section s E r ( L ) with jk(s)(x) = 0 and s(y) # 0. 0

It pays to compare the statement of Proposition (2.4) with what is known for k = 1. If L is 2-jet ample, then X does not contain lines; hence X y is nondegenerate. On the other hand, if L is simply 1-jet ample (i. e. very ample), then Xy can be degenerate. Note that n1 is the Gauss map; so if Xy is nondegenerate, then nl is birational [Z, Theorem 2.3, p. 21). In some special cases we have the following better results (2.6) and (2.7) on the birationality of nk. We will refer to them in Section 4.

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013

HIGHER ORDER DUAL VARIETIES OF SURFACES

(2.6) Proposition. If (Fe, L) is k-regular, then n k is birational.

Proof. By (0.4) we have L = [aCo + bf], with

Fix a general point x E IF,. To show that x k is birational it is enough to produce an element D E (L - (k + 1)xI having no singular point of multip1icit:r 2 (k + 1) apart from x. We do so by induction on k. For k = 1, such a D exists, since, by assumption, L is very ample.

Now let k > 1. Note that hO(Co + e f ) = e + 2 2 2 and reciill that the general element of (Co + e f ( is irreducible, the reducible ones consisting of Co plus e fibres, e > 0 [Ha, Theorem 2.17, p. 3791. So, since x is general, there exists a smooth curve I? E JCo + efl passing through x. Now consider the line bundle L' := L - [I?] = [(a - l)Co + (b - e) f ] . Note that (IF,, L.') is (k - 1)-regulirr in view of (2.6.1) and (2.2), hence by induction the general element R E JL1- kx( has no other points of multiplicity 2 k apart from x. We have D = I? + R E JL - ( , I : + l )x ( . To see that D has no other points of multiplicity (k + 1) apart from x is : t enough to show that we can choose R with no singular points on I? \ {x}. To see this consider the exact sequence

In view of (2.6.1) we can apply [Ha, Lemma 2.4, p. 3711 to see that hl( iJ ' - [r]) = 0. Then the exact cohomology sequence induced by (2.6.2) shows that tke restriction homomorphism I?(L1) -i I?(L;) is surjective. Since Lk S Opl(s) witk s = Rr we thus conclude that there are elements R E lL1( cutting out on r divisors of the form kx + y k + l + . . . + ye, the y,'s being distinct points, i. e. , elements R having no singular points along J? outside of x.

For k = 2 we can prove a similar result for elliptic conic bundles. Let p : S -i C be a P1-bundle over a smooth curve of genus 1. According to (0.4) if (S,L) is a conic bundle then L = [2Co +p*V], with lY E Pic(C).

(2.7) Proposition. Let (S, L) be a 2-regular conic bundle over an elliptic curve. Then ~2 is birational.

Proof. Let (S, L) be as above, let b = degD and note that deg LC, 2 3, since L is very ample; hence b 2 2e + 3, equality holding if and only if Co embedded by JLJ is a plane cubic. Assume that

Let x E S be a general point and let f be the fibre of p through x. Since b 2 2 by (2.7.1), there exists a divisor in ID) containing the point p(x); hence there is a divisor D E JLI containing 7 as as a component. We prove that ther~e exists such a divisor having a triple point at x and no other triple points on S.

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013


Write D = T+ R. Then R E IL'J, where L' := [2Co +p9(D - p(x))]. Combining (2.7.1) with the conditions in [BiLi, Theorem 6.31 we conclude that L' is very ample. Then its linear system contains an element R having a double point at z and being smooth elsewhere. So since Rf = 2, the divisor D has no triple points outside of x and we are done.

It remains to consider the case where b = 2e + 3. In this case, let y E Co be a flex of the plane cubic Co, and choose local coordinates (2, w) on S around y in such a way that z is a local coordinate on Co and w is a local coordinate on the fibre through y. Then a local computation immediately shows that the homomorphism

j z : HO(L) 8 Os + JzL

has an image of dimension 5 5 at y. But this gives a contradiction since (S, L) was assumed to be 2-regular.

As we said the notion of k-jet ample line bundle 00 an n-fold was introduced by Beltrametti and Sommese in [BS2]. A previous conjecture of these authors [BSl, (2.6)] formulated for k-spanned line bundles L on a surface S attaining the lower bound for hO(L) was studied by Ballico [Ba] for k = 2. However the original conjecture can be more appropriately stated in the setting of k-jet ample line bundles. The corresponding expected result is generalized to k-regular surfaces as follows.

(2.8) Proposition. Let S be a surface and let L E Pic(S) be a line bundle such that (S, V) is k-regular for some vector subspace V C HO(L). Then dim V 2 ('ik) and equality holds 27 and only if (S, L) = (PI Op2(k)) and V = HO(L).

Proof. Since (S, V) is k-regular we have a surjective bundle homomorphism

(2.8.1) X x V -+ JkL.

Since rkJkL = ('ik), this gives the desired inequality. Now assume that equality holds; then (2.8.1) is an isomorphism, hence the bundle J k L is trivial, and then the assertion follows from Lemma (0.2). 0

(2.9)Remark. In the special case where V = HO(L) and L is k-jet ample, the above result, though not explicitly stated, can be read between the lines in [BS2, p. 358 and bottom of page 364). The proof given there relies on a result of Sommese [So, Prop. 111) on projective manifolds with trivial jet bundles. Note that our proof uses simply the triviality of the Chern classes of JkL.

jFrom now on we consider surfaces S . If (S, V) is k-regular k 2 2 for some subspace V C HO(L), then according to Theorem (1.4) C2(JkL) is the degree of Sl, provided that Si is nondegenerate and T,, is birational. Our aim is to investigate this degree. Note that the vector bundle JkL is spanned, since (S, L) is k-regular, hence cz(JkL) 2 0. In fact we are going to prove a more refined result under the assumption that L is k-jet ample. This will prove the nondegeneratedness of Sl

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013


for k-jet ample surfaces. Moreover the proof will give as a byproduct several lower bounds for deg S Y , according t o the Kodaira dimension n ( S ) o f S , which we will discuss more closely in case k = 2 in the next Section. W e need the following lemma.

(3.1) Lemma. Let L be a k-jet ample line bundle (k 2 1) on a surface S .

( i ) Then k K s + 3 L is very ample unless ( S , L ) = ( p , O p z ( k ) ) .

(ii) Moreover ( k - l ) K s + 3L is always very ample.

(iii) Furthermore ( k - l ) K s + 3L is (2k + 1)-jet ample unless ( S , 1,) is one of the following pairs:

( 1 ) ( p , O p a ( a ) ) , with k 5 a 5 3(k - 1) ; ( 2 ) (P1 x P1, Opl x p ~ (a , b)) , with k 5 min{a, b) 5 2k - 2; ( 3 ) S is a P1-bundle over a smooth curve and L f 5 2k - 2 for euely fibre f .

Proof. Since L is k-jet ample we have that L := 3L is 3k-jet ample (13S2, Lemma 2.21, hence 3k-very ample [BSZ, Proposition 2.21. As k 5 3k - 1 it thus follows from [BS3, Theorem 3.11 that k K s + L = k K s + 3 L is very ample apart from ,ive possible exceptions. A close inspection o f them immediately gives ( i ) . E. g. , in case (3.1.1) o f [BS3] we get ( S , L ) = ( p , OPa(3a)), k <_ a 5 3k, but in fact we see that k K s + L = Opa (-3k + 3a) is very ample unless k = a , giving the only expected exception. In case (3.1.2) we have ( S , L ) = (P1 x P1, Opi ,pi (3a, 3b)) , with k < min{a , b) 5 2k. However in this case we see that k K s + L = x p ~ (-2k + 3a, -2k + 5b) is always very ample. T h e remaining exceptions (3.1.3)-(3.1.5) in [BS3] can be handled similarly.

The same inspection also gives ( i i ) . For instance, i f we are in cii.;e (3.1.5) o f [BS3], then S is a Del Pezzo surface and L = - y K s . It cannot be that S = P; otherwise, L could not be k-jet ample. Similarly it cannot be that S = E" xP1. So S has to contain an exceptional curve, say E . But then k 5 L E = - y K s ~ = k-l 3 '

a contradiction.

As to (iii) , since L is k-very ample, it also follows from [BS3, 'Iheorem 3.11 that (k - l ) K s + L is very ample apart from the five possible exceptional cases (3.1.1)-(3.1.5) in [BS3]. T h e first three o f them easily lead to the exceptions ( I ) , ( 2 ) , ( 3 ) listed above. As t o the remaining cases, the same exception (3.1.5) quoted above does not occur. Otherwise, as before, for L = -(k - l ) K s and 4 containing an exceptional curve E , we would get k 5 L E = k - 1, a contradiction. Moreover case (3.1.4) in [BS3] also does not occur since it represents an exceptional case only for the nefness and bigness o f k K s + L. Finally, since very ampleness is equivalent to 1-jet ampleness, by using again [BSZ, Lemma 2.2) we see that the line bundle ( k - l ) K s + 3 L = ( ( k - l ) K s + L ) + L + L is (2k + 1)-jet ample apa-t from cases ( I ) , ( 2 ) , (3). This proves (iii). 0

Now we can prove our second main result.

(3.2) Theorem. If L is a k-jet ample line bundle on a surface S , thel;, c z (JkL) 2 0 with equality i f and only i f ( S , L ) = (P, Opa ( k ) ) .

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013


Proof. By (0.1.3) we can write

Let us look at the two summands A and B separately. Since L is k-jet ample, Lemma (3.1), part (i), shows that

with equality if and only if (S, L) = ( p , O p ~ ( k ) ) .

Now let q : S -t So and s be as in (0.3), So being a minimal model. Then

If n(S) 2 0, we have

This inequality follows from the Castelnuovo-de Franchis theorem and the fact that Kio = 0 when n(S) = 0 , l ; while it is just the Miyaoka-Yau inequality if n(S) = 2. So, if n(S) 2 0, then the assertion follows by combining (3.2.2), (3.2.3), (3.2.4).

Now assume that n(S) = -co. If S = PI then the summand B is zero, while, as we said before A vanishes only if L = Opz(k). If S is a rational surface, S # p, we can repeat the same argument as above taking So = 1F,, as noted in (0.3). We have 3c2(So) - Kzo = 4 and then by using again (3.2.2), (3.2.3) we conclude that A + B > O .

To deal with the non-rational case, we first prove the assertion when S = So: i. e. , S is a Q1-bundle over a smooth curve C of genus q = q(S) > 0. Let e be the invariant of S . According to (0.4), we have L r [aCo + bf] where the integers a and b have to satisfy the following conditions:

(3.2.5) a > k and 2 b 2 a e + l .

The first condition is equivalent to L f 2 k, which follows from the k-jet ampleness of L, and the second one simply comes from the ampleness of L, regardless of the sign of the invariant e [Ha, p. 382). Since Ks [-2Co + (29 - 2 - e) f], after a straightforward computation we get

A + B = 3(3a - 2k)(2b - ae) + (q - 1)(4k(3a - 2k) - 1).

By applying (3.2.5) we see that this term is bounded from below by 3k + (q - 1)(4k2 - 1); whence A + B > 0.

Finally, assume that S is a non-minimal ruled irrational surface, and let rr : S -t C be the ruling over the smooth curve C. In this case, the summand B in

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013

HIGHER ORDER DUAL VARIETIES OF SURFACES 484 1

and s > 0, since S is non-minimal. Now look at the summand A. By Lemma (3.1), part (i) we know that kKs + 3L is very ample. So, by applying the basic A-genus inequality in (0.6) to the pair (S, kKs + 3L), we get

(3.2.7) (kKs + 3 ~ ) ~ 2 h O ( k ~ s + 3L) - 2.

Set .L := (k- l)Ks +3L. We know by Lemma (3.1), part (ii), that L is very ample. Hence, from the Riemann-Roch theorem and the Kodaira vanishing thelorem, taking into account that hO(Ks) = 0, we get

Take a smooth curve I' E 1/51 and look at the surjective morphism 7i : I7 + C. From Lemma (3.1), part (iii), we see that

deg air 2 2k + 1,

unless (S, L) is one of the exceptions in the list, which, however cannot occur in the present case. Therefore the Riemann-Hurwitz theorem applied to nir gives

(3.2.9) g(C) > (2k+ l)(q - 1) + 1.

Putting together (3.2.7), (3.2.8), (3.2.9)) we get

(kKs + 3 ~ ) ' 2 (2k + l)(q - 1) + 1 - p - 2 = 2k(q - 1) - 2

Finally, adding the summand B given by (3.2.6), we get

A + B > (2k-4)(q- 1) +4s - 2 > 49-2 > 0.

This concludes the proof. 0

As a consequence of Theorem (3.2) and Proposition (2.4) we get the following nice fact, extending what is known in the classical case.

(3.3) Corollary. Let L be a k-jet ample line bundle on a surface S' and suppose that (S, L) # (p, 0p (k)). Then Sl is nondegenerate.'~oreover, ij L is (k + 1)-jet ample, then cz(JkL) = deg Sl.

(3.4) Remarks. i) Note that the analogue of Theorem (3.2) and its consequence (3.3) are true for curves. In fact, for n = 1, using the exact sequence (0.1.1), we can see easily that cI(JkL) = v ( k K x + 2L). So we are led to conjecture that the same is also true for higher dimensions at least for k large. In this case we would get the following consequence: if X is an n-fold and L is a k-jet ample line bundle on X (k large), then either (X, L) = (P, Opn (k)), or X: is nondegenerate.

ii) It also seems natural to conjecture that (3.2) and (3.3) are true simply assuming that (S, L) is k-regular. As the proof of (3.2) shows, the delicate case 1s that where S is an irrational ruled surface, where the k-jet ampleness assumption played a crucial role. However this case can be done also for (S, L) k-reguliar a t least for k = 2,3 as we will show in Section 4.

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013


iii) In view o f the good properties o f k K s + 3 L shown by Lemma (3.1), part ( i ) , and Theorem (3.2), recalling also (0.1.2), we conjecture that i f L is k-jet ample, then JkL is not just nef (being spanned by global sections) but even ample, apart from the obvious case ( S , L ) = ( P , O p = ( k ) ) . This statement is true at least for curves, as we can see by induction on k , using the exact sequence (0.1.1).

Here we produce some functions o f k expressing lower bounds for c 2 ( J k L ) in some instances, e. g. , assuming that S is not ruled, or o f general type with L k-jet ample. T h e following proposition generalizes a result in [ L l , Proposition 1.6, b ) ] (see also ( 1 ) in Proposition (4.3)) .

(3.5) Proposi t ion. Let L be an ample line bundle on a nonruled surface S and set d := L2. Then c z ( J k L ) 2 3(3ak)d for k 5 6 and equality holds if and only if S is either Abelian or hyperelliptic.

Proof. Write c 2 ( J k L ) as in (3.2.1) and argue as in the proof o f Theorem (3.2): let 7 : S -+ SO, s and E be as in (0.3) with So a minimal model. Then K s = q*Kso +E and LE 2 s since L is ample. Moreover K s , is nef and K s = K i o - s. Hence

Combining this inequality with the expression for B given in (3.2.3), we get

Taking into account the Castelnuovo-de Franchis theorem, for k 5 6 this inequality shows that A + B >_ 9d, with equality i f and only i f S = So , with K s numerically trivial and c z ( S ) = 0. In view o f the Enriques-Kodaira classification, this statement is equivalent to saying that S is Abelian or hyperelliptic. 0

In a similar vein we have the following result.

(3.6) Proposi t ion. Let L be a k-jet ample line bundle on a surface S of general type and set d := L2. Then c2(JkL) 2 3(3:k)(4k2 + 15k -k 59) for every k 2 1. f i r thenore , i f equality holds for some k , then S is minimal.

Proof. Using the same notation as before, we have

A = K i + 2 K s ( ( k - 1 ) K s + 3 L ) + ( ( k - l ) K s + 3L)'.

Hence, recalling (3.2.3) and Noether's formula, we get

A + B = 2(k - 1 ) ~ ; + 6 K s L + ( ( k - l ) L s + 3L)' + 3 c z ( S )

= 2(k - 1)(KZo - s ) + 6(q*Kso + E ) L + ( ( k - 1 ) K s + 3 ~ ) ' +

3(cz(so) + 3). So A + B is given by

(2k - 5 ) ~ : ~ + (5 - 2k)s + 36x(OsO) + 6(7*KsO + E ) L + ( ( k - l ) K s + 3L)'.

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013


Now look at the individual summands in the last expression. First o f all, both x(Os,) and K i n are positive, So being a minimal surface o f general type. Secondly, since L is k-jet ample, for every irreducible component E o f E, we have L E 2 k; hence EL 2 ks.

Set again C := ( k - 1 ) K s + 3L. Since S is o f general type, we know by Lemma (0.7) that

(3.6.1) L' 2 2h0(C) - 3.

Moreover, C is (2k + 1)-jet ample, by Lemma (3.1), part (iii). This fact, combined with Proposition (2.8), gives

( ( k - 1 )Ks + 3 ~ ) ' = C2 2 2( (2kl 3, + 1) - 3 = 4k2 + 10k 4- 5.

Finally, note that h0(2Kso) 2 2, since So is o f general type. Hence )27*Ks,I is a pencil, and it is easy t o see that, i f D E 127*Ks0), then g ( D ) = 1 +3.F:io 2 4. So, i f D is irreducible, from the k-jet ampleness o f L , we get LD 2 6+k (BS3, Corollary 1.61. On the other hand, i f all the elements D E 12q*Ksol are reducible, we can argue as in [ L l , Lemma 1.91, looking at the irreducible components o f jSi. Let Dl be one o f them. Then LD1 2 k , since L is k-jet ample. Note that i f g(DI) 5 1, then Dl is a fixed component, since a surface o f general type cannot contaiu a pencil or rational or elliptic curves.

On the other hand, as already observed, ID( is a pencil. Hencx* one o f the irreducible components o f D , say D2 , must have genus g (Dz ) 2 2. Then by applying again [BS3, Corollary 1.61 t o D2 , we get

LD 2 LD1 + L D 2 2 k + g ( D 2 ) + k + 2 > 4 + 2 k .

Thus, since LD is even, in any case, we obtain L D 2 6 + k . Henct: 6?*Ks,L = 3LD 2 18 + 3k. Putting together all these inequalities, we get

Then the assertion follows from (3.2.1) because s 2 0. 13

4. O N THE DEGREE OF THE SECOND DUAL VARIETY

In this Section, we first prove some results on cz(J2L) for L a very ample line bundle on S , and then we apply them to investigate the degree o f the second dual variety.

First, let us determine when c z ( J2 L ) < 0. Let S be a smooth complex projective surface, let L E Pic(S) be any ample line bundle, and set d = L2. Recalling (0.1.3), we have

(4.0.1) c2(JzL) = 6Ox(Os) + 20KsL + 15L2.

Let n ( S ) denote the Kodaira dimension o f S . By Proposition (3.5) we know that c z ( J z L ) > 15d > 0 , unless n(S) = -m.

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013


Now assume that K ( S ) = -00. First suppose that S = P ; then L = Opa(a) for some integer a 2 1. Since d = L2 = a2, we get from (4.0.1) that

c2(JzL) = 15(a - 2)'.

Hence c2 ( J 2 L ) > 0 unless ( S , L ) is the Veronese surface.

Now assume that S is ruled, but not p. By applying (0.3) again, we find that S dominates birationally a P1-bundle So over a smooth curve o f genus q = h l ( O s ) , and that

K: = K:, - s = 8(1 - q ) - s ,

where s 2 0. Since ~ ( 0 s ) = 1 - q, we get from (4.0.1) that

cz(J2L) - 5d = 6 0 x ( O s ) + lO(2KsL + L 2 )

= 6 0 x ( O s ) + 1 0 ( ~ : + 2 K s L + L ~ ) - 1 0 ~ :

Assume for a moment that ( S , L ) is not a scroll. Then we have ( K s + L)2 2 0 , K s + L being nef, and so we get

cz(J2L) 2 5d + 20(q - 1).

Hence c2(J2L) > 0 unless q = 0 and d 5 4. Note that, i f L is very ample, this situation can be easily handled using the projective classification, since rational surfaces o f degree d 5 4 are well known. In particular we see that either h o ( L ) 5 5 or (S , L ) is a quartic rational scroll in p, which is excluded by our temporary assumption.

Finally suppose that ( S , L ) is a scroll. T h e n ~ ( 0 s ) = 1 - q . Moreover, since ( S , L ) has sectional genus q , the genus formula gives L2 + L K s = 2q - 2 . Putting this information into (4.0.1), we get that

Therefore, i f ( S , L ) is a scroll, then cz(J2L) is negative unless q = 0 and d 5 4.

I f L is very ample, then the discussion above proves the following result.

(4.1) Proposition. Let S be a surface endowed with a very ample line bundle L with hO(L) 2 6.

( a ) If K s + L is nef, then cz(J2L) > 0 . ( b ) If K s + L is not nef, then c2(JzL) 5 0 , and equality holds i f and only if

( S , L ) is either (P , Up1 ( 2 ) ) or a quartic rational scroll of p.

In particular, it is clear from Proposition (4.1) that , i f cz(J2L) < 0 , then ( S , L ) is a scroll with either q > 0 or h O ( L ) 2 7.

Recall that, i f ( S , L ) is 2-regular, we have h O ( L ) 2 6 by Proposition (2.8). So Proposition (4.1) proves the conjecture in (3.4,ii) for k = 2 . W i t h a little more work, we can also prove the conjecture for k = 3.

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013


(4.2) Proposition. Let (S, L) be a 3-regular surface. Then c2(J3L) 2 0, with equality if and only if (S, L) = ( p , Op(3)) .

Proof. As in the proof of Theorem (3.2), consider the summands

A = 9(Ks + L ) ~ and B = 3cz(S) - K:.

First of all, note that S can contain neither lines nor conics in view of Lemma (1.2). So Ks + L is nef and big, unless (S, L) = ( p , Opl (3)) by standard results in adjunction theory [BS4, Chapter 71. Hence A > 0, with equality only in this exceptional case. If S is nonruled, then B > 0; see the proof of Theorem (3.2). If S is rational, then either S = or S dominates a surface !Fe via .a birational morphism, and then either B = 0, or by (3.2.3) we have B = 4+4s > 0, respectively.

Finally, assume that S is a ruled irrational surface. First of all note that, if q = hl(Os) = 1, then A > 0; however, by (3.2.6), we have B 2 0. IVe can thus assume that q 2 2. Then by [SV, Main theorem] Ks + L is very ample. Since the morphism given by ( K s +LJ is an embedding and S has irregularity q ;> 2, recalling (0.6) we see that

(Ks + L)' > hO(Ks + L).

On the other hand, by the Riemann-Roch theorem and the Koda.ira vanishing theorem, we have hO(Ks + L) = g - q, where g is the genus of the general element D E I L( and q = hl (Us) is the genus of the base curve C of S.

Let T : S -+ C be the ruling projection. Then the restriction rlo : D -+ C is a morphism of degree at least 3 by Lemma (1.2). Hence the Riemmn-Hurwitz formula gives g 2 3q-2. Combining all this information and recalling tkae expression for B given in (3.2.6), we finally get

as desired.

Several papers investigate projective surfaces S with a very ampla line bundle L, according to the difference p - hd for h = 1,2,3, where p = cz(J'1 L) = deg Sy is the class of (S, L) (e. g. , [Ll], [LT]). Keeping the classical case in mind here, we proceed similarly for 2-regular surfaces (S, L), although sometimes L is assumed 2-jet ample. More precisely, we investigate lower bounds for the difkence v - hd where h is a suitable positive integer and

In addition, we characterize pairs (S, L) whose differences are netar these lower bounds.

Recall (see Propositions (2.4) and (2.6)) that, if (S, L) is 2-reg,ular and TZ is birational, then v is the degree of the second dual variety Si of S emhedded by IL1,

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013


apart from the obvious exception given by the Veronese surface. So assume that ( S , L ) is 2-regular and that

Then by (4.0.1) we can write

As we already observed, if n ( S ) 2 0, then by Proposition (3.5) we have

with equality if and only if S is either an Abelian or a hyperelliptic surface.

Now assume that v - 15d > 0. Recall that K s L 2 0 and ~ ( 0 s ) 2 0 because n ( S ) 2 0. Then v-15d 2 20 by (4.3.2), and equality occurs if and only if x (Os) = 0 and LKs = 1. We show that this cannot occur if ( S , L ) is 2-regular.

Let q : S 3 SO, s and E be as in (0.3). Recall that L E 2 2 for every irreducible component of E by Lemma (1.2). Since Kso is nef, we thus get

with equality if and only if Kso is numerically trivial. So s = 0 and LKs = 1. Note that n(S) # 2 because ~ ( 0 s ) = 0. Hence the condition LKs = 1 says that S is an elliptic surface. But since ~ ( 0 s ) = 0, we conclude that S is an elliptic quasi-bundle. Then the same argument as in [L2, (2.2)] shows that the base curve of S has genus 0 or 1. In both cases,

since L is very ample (L2, Proposition (1.5) and end of (2.2)]. This inequality gives a contradiction. So v - 15d 2 40 by (4.3.2).

Assume that equality holds. Then x ( O s ) = 0 and LKs = 2. Consider again the birational morphism q : S -+ So as in (0.3). We have 2 = LKs = Lq*Ks, + L E 2 2s with equality if and only if Kso is numerically trivial. As before, if s = 0, then S is an ellipic quasi-bundle satisfying (4.3.3), and we get a contradiction. On the other hand, if s > 0, then s = 1, E = E is an exceptional curve with LE = 2 and Kso is numerically trivial. Combined with ~ ( 0 s ) = 0, this conclusion implies that So is either Abelian or hyperelliptic. This argument proves the following result.

(4.3) Proposition. Let L be a very ample ample line bundle o n a surface S of non-negative Kodaira dimension. Let d = L2 and v = cz(J2L).

(1) T h e n v - 15d 2 0 with equality if and only i f S is Abelian or hyperelliptic. (2) If the inequality i n (1) i s strict and ( S , L ) is 2-regular, then v - 15d 2 40,

with equality i f and only i f S is a n Abelian o r a hyperelliptic surface So blown-up at a single point p, and L = q f L o - 2 E , where Lo is a suitable line bundle o n So, where q : S -t So is the blowing-up and where E denotes the exceptional curve corresponding to p.

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013


Note that the situation in (2) occurs. In fact, by [L2, Theorem 3.41, it is possible to find a very ample line bundle to on an Abelian or a hyperelliptic suiface So such that the corresponding line bundle C := q*CO - E is very ample on S. So, by (BS2, Lemma 2.21, the line bundle L := 2C is 2-jet ample on S. Hence (S, L ) is 2-regular and v - 15d = 40.

Restricting our attention to surfaces of general type we, get the follciwing result.

(4.4) Proposition. Let L be a 2-jet ample line bundle on a surface S of general type. Then v - 15d 2 140, and if equality holds then S is minimal.

Proof. Since n(S) = 2, we know that ~ ( 0 s ) > 1. Let q : S + So, s and E be as in (0.3). Since L is 2-jet ample, by the argument in the proof of (3.6) we know that L7*Kso = i L D 2 4. Hence LKs = Lq*Kso + LE 2 4 + 2s. Therefore by (4.3.2) we get v - 15d 2 60 + 20(4 + 2s) 2 140. 0

Assume that n(S) = -00. First suppose that S = p. Then L = O p ( a ) for some integer a 2 3 because of (4.3.1) and the 2-regularity assumption. Since d = L2 = a2, we get from (4.3.2)

This equation shows that Y -5d > -30, with equality if and only if a = 3. Moreover, if a 2 4, then

= -20, if a = 4; v - 5d ( 2 10, if a 2 5, with equality for a = 5.

Now assume that S is ruled, but not p. Then (4.0.2) gives

Let us look at the three summands above. Recall that (S, L) cannot oe a scroll by Proposition (1.3) since (S, L) is 2-regular; so (Ks + L)2 2 0, with equality if and only if (S, L) is either a conic fibration or a Del Pezzo pair. The secmd summand is nonnegative unless S is rational. The third summand is zero if and only if S is a P1-bundle, and it is a positive multiple of 10 otherwise. All this shows that

with equality if and only if S is a rational PI-bundle, and at the same time (S, L) is a conic fibration or a Del Pezzo pair, not containing lines. In other words, equality occurs in (4.5.2) if and only if S = IF, for some e and L = [2Co + bf], where, according to Remark (2.2), b is an integer satisfying the condition b 2 2e + 2. Note that this description also includes (P1 x P1, Up x p ~ (2,2)).

Now assume that (4.5.2) is a strict inequality. Then looking again at the three summands in (4.5.1), we see that

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013

4848 LANTERJ AND MALLAVIBARRENA

If equality occurs, then q = 0 and ((Ks + L) ' , s ) = (0 , l ) or (1,O). The former case cannot occur; otherwise, (S, L) would be either a conic fibration with a reducible fibre or a nonminimal Del Pezzo pair; in both cases, (S, L) would contain a line, contradicting the 2-regularity by Proposition (1.3). In the latter case, S = Fe for some el so that [BS4, Theorem 10.2.31 implies that (S, L) contains some (-1)-line. This contradicts the 2-regularity by Proposition (1.3). Therefore (4.5.3) is a strict inequality, and then (4.5.1) again gives

Even in this case, we can characterize equality. Indeed, equality in (4.5.4) implies either q = 1 and (Ks+L)' = s = 0, so S is a P1-bundle over an elliptic curve, L giving it the structure of a conic bundle, or q = 0 and ( ( K s + L ) ~ , s ) = (0,2), ( I l l ) or (2,O). In the first case (S, L) is either a conic fibration with 2 reducible fibres or a nonminimal Del Pezzo pair. But then the existence of lines contradicts the 2-regularity. In the second case, we have Ki = Ki0 - 1 = 7. So by [BS4, Theorem 10.2.31 we see that (S, L) contains a (- 1)-line, but this statement contradicts again the 2-regularity. Finally, in the third case S = F, for some e , and then [BS4, Theorem 10.2.31 again shows that (S, L) = (P' x P1, 0 p 1 (3,3)).

Summing up the above discussion, we finally get the following theorem.

(4.5) Theorem. Let ( S , L ) be a 2-regular surface with L2 = d. Assume that (S, L) is not the Veronese surface and let v = cz(J2L). Then

(1) v - 5d = -30 if and only zf ( S , L) = (p, Op2 (3)); (2) v - 5d = -20 if and only 2 f (S, L) is either (PI Opz ( 4 ) ) or (Fe, [2Co + b f ] )

with b 2 2e t 2; (3) v - 5d = 0 i f and only if (S, L) is either (P' x P1, 0 p 1 (3,3)) or a conic

bundle over an elliptic curve.

Apart from these cases, we have v - 5d 2 10.

Note that, in all the exceptional cases listed above, 7 ~ 2 is birational by Proposi- tions (2.4), (2.6) and (2.7). So the above result can be viewed as a 2-regular analog of what was proved for deg Sy - d by Marchionna and Gallarati in the 1950s (see [Ll, (0.1) and (0.2)]).

In this Section, we make some remarks about examples of non k-regular pairs (XI L) to put in perspective our results and to suggest further possible investigations of analogous problems.

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013


(5.1) Pairs (X,V) with V C_ HO(L), which are generically k-regulzn-. We point out here the analogy with the theory in [LPS]. Namely, X l is the ;analog of the main component of the discriminant locus. Denote by Dk(X, V) the locus in IV) parametrizing the elements having a (k + 1)-tuple singular point. Then it should be possible to analyze the irreducible components of Dk(X, V) other than X l .

As an example, consider a smooth curve X C = P(V) for N 2 3 with L simple flexes. Then (X, V) is generically 2-regular, since it is 2-regular on the complement of the set of flexes. If N = 3, then X$ C pV is a curve, auld using [Pl, Theorem 5.1 and Theorem 3.21, we find deg XZV = 3(d + 29 - 2) - L, wh~ere d = deg L and g stands for the genus of X . On the other hand, any plane of containing the tangent line to X at a flex x gives rise to plane section having a. triple point at x. For every flex x, such planes form a line in pV. So D2(X, l J ) consists of X i and L linear subspaces of codimension 2. Thus there are extra components in Dk(X, V) if (X, V) is not k-regular. To relate the degree of X i with (-1 (JzL), note that cl(J2L) = 3Kx + 3L by (0.1.1). Since degcl(J2L) = degX$ -- L, therefore deg cl(JzL) is the degree of the whole second discriminant locus Dz(X, V).

As an example of surfaces, consider the general complete intersection S of three quadrics in P. Its hyperplane bundle L is 2-spanned [Ba], hence 2-very ample; whereas (S, L) is not 2-regular by Proposition (2.8), hence L is not 2-jet ample. However, by [BSl, Proposition 2.51, (S, L) is generically 2-regular. The locus of points x E S where the d im0sc ; (~ ) < 5 is a curve C E (6L( by I:Sh, (0.3),(a)]; hence its degree is 48. As c2 (Ja L) = 240, we see that D2 (X, L) C I*" contains a number of lines corresponding to the points y E C where dirnOsc~(5;: = 3, instead of 4. Note that, in this case, the second dual variety does not exist b:y (1.1.1).

(5.2) Pairs (X, V) which are nowhere k-regular. This case is typi1:al of scrolls. Descriptions of higher order dual varieties of rational or elliptic normal scrolls and scrolls in p are given in [P2], [PSI and [MP], including some computations for d i m x i and degX$. For surfaces in p, we quote here the result of Shifrin, who proves, among other things, that the quartic rational scroll (P1 x P1, Up1 x P ~ ( l ,2) ) is the unique uninflected ruled surface S c [Sh, Theorem 4.31. firthermore, S is also the unique perfectly hypo-osculating surface in p, i. e. din10sc; = 4 at every point x E S. This fact, conjectured by Shifrin, was proved by 13iene and Tai [PT]. Moreover, in [PT] and [BaPT] this result is generalized by a cimracterization of balanced rational normal scrolls in terms of their osculating spaces.

[ACGH] E . Arbarello, M. Cornalba, Ph. Griffiths, and J . Harris, Geometry of Algebmic Curves, Volume I , Springer-Verlag, New York, 1985.

[Ba] E . Ballico, A characterization of the Veronese surface, Proc. Amer. Math. Soc. 105 (1989), 531-534.

[BaPT] E . Ballico, R. Piene, and H.S. Tai, A chamcterization of balanced mtiontrl normal surface scrolls in terns of their osculating spaces, II, Math. Scand. 70 (1992), 204-206.

[Be] A. Beauville, L'application canonique pour les surfaces de type g i n h l , Invent. Math. 55 (1979), 121-140.

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013

LANTERI AND MALLAVIBARRENA

M. Beltrametti and A. J. Sommese, On k-spannedness for projective surfaces, Algebraic Geometry, Proc. L'Aquila, 1988 (A. J. Sommese et al., eds.), Lecture Notes in Math., vol. 1417, Springer, 1990, pp. 24-51.

, On k-jet ampleness, Complex Analysis and Geometry (V. Ancona and A. Silva, eds.), The Univ. Series in Math., Plenum Press, 1993, pp. 355-376.

-, On the preservation of k-very amplene~s under adjunction, Math. Z. 212 (1993), 257-283.

, The Adjunction Theory of Complez Projective Varieties, Expositions in Math., vol. 16, De Gruyter, 1995.

A. Biancofiore and E. L. Livorni, On the genw of a hyperplane section of a geometrically ruled surface, Ann. di Mat. Pura ed Appl. (IV) 147 (1987), 173-185.

M. Demazure, Surfaces de Del Pezzo, I-V, Sdminaire sur les Singularit& des Surfaces, Lect. Notes in Math., vol. 777, 1980, pp. 23-69.

L. Ein, Varieties with small dual varieties, I, Invent. Math. 8 6 (1986), 63-74.

[FKPT] W. Fulton, S. Kleiman, R. Piene and H. Tai, Some eztrinsic chamcterizatioru of the projective space, Bull. Soc. Math. fiance 113 (1985), 205-210.

Ph. Griffiths and J. Harris, Principles of Algebmic Geometry, Wiley-Interscience, New York, 1978.

R. Hartshorne, Algebmic Geometry, Springer - Verlag, New York - Heidelberg - Berlin, 1977.

P. Ionescu, Embedded projective varieties with small invariants, Algebraic Geometry, Proc. Bucharest, 1982 (L. Bgdescu et al., eds.), Lecture Notes in Math., vol. 1056, Springer, 1984, pp. 142-187.

S. L. Kleiman, Tangency and duality, Proc. 1984 Vancouver Conference in Algebraic Geometry (J. Carrel1 et al., eds.), Canad. Math. Soc. Conf. Proc., vol. 6, 1986, pp. 163- 225.

A. Lanteri, On the class of a projective algebraic surface, Arch. Math. 45 (1985), 79-85.

, On the class of an elliptic projective surface, Arch. Math. 6 4 (1995), 359-368.

A. Lanteri and F. Tonoli, Ruled surfaces with small class, Comm. Alg. 24 (1996), 3501- 3512.

A. Lanteri, M. Palleschi and A. J. Sommese, On the discriminant locw of an ample and spanned line bundle, J. reine angew. Math. 4 7 7 (1996), 199-219.

R. Mallavibarrena, R. Piene, Dudity for elliptic normal surface scrolls, Enumerative Algebraic Geometry (S. L. Kleiman and A. Thorup, eds.), Contemporary Mathematics, vol. 123, Amer. Math. Soc., 1991, pp. 149-160.

G. Ottaviani, Varieth proiettive di piccola codimensione, Notes of INDAM course, 1st. Naz. di Aka Matem., Roma, 1995.

R. Piene, Numerical characters of a curve in projective space, Real and Complex Singu- larities (P. Holm, ed.), Proc. Symposium in Math., Oslo, 1976, Sijthoff and Noordhoff, 1977, pp. 475495.

, A note on higher order dual varieties, with an application to scrolls, Singu- larities (P. Orlik, ed.), Proc. Symposia Pure Math., vol. 40, Amer. Math. Soc., 1983, pp. 335-342.

R. Piene and G. Sacchiero, Duality for mtional normal scrolls, Comm. Alg. 12 (1984), 1041-1066.

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013


R. Piene and H. S. Tai, A chamcterization of balanced rational normal scm11s in terms of their osculating spaces, Enumerative Geometry, Proc. Sitges, 1987 (S. Xarnlm-Descarnps, eds.), Lecture Notes in Math., vol. 1436, Springer, 1990, pp. 215-224.

T. Shiirin, The osculatory behaviour of surfaces in P, Pac. J. Math. 123 (1986), 227- 256.

A. J. Somrnese, Compact complez manifolds possessing a line bundle wlt4 a trivial jet bundle, Abh. Math. Sem. Univ. Hamburg 47 (1978), 81-91.

A. J. Somrnese and A. Van de Ven, On the adjunctron mapping, Math. A n ~ i . 278 (1987), 593-603.

F. L. Zak, Tangents and Secnnts of Algebmic Varieties, Translations orf Math. Mono- graphs, vol. 127, Amer. Math. Soc., 1993.

Received: S,eptember 1998

Revised: Oclober 1998

Dow

nloa

ded

by [

Geo

rge

Mas

on U

nive

rsity

] at

02:

27 1

1 M

ay 2

013

Date post:	09-Dec-2016
Category:	Documents
Upload:	raquel
View:	215 times
Download:	0 times

Higher order dual varieties of projective surfaces

Documents