Linear series on k -gonal curves

Ann. Univ. Ferrara - Sez. VII - Sc. Mat. Vol. XLVII, 1-8 (2001) Linear Series on k-gonal Curves. E D O A R D O B A L L I C O (*) - C L A U D I O F O N T A N A R I ( * * ) SUNTO - In questo lavoro si dimostra il seguente teorema. TEOREMA 1.1. Sia X una curva proiettiva ridotta e irriducibile di genere aritmetico g e k >t 4 u n intero. Si supponga l'esistenza di L ~ Pick(X) con h~ L) = 2 e L generato. Si fissi un fascio senza torsione di rango uno M su X con h~ M) --"r + + 1 I> 2, h I(X, M) >>-2 e M generato dalle sue sezioni globali. Si ponga d := deg (M) e M | (L*) | > 0}. Allora si verifica uno dei casi seguenti: s := m a x { n I> 0 : h~ (a) M --- L | ; (b) M ~ il sottofascio di ( Z x | | t:=g-d+r-1, generato da H~ Wx| (L .)~t); F) ~<k - 2 (c) esiste u n fascio senza torsione di rango uno F su X con 1 ~<h~ tale che M -- L | | F. Inoltre, se si fissa u n intero m con 2 <<.m <~k - 2 e si suppone r;~ (s + 1 ) k - (ns + n + 1) per ogni 2 <<.n<~m, si ottiene h ~ Si ricavano anche altre maggiorazioni su h~ ABSTRACT - F). Here we prove the following result. THEOREM 1.1. Let X be an integral projective curve of arithmetic genus g and k >t >I4 an integer. Assume the existence of L ~ Pic k(X) with h o(X, L) = 2 and L spanned. Fix a rank 1 torsion free sheaf M on X with h~ M) = r + 1 I> 2, h i ( X , M) >I2 and M spanned by its global sections. Set d : = d e g ( M ) and s:= max{n~>0: h~ M | (L*) ~n) > 0}. Then one of the following cases occur: (a) M - L | (b) M is the subsheaf of W x | | t:=g-d+r-1, spanned by H ~ X, ~ z | (L*)| F) <<.k - 2 such that (c) there is a rank 1 torsion free sheaf F on X with I <~h ~ M ~- L | | F. Moreover, i f we f i x an integer m with 2 <~m <<.k - 2 and assume r~(s+l)k-(ns+n+l)perevery2<<.n<~m, We find also other upper bounds on h ~ wehaveh~ F). (*) Indirizzo dell'autore: Dept. of Mathematics, University of Trento, 38050 Povo (TN), Italy. E-mail: [email protected] (**) Indirizzo dell'autore: Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy. E-mail: [email protected] 2 1. - EDOARDO BALLICO - CLAUDIO FONTANARI Introduction. Our research was suggested and stimulated by the reading of the two papers [CM1] and [CM2]. The following theorem, which arose as a generalization of [CM1] (1.2.2), is only a first step towards a classification of (generalized) linear series on a (singular) k-gonal curve. THEOREM 1.1. Let X be an integral projective curve of arithmetic genus g and k >I 4 an integer. A s s u m e the existence o f L ~ Pick(X) with h ~ L) = 2 and L spanned. Fix a rank 1 torsion free sheaf M on X with h ~ M) = r + + 1 >I 2, h i ( X , M) >t 2 and M spanned by its global sections. Set d := deg (M) and s := max{n t> O: h ~ M | (L * ) | > 0}. Then one of the following cases occur: (a) M ~ L | H~ (b) M is the subsheaf of w x | W x | (L*)| | t:=g- d +r-1, spanned by (c) there is a rank 1 torsion free sheaf F on X with 1 <. h ~ F) <~k - 2 such that M -- L | | F. Moreover, i f we f i x an integer m with 2 <~m <. k - 2 and assume r # ( s + l ) k - ( n s + n + l ) for every 2<~n<~m, we have h~ F) <<.k - m - 1. For other upper bounds on h ~ F) see corollaries 2.3 and 4.3. Of course, this kind of results is well known for low gonality; when k = 2 (see [ACGH], I, ex. D - 9 for the smooth case and [B2], prop. 2.3, for the Gorenstein case), k = 3 (see [M] or [MS], prop. 1, for the smooth case and [B2], th. 2.8, for the Gorenstein case) or k = 4 (see [CM1], th. 1.2.2); for k I> 5, however, it seems to have been unheard - of even in the smooth case. In order to prove our main result we will need a generalization of the following theorem, which is attributed to Eisenbud (see [E], 5.2, where it appears as a corollary of a much deeper result, and also [CM1], (1.2.1), where a direct proof is given at least in the smooth case): THEOREM 1.2 (Eisenbud). Let C be a reduced irreducible projective curve. I f g,~ and g~(r, s >I 1) are complete and base point free linear series on C such that Ig,~ + g~ml has the m i n i m u m possible dimension r + s, then there is a complete and base point free linear pencil gl on C such that g,~ = rg I and g,~ = sg I . We may ask what happens if g~, g~ are not linear series, but generalized linear series, corresponding to isomorphism classes of non locally free, but torsion free sheaves. It is well known (the standard reference here is the pa- LINEAR SERIES ON k-GONAL CURVES 3 per [H]) that we can add two generalized divisors only if at least one is Cartier. So even to pose the problem we must assume that one of the two series (say g~) is a base point free linear series. The other one, g~, will be a priori only a generalized linear series, free in the sense of Coppens (see [C] for a precise definition, but roughly speaking it means that the corresponding sheaf is spanned). We claim that, in the hypotheses of Eisenbud's theorem, also g~ is forced to be a base point free linear series (see proposition 2.1 for a precise statement). We work overan algebraically closed field K. This research was partially supported by MURST. 2. - E i s e n b u d ' s type results a n d t h e c l a s s i f i c a t i o n t h e o r e m . In this section we develop the project outlined in the above introduction. PROPOSITION 2.1. Let C be a reduced irreducible projective curve and let g~ = IRI and g~m= IS I, where R and S are spanned torsion free sheaves. Assume that R is locally free and that Ig~ + g~ I has the m i n i m u m possible dimension r + s. Then also S is locally free and g,~ and g,~ are composed of the same rational involution. PROOF. If g~ is base point free, the thesis follows from Eisenbud's theorem 1.2. Assume on the contrary that S is non locally free. The idea is to adapt to this situation the beginning of the proof of [CM1], (1.2.1). Define k = = min ( r , s} and let z : C--->C' r the morphism associated with the g~. Fix Q1, ..., Qk general points of C ' ; for a general hyperplane H o P ~ with (Q1, ..., Qk) c H we have z - l ( H n C ' ) n Sing(C) = •. Choose P i e Creg with z(Pi)=Qi, i=l,...,k. Let D e l g ~ ( - P 1 . . . - P k ) l and D ' e l g , ~ ( - P l . . . - Pk) I, with D, D ' general (notice that one of them, according to k = r or k = s, will be the only divisor in its series). We have g~ = ID + P1... + P~ I ; since S is assumed non locally free and P 1 , .--, Pk are smooth, D is not Cartier and must contain at least a singular point. On the other hand, by the choice of the Pi, D ' does not contain any singular point and it is Cartier. Hence we may define the greatest common divisor D n D ' of D and D ' and we have D n D ' ~ D. From now on we will copy almost verbatim [CM1], top of page 6. We have D = (D n n D ' ) + F , D ' = (D n D ' ) + F ' for some effective divisors F , F ' of C such that F n F ' = 0 and F ~ 0. Since D + F ' = D ' + F (notice that both sides are well defined because D ' and F ' are Cartier), we obtain g~ + F ' = ID +P1... + +Pkl+F'c_lD+Pl...+Pk+F'l=lD'+P1...+Pk+rl=lg~+FI. Since F n F ' = 0 it follows that gn~ + F is not complete (i.e. F ~ 0 is not a fLxed divisor 4 EDOARD0 BALLIC0 - CLAUDI0 FONTANARI of the complete linear series Ig~ + F I ). Thus dim Ign~ + F I > dim (g~ + F ) = r and so d i m l g ~ + g ~ - P 1 - . . . - P k l = d i m l g , ~ + D l > ~ d f m l g ~ + F l > r . But then dim Ign~ + g~ I > r + k (recall the Pk 9C are general). If k = s, we have just reached a contradiction. Assume now k = r. As before, since D + F ' = D ' + F we obtain g ~ + F = ID' +P1... +P~I + F o l D ' + P 1 . . + P ~ + F I = ID +P1... + + P~ + F ' I = Ig~ + F ' I" Since F N F ' = 0 it follows that g~ + F ' is not complete. Thus dim Ig~ + F ' I > dim (g~ + F ' ) = s and so dim Ig(~ + g~ - P1 - . . . - P r I = = dim Ig~ + D ' I >~ dim Ig~m+ F ' I > s. But then dim Ig~ + g~ I > s + r, a contradiction again. LEMMA 2.2. Let X be an integral projective curve and L 9Pic (X) with h~ L) = 2 and L spanned. Let F be a rank 1 spanned torsion free sheaf with r:=h~ F ) - 1 > 0 . Assumed s : = m a x { n ~> 0: h~ F| | > > 0 } >t 1. Then 0 < h ~ F@ (L *) | <~r + 1 - h ~ F @ L *). PROOF. Set V:= H ~ L). Consider the multiplication map u : F| - - ) F | L . Since Vspans L and L is locally free, a local computation gives the surjectivity of the map u. There is a non trivial map u : F @ L * - - ) K e r (u). Since F | L * is torsion free, v is injective. Since u is surjective and L is locally free we have deg (Ker (u)) = 2 deg (F) - deg (F) - deg (L) = deg ( F | L * ). Hence v is an isomorphism. Thus we obtain an exact sequence O--->F | L * --~F | V---) F | L --* O . (1) If/~ : H ~ F| dim Im(/~)<~h~ F| nh~ (L*) | h~ F| | follows that 0 < h~ *)@ V--)H~ F) is the multiplication map, obviously so b y tensoring (1) with L * we obtain h~ F| *) - (r + 1 - h ~ F| and by induction on F| | ( r + 1 - h~ F| It ~> h~ F | (L*)| ~<r + 1 - h ~ F| as claimed. - PROOF OF 1.1. The cases 1 ~< r ~<k - 3 are trivial: if s = 0, just take F := := M - M | (L *)| if s I> 1, we can, apply lemma 2.2 with F := M and obtain 0 < h~ M| | ~<r+ 1 - h~ M| ~< k - 2 - h~ M| <. ~< k - 2. So we may assume r 1> k - 2. We have h ~ M| < h~ M) = = r + 1. I f h o(X, M | L *) = r we are in case (a) by proposition 2.1; we turn next to the case h o(X, M | L *) ~< r - (k - 2). By the exact sequence (1) with F := := M w e obtain h~ F| >~r+ k and so by Riemann-Roch and the duality on locally Cohen-Macaulay schemes ([AK]) we obtain h ~ Hom ( F | Wx)) I> >~g-l-(d+k)+(r+k)=g-d+r-l=h~ Hom(F, w x ) ) - l . Since Hom(F| fox) - Horn (F, W z ) | and Hom (F, Wx) is torsion free, using LINEAR SERIES ON k-GONAL CURVES 5 proposition 2.1 we obtain t h a t the subsheaf, A , of H o m ( F , Wx) spanned b y Horn ( F , w x)) is isomorphic to L | with t = g - d + r - 1. Since H o m ( H o m (B, Wx), Wx) ---B for e v e r y torsion free c o h e r e n t s h e a f B on X b y M| local duality ([AK]), we are in case (b). N o w a s s u m e h~ - (k - 3). I n this case certainly s I> 1, so we can apply l e m m a 2.2 with F:= M and obtain 0 < h ~ 1 7 4 1 7 4 1 7 6 1 7 4 - (k - 3 )) = k - 2, with equality (possibly) holding only if h o (X, M @ L * ) = r - (k - 3). So we are in case (c); in o r d e r to prove the final r e m a r k , we set m : = 2 (the g e n e r a l case is not m o r e difficult, only notationally messier) and a s s u m e F ) = k - 2. I f s = 0 and G is the s p a n n e d p a r t o f F , on the c o n t r a r y t h a t h~ we have G ~ M so r + 1 = h ~ M) = h~ G) = h~ F) = k - 2 , which implies r = k - 3, contradiction. N o w a s s u m e s > 0. I f M ~ L | | F we have h~ unless h ~ 1 7 4 1 7 6 1 7 4 unless r - ( k - 3 ) = So, b y induction on s, we have h ~ = h 0 (X, L | 1| = sk - (2 s + 1 ) + 1, which implies r = (s + 1 ) k - (2 s + 3). This contradiction ends the proof. H~ REMARK 1.3. lowing invariant: (2) U n d e r the h y p o t h e s e s of t h e o r e m 1.1 w e introduce the fol- v(M, L) := h~ - h~ | *) F r o m the p r o o f of t h e o r e m 1.1 we easily obtain: (i) M is in case (a) r v(M, L) = 1. (ii) M is in case (b) r v(M, L) >t k - 1. (iii) M is in case (c) ~ 2 ~< v(M, L ) ~< k - 2. Moreover, if we are in case (c), we have 1 ~< h ~ F) <<.v(M, L). Finally, f r o m t h e p r o o f of l e m m a 2.2 we see t h a t s > ~ [ h ~ 1 7 4 Thus in the s t a t e m e n t of t h e o r e m 1.1 we m a y take -h~174 s >I [r + 1 - v(M, L)/v(M, L)]. COROLLARY 2.3. With the notations of theorem 1.1, we have h ~ ~< min {k - 2, r - s}. F ) ~< r+I=h~176174176174 PROF. W e have + h o (X, F ) - 1 = s + h o (X, F), so h o (X, F ) ~< r - s + 1. I f the equality holds, b y the proposition F = L ~r-~, so M = L | and we are in case (a). H e n c e h~ F)~< r - s , and the claim is proved. 6 EDOARDO BALLICO - CLAUDIO FONTANARI 3. - A r e m a r k o n l i n e a r s e r i e s o n s i n g u l a r k - g o n a l c u r v e s . The proof of [B1], Prop. 1, gives verbatim the following results 3.1 and 3.2. THEOREM 3.1. F i x positive integers g, k, a, with a >13, k I> 3, and (a - 2)(k - 1 ) < g <~ (a - 1 )(k - 1 ). F i x an integral curve C r P 1 • p 1 of type ( k , a ). F i x S c Sing(C) with card (S) = ( a - 1 ) ( k - 1) - g and assume that C has only ordinary nodes or ordinary cusps at every point of S. A s s u m e that S imposes independent conditions to f o r m s of type ( k - 2 , 0), i.e. that the points of S have different images under the projection P 1 • p i __.>p 1 on the second factor. Let ~ : X---* C be the partial normalization of C at the points of S; hence pa (X) = g. Let L 9Pic k(X) be the degree k linear series on X induced by the composition of ~ and the projection P 1 • p 1..~ p i on the second factor. Then for every integer r with 0 <~r <~a - 2 we have h ~ L | = r + 1 and for every integer x >l a - 1 we have h ~ L | = kx + l - g, i.e: hi(X, L | = O. Notice that the values for h~ L| and h~ L| are the minimal ones compatible with Riemann-Roch and the assumptions h~ L)t>2, deg(L) = k and pa(X) = g. THEOREM 3.2. Make all the assumptions of 3.1. F i x an integer y with O<~y<.r-2. (a) A s s u m e also that for every O-dimensional subscheme J of C with S c_ c_J and lenght (J) = card(S) + 1, J imposes card(S) + 1 independent conditions to f o r m s of type (k - 2, r - 2 - y). then for every rank I torsion free sheaf F on X with L | and d e g ( F ) = k y + l we have h ~ 9(b) A s s u m e also that for every O-dimensional subscheme T of C with S r c T and lenght ( T ) = c a r d ( S ) + 2, T imposes c a r d ( S ) + 1 independent conditions to f o r m s of type (k - 2, r - 2 - y). Then for every rank I torsion free sheaf G on X with L | and d e g ( G ) = k y + l we have h ~ REMARK 3.3. Fix integers g, k, t with 0 ~<t ~<g. One may consider the general k - gonal irreducible curve of genus g with exactly t ordinary nodes as only singularities. In this case, as in [CM1] (1.2.7), theorem 3.2 gives the dimension of all the multiples of the linear series described in (a), (b) of theorem 1.1. We believe that the property stated in part (a) of 3.2 is the good notion of primitivity for Gorenstein curves and hence we will intoruce it as a formal definition (see [CKM] for the smooth case). LINEAR SERIES ON k-GONAL CURVES 7 DEFINITION 3.4. Let Y be an integral projective curve and A a rank 1 torsion free s e h a f on Y. We will say that A is primitive i f it is spanned by its global sections and f o r every rank i torsion free sheaf B on Y with A r B and deg(B) = deg(A) + 1 we have h ~ B) = h~ A). REMARK 3.5. L e t Y be an integral projective Gorenstein curve. By Riemann-Roch and Serre duality a rank 1 torsion free sheaf A on Y is primitive if and only if the torsion free sheves A and Hom (A, ~ y) are spanned by their global sections. REMARK 3.6. It is easy to check that the thesis of part (b) of 3.2 is equivalent to the very ampleness of the line bundle W y | (L | 4. - Clifford's type results and d i m e n s i o n bounds. In this section we collect some easy remarks in order to improve classical Clifford's inequality in the case of a general smooth k-gonal curve. PROPOSITION 4.1. Let (X, L ) be an integral projective k-gonal curve. W i t h the notations of theorem 1.1, i f d <. g - 1 and M is of type (a) or (b) we have k r <<.d. PROOF. In case (a) we have M - L | so d = kr. If we are instead in case (b), by taking degrees we obtain 2g - 2 - d >I kt = k(g - d + r - 1), hence d - k r >t (k - 2)(g - 1 - d) I> 0 and the proof is over. PROPOSITION 4.2. A s s u m e c h a r ( K ) = 0. f i x integers g, k, d with g >I 6, k >I 4, k <~d <~g - 2. Let X be a general k-gonal curve of genus g and M e ~ Picd(X). then 4(h~ M) - 1) ~<d. PROOF. I f k = 4, this is [CM1], (2.3.6). Assume k t> 5. The closure in Mg of the locus Ik of k-gonal curves contais the locus/4 of 4-gonal curves. If the result were false for the general k-gonal curve of genus g, then, by the projectivity of the relative Picard variety over a finite covering of Ma and semicontinuity, there would exist a general Y ~ I 4 and N E Picd(y) with 4(h~ N) - 1) > > d, contradicting [CM1], (2.3.6). COROLLARY 4.3. Let X be a general k-gonal curve o f genus g with g >16, k >I 4. W i t h the notations of theorem 1.1, i f d <~g + ks - 2 we have h ~ F ) <. ~< rain (k - 2, r - s, (d - ks)~4 + 1 ). 8 EDOARDO BALLICO - CLAUDIO FONTANARI PROOF. Since M ~ L | | F we have deg ( F ) = d - ks, so d <<.ks - 2 implies deg (F) ~<g - 2. All the hypotheses of proposition 4.2 are satisfied, hence we obtain 4(h~ F ) - 1) <~d - ks. The thesis now follows from corollary 2.3. REFERENCES [ACGH] [AK] [B1] [B2] [C] [CKM] [CM1] [CM2] [E] [H] [M] [MS] E. ARBARELLO- M. CORNALBA- P. A. GRIFFITHS- J. HARRIS,Geometry of algebraic curves, I, Grund. der Math. Wiss. 267, Springer-Verlag (1985). A. ALTMAN - S. 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