Some Projective Geometry Associated with Unramified Double Covers of Curves of Genus 4 (*)

Some Projective Geometry Associated with Unramified Double Covers of Curves of Genus 4 (*).

A~DI~EA ]:)EL CE~TINA (Firenze) (~) - SEVIN I~ECIL]SAS (Mexico) (~)

Sunto. - Sia ~: 0 -~ C il rivestimento doppio non diramato di una curva di genere quattro de]i- nita su C e a moduli generali, assoeiato al punto di 2-divisione a~Pi%(C). C ha esatta- mente due g~ (autoresidq~e rispetto alla serie canonica) ed ammette q~n unico modello piano quale sestiea con tre punti doppi ordinari non aUineati. L'involuzione v su 0 che dh ~ ~ su Y la restrizione della tras]ormazione Cremoniana centrata in detti punti. 0 nel suo spazio ca- nonieo IK~l* giaee su una super]ieie di del t)ezzo S (6) che si proietta 2 a 1 da ]Kc| ~1" in IK~]* sulla cubica di Cayley C m) associata a ~. L'ideale di 0 ~isulta generato da qnadriehe di ranyo quattro. Si costruisee poi, in modo proiettivo, la curva di genere tre la cui Jacobiana la variet5 di l~rym di ~: C--~ C.

1. - Introduction and statement of main results.

Let s: C--> C denote an unramified double cover of a general curve of genus four defined over C. The aim of this paper is to give an explicit description of the projective geometry associated with s. Suppose z associated with the 2-division point ~ epic0 (C), let denote v the involution on 0 associated with s and also denote with Ko, Kv the canonical sheaves of C and C, and with KsQ ~ the Prym-eanonical sheaf of C. First of all in w 2, we show that ~ carries some particular linear series, precisely: three linear series of degree ~ and dimension 1, ~g i = 1, 2, 3~ and two linear series of degree 6 and dimension 2, g~ and g~'. They are such that v(~g) = ~g i = 1, 2, 3,

The plane model :Y of C associated with the simple g~ is a sextic with three not aligned double points. Let denote with IKcl*, ]K~.[* and IKo| a[* the canonical spaces of C and 0 and the Prym-canonical space of C. IK~]* and IK~| are naturally embedded in IK I*. In w 3 we prove that the canonical model of C lies in ]Kol* on a del Pczzo surface S(% which projects from ]K~Q ~]* in IKal* two--to--one on the unique Cayley

(*) Entrata in Redazione il 24 aprile 1982; versione riveduta il 22 settembre 1982. (1) During the preparation of part of this paper the first Autor was a visiting researcher fellow at Instituto de ~atem~ticas of the Universidad Nacional Aut6noma de ~6xico. (2) Partially supported by I.A.G.A. of the C.N.R. J~6 A~D~EA DEL CE~m~NA - SEVEN REC~LLAS: Some projective, etc. surface g(3) associated with ~ (as in [1~]). This morphism ramifies exactly at the four nodes of g(s). ~ extends in a unique wise to (g~)* = P~ as Cremona transformation with respect to the triangle of singular points of Y. Let us denote by V~c IKvI* the projectivized tangent cone to W~c J(C) at ~g, i = 1, 2, 3 ; for any i, j e {1, 2, 3} we have V~. Vj = S (s), and the rank four quadrics through the V~'s span an 8-dimensional family whose intersection is S (s), in w ~ we prove that ~ = S (s)'0, where ~) is the rank four quadric which is the cone, with vertex at [K~@ ~1", over the unique quadric Q in [K~]* such that C = c#(S).Q. So the ideal of C in ]Kvl* is generated by rank 4 quadrics. It is well known that the Prym variety of ~: C -~ C is isomorphic (as principally palarized abelian varieties) to the Jacobian of a curve of genus three X, and there are several construction of such X ([B], [D], [D-S]); here, in w 5 we give a projective construction which arises from the purticular linear series on ~. Under the hypothesis that both canonical and Prym-canonical maps are birational~ there are two types of special curves which do not have the same behaviour of a general curve. Seem to us of some interest to give, in w 6, a short account of those cases.

2. - Special linear series on C. Let C denote a complete, irreducible, non singular, non-hyperelliptic curve of genus 4 defined over C. Let us consider now an unramified double cover C-2> C, and denote by ~ e Pic ~ (C) the corresponding 2-division point and by ~: C -~ C the involution which gives z. Recall that g(C) = 2g(C)- 1 = 7. In what is following, except in w 6, we will suppose C a general point in J//~, the moduli space of curves of genus four. Let us denote by W,,c Pic~ (C) the image of the Abel map r C (~') -+ Pic~ (C), that is, the set of classes of effective divisors of degree n on C. For a general C, as above, considering the intersection W~ n (W~ @ ~) we ge~ exactly 6 points, which can be called in such a way ([R]) that: 3 r ~- P;) ~-- r ~- q',) -~ ~, i --~ 1, 2, 3, and ~ (p, ~- p'~) e ]Ka] i=l

(note that no p, is equal to a q~ since this would imply that C is hyperelliptic). When we lift these divisors up to C we get z*(p, + p'~.) ~ z*(q~ + q~) and, being they dis- jointed, we get on C three complete linear series of degree ~ and dimension 1, ~g, i = 1, 2, 3, (we have ~*(p~+p~), 7~*(q~+ q',) e,g, i = 1, 2, 3) which do not have fixed points. Moreover any existing g~ on C has precisely this form [D], and so the ~g are the only g~'s on C. Moreover they are such that lg ~-2g ~, 8g -~ IK~[, v(ig) = ig, 7~*(pi -~ p'i) and u*(q~-~ q'~) being the fixed divisors under ~ on each ~g, i -~ 1, 2, 3. Let us consider the r~tion~l map a: C -* Pa given by the g] ~ ]~g -~ ~gl (it will be clear later on that there is no loss of generality considering this particular couple). A:ND:~EA DEr~ CE~T~h - SEV~ RECkLeSS: Some projective, etc. 127

We claim that such map is birational. If it were of degree 2, then the image curve in P3 would be of degree 4, so it would be an elliptic or a rational curve. The elliptic case would imply that C has an infinite number of g~'s, which is not the case; the rational case implies C hyperelliptic which implies C hypere]liptic, so this case can be also excluded. If the degree were 4, then C would be mapped Ion a conic, i.e. a plane curve, which is an absourd. By standard formulas, the number of common pairs of a g~n and a g~ is (m -- 1 )(n -- 1) -- g. We note that since the genus of C is 7, ~g and jg have two couples in common and by the invariance under v, if we denote by s~-~ s~ one common couple, then v(s~-~ s'~)is the other one (here {1, 2, 3} = {i, j, k}). We note also that v(s~)#s~ being C general, i.e. not hyperelliptic and @(W~.(W2-~ (~))-~ 6. So the curve a(C) is a space curve of degree 8 contained in a quadric and of type (4, 4) with two double points (corresponding to the common couples s3-~-s'a and ~(~ + 8~)). One observes that these two double points can not be on the same line on the quadric, otherwise lg and ~g would coincide. If we project from a general point on the quadric surface we obtain a plane curve Of degree 8 which has two ordinary singularities of order 4, Q~ and Q~, and two ordinary double points (those must be ordinary by the genus formula for plane curves) which by abuse of language we will denote by s3 :~ s'3 and v(s~ ~-s'3). The lines through each 4-fold points cut ~g and 2g, respectively, and the conics through the four singular points cut 3g. This can be since lg ~- ~g -~ sg ---- ]K~l and since all the lines through the singular points, except the one determined, by s3 + s's and v(s8-~ ss),! form an adjoint curve of degree five which is a canonical adjoint. Since the pair of lines determined by the pairs of points QI~ s3 ~-s's and Q~, v(sa ~-s'a) respectively, is a conic through the four singular points and since each line cuts a divisor on ~g and a divisor on :g, respectively, we have that the other two points of intersection of each line with the curve must be common couples to lg and ~g, and ~g and ~g, respectively (call them v(s~ + s'~) and s~ + s'l, respectively) and, since the same happens for the lines determined by the pairs of points Q1, v(sa~ s~)and Q~, s~-s~! we finally get the following property of the common couples:

! f !

! l I ! (*) 81 + sl + ~(s3 + ~1; ~(sl§ s~) § s~§ s.e2g

[ ! 81 + sl! § ~(s~ + s~);l

We now describe the linear series of degree 6 and dimension 2; one is, g~= ]lg ~-2g-- ~(s3+ s'~)[ = tlg +3g-- u(s~+ s'~)I = ]~g + ~g-- ~(sl+ s'l)] =

= Ilg + ~1 + ~'11 = I~g + ~ + ~'~t = J~g + ~ + ~'~l

8 - AnnaZi eli Matematica 128 Al~Dm~i D~L CEN~I~A - SEV~I~ RECILLAS: Some projective, etc.

and the other one is

Such series are simple since the map on p2 given by them is just the projection of a(C), from one of ~ts double point. If we denote by Nm: Pic~(C)-~Pic.(C) the norm map, we note that since the g~ is not invariant, then dim (Nm g~)> 2 ([DC]), hence Nm g0--~ IKot. Suppose having another g~ on C, then it must be simple, otherwise the rational map that it gives into p2 would be of degree 2 or 3 and one excludes such possibilities in the same way as done before with the g~. Hence such g~ gives a morphism of C onto a plane sextic Y'. Since this plane curve I r' is of degree 6 and genus 7 it must have some singularities, and the following list exhausts all possibilities:

1) I ;' has a triple point analytically equivalent to ya_ x3_= 0;

2) Y' has a triple point analytically equivalent to y3_ xd= 0;

3) IT' has a double point analytically equivalent to y2 X6= 0;

4) :g' has two double points, one analytically equivalent to y2_ x 5 = 0 or to y~-- x ~ = 0, and the other one a node or a cusp.

5) :g' has three double points all of them nodes or cusps.

Cases 1) and 2) are excluded since the lines through the triple point would cut a g~ on Y', that is C would be trigonal and this is not compatible with the existence of a fixed point free involution on C [DC-I~]. In ease 3), the unique tangent T a at the double point a has an order of contact equal to six, hence the conics through the double point a and tangent to T a eat a linear series of order 6 and dimension 3 on it', hence :g' is a hyperelliptic curve in this case. So this one is a!so excluded. Case 4) is exehded because the two double points of a(C) are not on the same line on the quadrie. So the only remaining possibility is case 5) and, considering as above a line join- ing two double points, one shows that this g~ must be one of the previous ones. We can summarize the above results as follow:

LEN-~A 1. -- For C a general curve, ~ has exactly two complete linear series o] degree 6 and dimension 2: g~ and g~', and exactly 3 complete linear series o] degree 4 and dimension 1 : lg, ~g and ~g. Moreover they arc such that lg @ 2g @ ~g = g62 @ g~2' = = [K~,], ~(g~) = g~' and u(~g) = ~g, i = 1, 2, 3. We now describe in more detail the geometry of the plane sextie. ANDREA DEL CENTINA - SEVIN I:~EGILLAS: Some projeetive, ete. 129

l~irst of all let denote by ~, ~ and ~ the invertible sheaves associated with s,-~ s:, ~g and g~ respectively. Moreover let denote by ~, k ---- 1, 2, 3, sections of s such that (a~) = s~- s~. By means of the property (.) of the divisors of common couples to the ,g's we have a plane sextic with three ordinary double points. These are not aligned. In fact in this case g~ ~nd g~' would coincide and 12g~]---- [K~I, so that g~ : z*g], This implies [DC-1%] (see also w 6) that C lies on a cone, i.e. C is not genera]. Again we will denote by s~ ~ s~,! s~ ~ s~,f s3-~ s3l these double points. If we choose a coordinate system (x~; x2; x~) in (g~)* --~ P~ in such a way that the line {x~ : 0} l l is the line which joins s~ -[- sj and s~ ~- s~, where {i, j, k} : {1, 2, 3}, we observe thar the residual intersection of the line {x~ z 0} With the sextic is the divisor (s, + i = 2, 3. Recalling the definition of ~i we have that

~, a~e H0(C, ~), (i,j, k} ---- {1, 2, 3}

form a basis (here ~e H~ ~-

one is invariant and the other one is antinvariant with respect to v, that is, their divisors are the two divisors of the 7cg left fixed by the involution. If one multiplies Such a section by ~ one observes *hat those divisors are on the lines X~4-X2-~ O, X~=kX3= 0 and X2• ~-O, which clearly meet in 4 points. Observe that none of these points lies on the curve Y since they are invariant under the involution r and such an involution is fixed point free. So we proved the following:

Llama 2. - The plane representation o] ~ given by the g~ is a sextic curve Y c c (g~)*~ p2 with three no aligned double points (nodes or cusps) and the involution applied to the divisor corresponding to a double point gives the residual intersection of the opposite line with the sextie. Moreover the 6 lines (2 through each double point) which correspond to the ]ixed divisors of the ifs meet three at a time in/our points.

3. - The del Pezzo surface S (6) associated with C and its projection from [Ko | (r[*.

Now that we have this planar description of C we will study the associated sur. face S Is) in the canonical space ]K;[*~_P 6 of C~ and, recalling that the canonical and Prym canonical spaces of C are naturally embedded in [K;I*, we will also describe the projection of S (6) from [KaQ al* into ]Kc]*. 180 ANDREADEL CENTINA - SEVIN RECILLAS: Somv projective, etc.

It is known that given three points in the projective plane which are not on a line, the system of eubics through the three points give a rational map r P~ pG whose image is a smooth surface of degree six, a del Pezzo surface S (6). Such a surface contains exactly 6 lines (the blow-ups of the three points and the image of the lines which form the sides of the triangle) and such lines form a hexagon which spans a hyperplane in p6. Under r a line through a vertex gets transformed onto a conic on the surface S 16) which meets the blown up line corresponding to the vertex and the line which is the image of the opposite side of the triangle. It is known that certain projections of p6 onto p3 give degree 2 morphisms of S (6) onto a Cayley surface [S-R]. Here we will give an explicit description of the projection and then recalling the definition of the Cayley surface C(~81= [Kof* associated with o, we will show that ~13) = ~. From the description of Y given in lemma 2 it follows that the pair of meeting lines in S (6), cut on C divisors of the ~g's (the span of a corresponding divisor being a plane) as in the figure 1, where with ~ and ~, i = 1, 2, 3 we have denoted the six lines on S 161 (~ the blows-up of the singular points of Y), ~ the transformed of the lines through the singular points s4-}-s~ and s~-]-s~ of Y.

\ / \ / \ /

,^,/\ / 8z

r 81 r 8z

t s 8s -

/ \ / \ / ~,~ \ / ~3 \ / \~

Fig. 1. ANDREA DEL CENTINA - S:EVIN RECILLAS: Some projective, etc. 131

Moreover the conics which are the image of the lines X~+Xj = O, i

(X~; X2; X3) --, (X1X~X3; X~X~; XlX~; X~X3; XlX~; X~X~; X2X~) = (~o; ...; ~6),

this follows from the fact that seven entries on the right form a basis of the canonical adjoints of 17. In order to construct a corresponding basis of H~ K~), let us write ~= ~2~H~ ~1), and, similarly, ~ = ~1~ and o3 = ~. So let

XxX2Xa ~i 0~2 0~S X~X~

O~10~ 2 0~3

and, similarly, w2 = 5[~,3, w3 = 515~5~s, wa = 61 o2 5~, w~ = 51 o2o3 and w6 = ~x~2 ~a.

Can be immediately seen that w~ = We; wlv = w6; w2 _-- w4 and w3v = Wh, hence we, wl ~ w6, w2 -~ w4 and w3 ~- w5 give a basis of the invariant part of H0(C, KS) and wl--w6~ w2--w4 and w3--w~ a basis of the anti-invariant part. Hence ]K~]*= : {#1-- #6 :/~2-- #4 = #3-- #5 : 0} and [g~@ ~[* : {#0 : #1 @ #6 =/~2 + #, : #3 + -b #5 = 0}. So the projection from ]Kz,l* onto IK~I* with center IKo| gl* can be given explicitly as (#o; ... ;#6)-s (#o; #1 @/h;/h @/~; #3 ~- #5); recall that this projection is such that PI~. = ~: 0 -+ 0. Using the explicit forms of r and P, one shows that S(6) does not meet the center of projection IKz@a]* and tha~ r X2; X~)eS(6)~ ]Kcl*.e:~X~-= Xg= X~ so this intersection is precisely {r r ~5(a3); r = {(-- 1 ; 1 ; -- 1; 1; -- 1; 1; 1), (--1;--1;1;1;1;1;--1), (--1;1;1;--1;1;--1;1), (1; 1; 1;1;1;1;1)} i.e. the points which the conics C~ have in common. If we compose both maps we get a rational map:

V~-~ P~; (Xl; X~; Xs) -, (XiX2X~; X~(X~ + X~); X~(X~ + X~); X,(X~ + X~)) = = (:go; :gl; :g,; :g~)

whose image is the quadrinodal cubic surface r given by the equation:

det|:g2 2:go I73 =0.

:gl I73 2 :go 132 ADIDI~EADEI~ CE2qTII~A - S:EvI~ I~EClLLAS: Some projective, etc.

The involution v on ~ induces a projective automorphism on IK;.I*, again denoted with v. v maps linear subspaees onto linear subspaces, hone% since the lines in S(6) are determined by the divisors s~ § s~t and v(s~ § sk)! , k = 17 2, 3, it follows that v(S(6)) is another surface which contains those lines. Moreover, any point # ~ S (6) not on a line of the surface is such that # -- r with X not on the triangle. We have ~(xix~x3; x~x~; xlx]; x~x~; x~x~; x~x~; x~x~) = (x~x~x~; x~x~; x,x~; x~x~; X~Xg; X~X~ i X~X~) -----r X~X3; X~X~), hence if we denote by T: P~--->P~ the Gremona transformation (X~; X2; Xs) --> (X~Xs; XIX3; X~X2), we have shown that ~(r : r e S ~6~, hence ~(S ~6~) = S 16~, i.e. ~ induces an automorphism of S ~61. T and ~ are biregular automorphisms of S (~ which coincide almost everywhere, so coincide. So we h~ve shown that P: S (~ ---> ~(~ is the quotient by an involution, hence of degree 2, ~nd it ramifies at the points which correspond to the fixed points of the Gremona transformation on (g])*, that is, ~t the nodes of ~(~. We can describe ~d~ in a different way. Let H ~ denote the cubic hypersurface which is the projec~ivized tangent cone at the singular point on the there-divisor of J(0) corresponding to the g~, such hypersurface is in fact the chordal variety of the del Pezzo surface S (~ [S-1~], hence U (x, vx) c H/~l ((x, vx> denotes the line ~eS(~) which contains x and vx) and so ~d(3) = [Ko[*. U (x, vx) c IKoI*.~(*,. mhe next com- me8(~) putation will show tha~ we have an equality. Since ~, ~, ~3 form a basis of H~ ~) it follows that ~3, ~x~ ~, ~a~ is a basis for the sections of the sheaf ~: K~ @ ~-~, ~ short computation using those basis shows that the equation of H (~) is dot M = 0 where

M = /~ #o #~

#~ /~o

In order to get the equation of [Kcl*.H(3) we need to decompose M in its invariant and anti-invariant parts (that we will denote by M + and M- respectively):

~ul+m m§ 2m / m-m m-m 0

2 Yo Y2 Y1 t t So: IK~I*'H~3)= \ Yl Y~ 2 IZo AI~I)I~E)~ ]:)EL CE~TIS~A - qEVII~ ]~ECILLAS: ~ome projevtive, etc. 133

Let us observe that H (~)" [K~(~a]*----3 lines which are the (( diagonal lines ~) of the Wirtinger sextie defined here below. Let denote by I~/~c Pit (0) che image of the Abel map tn: ~(~) -+ Pit (~). From these information we can reconstruct the curve C, in fact we can write O = P(T~0,~) .P(Tm~.~:). l~ow we will show that this Cayley surface (d(~) is the one corresponding to o. First we recall the description of the Prym-canonical map C-~ [K~@a]* ([B-g], [1~]). The image is a curve of degree 6 and since r p~)_--r q~)'+ ~, i = 1~ 2, 3, each el those divisors gives a double point, moreover such double points are the intersection of 4 lines, none of which is a tangent (nor the diagonal lines): call this figure the quadrilateral in [K~@ a]*. This sextic is ca]led the Wirtinger sextie [Co] and one can see that the lines of the quadrilateral taken three at a time give ~ basis of the adjoints of order 3 which cut the canonical system on C. This gives a rational map [Ko@ a[*-+ [K~]* whose image is a Cayley surface ff~). Such a map blows-down each of the 4 lines of the quadrilateral to the nodes of if{) and blows-up each of the vertices of the quadrilateral to a line which joins two nodes, so we get a configuration of 6 lines inside the quadrinodal cubic surface ~,~<~). Moreover each el the diagonal lines gets transformed into a line and all three are on a plane H which is tangent to the Cayley sfirface in 3 points (che image of the intersection of the diagonal lines). Let consider the Yrym-canonical curve and observe that i~ we take two of the lines of the quadrilateral and the diagonal line through their intersection ' we obtain an adjoint of order 3, hence a canonical divisor on C; such divisors are:

2(p, + p:) + ~, + ~: ] , i / ' i=1,2~3. 2(q~ + q~) ~- ~-}- ~

hence the two planes in IKGI* which cut those divisors contain the line determined by s~ ~- s'i which we will call a diagonal line ms of c#(3), and touch c~(8) along an edge of the tetrahedron (look at the figure 2). So the pencil of planes in [Ks]* which contain mi gives rise to a g~: the one ob- tained by considering twice the divisors pi ~ PI" q~ ~ q~ ~ a. Call those linear series ,h; 2p,-{-2p',, 2q~-[-2q~ G ,h. Observe also that the plane H is a plane for all ,h's, hence ~ q- ~:q- 5 -{- ~G ~h, {i, j, k} = {1, 2, 3}. We are now ready to show that ~(3)= ~(3>. We know that P(S ~6)) = ~(~) is a Cayley surface and a direct computation using the explicit forms of r and /) shows that P(C~,~), e < m G {1, 2, 3, 4} are the edges of the tetrahedron of ~#(3) and that P(~)----/~(V~), i--~ 1, 2, 3, are the diagonal lines of ~m). Let us prove now that ~(3) and ~) have the same 9 lines:

1) ~dI~l and (d(~3) have the same diagonal lines: first observe that on C, z(z*(p~ ~- pl)) = 2p~ ~ 2p'~ and z(z*(q~ -~ q'~)) = 2q~ -k 2q'i so l~m ig = ih, and next, 134 A~D~EA DEL CE~I~A - SEvI~ ~EC~LAS: Some projective, etc.

/ 4- Fig. 2. since ~(s,4- s'~)4- 2(p~4- p~) and ~(s~+ s~)4- 2(q~4- q',) are elements of Nmg~= ]Kol, it follows *hst the plane in [Kr which contains the line P(rh~) = P(vrh~) (the one determined by ~(s~ + s'~)) cut the linear series ~h. So the previous geometric description of ~h implies that P(~)~ m~. In particular we have that ~(~-}-s~)=f r = ~+ ~,, i = 1,2,3. 2) (d/3) and (#~> have the same tetrahedron: The equality ~ 4-~4- 2p~+ 4- 2p; = ~(s~ @ s; + 7~*(p~4- p;)) shows that the edge of the tetrahedron on (d(~) determined by p~ 4- p'~, coincides with the edge of the tetr~hedron on ~(~) determined by the conic C~ in S (6) corresponding to the divisor n*(p~ 4- p~)e~g.

Our situation now is as follows: (d(3) and ~(3)--ff are irreducible cubic surfaces in P~ which have in common 9 lines and a sextic curve C, so the only possibility is that c#(~: (~). We have shown the following:

TttEOtCE3~ 1. -- I) There exists a unique representation o] ~ as a plane sextic Y such that the adjoint system of order three (which is the canonical system) gives a rational map r (g~G)* = p6 _~ IK;I, whose image is a smooth del Pezzo sur]ace S (e). The involution v can be extended in a unique wise to P~- in the ]orm o] Cremona rans/ormation with respect to the triangle o/ singular points o] Y. ][I) The projection of (K~(* onto (K~I* with center [K~Qa[* gives a degree two morphism of S (6) onto a quadrinodal Cayley sur]ace ~(~)c IKeI* (the ones associated to a) and such morphism ramifies exactly at the nodes o/ ~(~). A~ImEA DEL CE~TI~*A - SEVI~ REOILLAS: Some projective, etc. 135

4. - Rank four quadrics through S 16).

Let us denote by Vic IK;.]* the projectivized tangent cone to l~ac Pic4 (C) at ig, i = 1, 2, 3 ; the V,'s are rational normal scrolls of dimension 3 and degree 4 in P~ (see for instance [A-H]). We claim that each of the V~'s contains the del Pezzo surface S (3~` To prove this, observe that under the map r lines in P~ through the node s,+ s'i of Y are mapped into conics on S (G) and the corresponding scroll V~ is swept out by the two dimensional linear space spanned by the four points of intersection of the line with Iz outside the node, i, e, by the planes of the conics; this shows that S (6) c Vi. Moreover, the intersection of any two such scrolls is S (6~. Let us prove this (as in [A-HI) for the couple V~, V~: consider a generic 4-dimen- sionM linear space X in P~ containing some point p ~ P~\S (3). Each of the scrolls V~ intersects X in a rational normal quartie curve F~ and F~= F2 since V~ V2: X.S(~)c F~.F~, but if F~.F: contains one more point than the six in X.S (6~, then F~ would be equal to F~, so X-S (3) ----F~'F2, hence p 6 V~. V~. This shows that S (6) = V," V~ for any i < j e {1, 2, 3}. We now compute the dimension of the family of rank four quadrics through S (3). Let ~,~H3(O, ~,) be such that (a,) ---- s~+ s', and ,~= ~eH~ ~i), j < k; i-~1,2,3 as in w 2. Observing that ~i, ~ is a basis of H~ ~), i = 1, 2, 3 and that ~, ~., o~,~ and ~ is a basis for H~ K~ ~-~), and recalling that Wo = ~o~3, w~---- o~o~, ~W2= 61 3 8, W3: 61V263~ W4 = 6162085 ~5:O~d2d3 and ~)6 616268 form a basis of H~ then, in the corresponding projective coordinates (/~0; ...;/~) of [K~[*, the rational normal scrolls have the equations:

V2:rk("e"'"3"~ 1 \#o~3#2#1/-- '

\#~#3#o~1/--

From this one can easily see that every V~ is contained in six independent quadrie hypersurfaces of rank 4, and among all rank 4 quadrics which come from the 2 • 2 minors of the above matrices nine of them are lineraly independent. The intersection of these nine quadrics is the del Pezzo surface.

Ono o ooni , independent quadrie hypersurfaces; hence, since ~ is of genus 7, it is contained in 10 136 A~D~EA DEL CENTINA - SEVIN REGLLLAS: Some projective, etc. linearly independent quadric hypersm's in [K-f*. Above we h~ve 9 o~ them, all of rank 4; we now construct another independent one: let Q c !Kc[*~ P~ denote the unique quadrie surface which contains C and denote by Q c [/i:~ 1" the cone with vertex ]Kc (~ ~]* based on Q. Clearly 0 is a quadric hypersnrface of rank 4 (since Q is of rank @: and dim IK~)(r[*= 2) which does not contain S (~) and is such that c Q. So we can choose all 10 independent quadrics to be of rank 4 and since is o~ degree 12, we have ~ = S(~).~. Let us observe that if we denote by X, X~e e Pic~ (C), the two g~'s on C, and set y = zc*X, y'= ~*X'ePic~(~), then O is the projectivized tangent cone to Wac Pica (C) at X (or X') and Q is the projeetivized tangent cone to W~ c Pies (~) at y (or y'). Thus we h~ve proved the ~ollowing: TI~O~EI~ 2. - There exist 9 linearly independent quadrie hypersur]aees of rank 4 in [K~]* which contain the del _Pezzo surface S (~) and for any couple of the rational normal scrolls V, associated with the gl's~ of ~, we have S (~) = V~. Vr The canonical curve c IKb[* is the intersection ~ = S (~) .~ and its ideal 1.~ is generated by rank 4 quadries.

5. - On the construction of the curve X of genus three whose Jacobian is the Prym of~-~ C.

Let us denote by Mg the moduli space of unramified double covers of curves of genus g and by A~_~ the moduli space of principally polarized abelian varieties of dimension g- 1. The Prym morphism Mg ~>~ Ag_l is the one that associates with each unramified double cover C-5> C its Prym variety (P, [~]) [D-S]. In the ease g = 4, for a generic C the Prym variety is isomorphic to the Jaco- bian variety (Jx, Ox) of a curve of genus 3 and in fact in this ease the fiber P-~(Jx, Ox) can be identified with an open set of Jx/• [1%]. There are several ways to construct such X ([B], [D], [D-S]); here we give a projective one which comes from C. Let g~ be the linear series of dimension 2 and degree 6 oll C as defined in 1). Since g6 § vg~--2 [K~[, we can define the morphism fi: g~--> IK~[ as D ~D§ De- note by W c tK~I the surface which is the image of this morphism ft. As in theo- rem 2, let Q c IK~[* be the rank r qu~dric associated with the g~ on C which is the pullback of a g~ on C, that is, the projectivized tangent cone to 1~6 at this g~, and denote by Q*c IK~[ the dual hypersurface which is also a rank 4 quadrie. Taking duals in ]K~,[*, one observes that IKc] and [Ka@ a] can be identified as subspaees o~ [K~] via the application z*, moreover using coordinates (~o; ...; ~6) on tK~] dual to the ones given by (/~0; ...; #6) on ]K~I*, one has that {).6= X~§ 26= --~§247174 and {).~--)~6=~--~4=~3--25 0}=~]gc]. The projection P': [K~ [ -> IKc t can be written as (~o; ; )~6) -~ (),o; ~ § )~; ~2 § 24; 2a § ~) and the composition P~ofl: g~-~ IKcl can be written as

(P%/~)(D) = P'(D § vD) = ~(D). ANDI~EA DEL CENTINA - SEVIN I~ECILLAS: Some projective, etc. 137

Recalling the notation of w2, a divisor/) in g~ is the zeros of a section ] ---- a~a ~- b~a-~ e~ of H~ ~), hence vD is the zeros of a section

of H~ ~), hence D-~ vD is the divisor of the differential

].vf = (a 2 + b ~ -[- c~)#o + av#~ + bv/z~ -[- ab/~a + bv~t~ -[- abl& + ac/zs ,

so the morphism can be written explicitly as

fl: p~__> p6; (a; b; c) --> (a~-~ b~ ~- e~; ae; bc; ab; be; ab; ae) .

From here it follows immediately that

fact that we could have predicted, since D-{-vD = :~*(:r(D)). In fact, this surface W c [Kv[ is the dual surface of the Cayley surface ~(~)c [Kv]*, and this can be seen as follows:

to D ~ g~ corresponds a line lv in the plane (g~)* of the curve 17 which is such that l~. Y= D. The image ?~ of this line under the rational map r (g~)* -+ IK~]* is a rational normal cubic curve which lies on S (6) and is such that y~. C = D. When we project onto lKc]*, P@v) is a cubic curve which lies on ~m) and is such that /~@~). C = z(D). Since ~r(D)e IK~[, this divisor is cut by a hyperplane section of r hence the curve _P@v) is on a plane //(~) of [K~I*, but since it is rational and of degree 3, it must have a double point, i.e., the plane H(~) is tangent to the Cayley surface cd(a) at this double point of JP(y~). Hence we have shown that any divisor ~r(D) with D e g~ corresponds to a tangent plane to cd(~). This shows that W is contained in the dual surface 5 p(4) (a Steiner surface)of (#(a), but since both are irreducible and of dimension 2, we must have 5 p(4) = W. Using the previous identifications one can see that the quadric hypersurface (~*c [K~] is in fact the cone with vertex on IKzQg[ over the quadric surface Q*c [KG[ dual to the unique qnadrie surface Q on IKa]* which contains C. So we are in the following situation:

= IKoI* and x = IK I.

Its shown elsewhere [R] that under this conditions the Jacobian variety of X is isomorphic (as p.p.a.v.) to the Prym variety of (C, a). We have shown the following: 138 A~DREA DEL CEh'T~A - SEV~ REO~LL~S: Some p~vjective, etc.

TttE01~E~'I 3. - The intersection W.@c IK;[ is a curve o/ genus 3 whose jacobian variety is isomorphic (as principally polarized abelian variety) to the Prym variety o~ O~ c.

6. - Special cases.

Once given the description of the generic case, we will like to give s short description of the special cases in which both, c~monieal curve and Prym-canonical curve, are birational to the given curve C, and this happens only when C is not hyperelliptic and dim W2" (W~ q- a) = 0. Under this hypothesis there are two possible cases [B-R].

Case I). - This case occurs when at some point of W~.(W~ q-a) the intersection is not transversal, in fact there exist curves of genus four (which are special in the sense of moduli) such that for x ~ W~. (W~ q- a), dim (T~,~. T~+o,,) = 1. Let us to describe the Prym-canonical curve under this assumption. Denote by D, E e C (~) the divisors such that x = ~odD ) (~0d is the natural morphism C ~/-+ Pica (C)) and D ~ E @ ~, then the above assumption is equivalent to h~ Ko(--D--E)) =1, let FEC (2) be the divisor such that D~E@IT~]Kc]" Since h~ K~| ~) = 3 and h~ G| ~(-- D)) = h~ G| ~(-- E)) = 2 it follows that there exists a divisor G ~ C (2) such that D 4- E q- G e IKc,@ af, hence F ~ G q- r Observing that 2D q- G, 2E q- G, D @ E q- G E iKc@ (r I we can now describe the Prym-canonical curve: it is a plane sextic with four double points, the two corresponding to D and E are tacnodes (anMytic~lly equivalent to y2q_ x 4 = 0) whose tangents intersect on the ordinary double point corresponding to F and the ordinary double point which corresponds to G is in the line which joins the two tacnodes. The map [Kc@al*-+ IKcl* given by the linear system of [canonical adjoint curves can be describe as:

and the cubic surface cd(a) has the equation:

Let us observe that this surface has only three singular points, and its automorphisms group is not finite. In this case, the curve C has only two g~'s, precisely g- ]~r-*D t = lu-lE] and h = Izr-*TI = I~-IGI, moreover they are such that 2g @ h = [K~[. In order to construct the corresponding g~'s let us observe that the common couples to g and h are G, and vG~, where G~ and ~G1 are such that G~q- vG~ = ~-*G. ANDREA ])EL CENTINA - SEVIN I~ECILLAS: Some projective, etc. 139

So the g~'s can be described as follows: go~ ----- Ig 4- 0~] = Ih 4- LI and ~g~= IK~-- g~I = = [g 4- vG] = [h 4- vL I where Z ~ ~(~) is such that ~(L) 4- F 4- G e IKc,@ a I. From 2G~ 4- vL e g~, it follows that the plane sextie representation Y of ~ has two double points: a tacnode (analytically equiva!ent to y~4- x~= 0) and a ordinary double point not on the tangent line of the tacnode. The rational map r (g~)*~--P~->l ~1' given by the canonical adjoint curves of Y can be describe as follow:

(x; x:; x~) -~ ((x~ - x~) xl; - (x~ + x~)~ x~; x~ xl; (x~ - x~) (x~ + x~) x:; - (x~ + x~)x~x~; x~x2(x~ + x~)) = (~o; ...; ~,0)

The image S (6> of the plane by r is again adel Pezzo surface which is intersection of nine independent rank 4 quadrics. So 0 = S(6)'0~ where Q is the projectivization of Tv7.... (gD, in particular the ideal of C is generated by rank 4 quadrics. The projection of IK~[* onto IKI* from IKo@al, given explieitely by (#o; ...; ..; #0) --> ()uo;/h; #2; #5 4-/~0) expresses (as in the general case) the cubic surface 5 ~) as the quotient of S (~ by an involution which extends v. Again in this case the chordal variety H (3) of S r176coincides with the projectivization of the tangent cone T~w~,g~, and C (3~ : H r ]K~[*. The intersection H (~.lKo@al consists of the tangent at the taenodes and the line which joins them. So we can write C = P(T~,,,,).P(Tw,g,). Case II). - This ease occurs when Sing (Ws" (W3 4- a)) r 0 and this is equivalent to any of the following (see [M], [B-I~],[DC-R]): i) C has only one g~ and a unique divisor D = ~1 ~ p~ 4- P3 such that g~ @ ~ ~-- D; if) C has a unique g~ which is ~*g~ (hence autoresidual with respect to K?); iii) the Prym variety of C-> C is the Jacobian of a hyperelliptic curve of genus three; iv) the Prym-eanonieal curve has a triple point.

Since ~*gl = g~ it follows that if H (3~ is the projectivization of the tangent cone Tw,,q,~', then H ~). IKvl* = I~ 4- Q where L is the tritangent plane to C corresponding

to D and Q is just the rank 3 quadric which is the projeetivization of T Ws~ff a~ So is this case we do not have enough information to reconstruct the curve. To finish we describe the Prym-canonicM curve, the sextic plane curve associated with the g~ and the corresponding del Pezzo surface. Denote by E~ C (2), i ---- 1, 2, 3, the divisors such that E~ 4- p~e g~, i = 1, 2, 3, so E~pj -4- pT~ 4- a {i,j, k} = {1, 9, 3}. So the Prym-canonical curve is a sextic curve with ~ triple point (corresponding to D) and three double points (corresponding to Ed. 140 ANDREA])EL CEI~TINA - SEVIE I~ECILLAS: Some projective, etc.

Moreover the three double points lie on a line (since E~+ E2+/~3e IKc~)a]) and each tangent at the triple point passes through a double point (since 2p~ +

The plane curve Y c (g~)* is ~ sextic with three double points (corresponding to z-~p~). The involution on ~ induces an involution on (g~)* which is this case is a projectivity which has the line of double points as fixed line (the intersection of the lines which cut z-~(E~ + p~) is the fixed point). The corresponding del ]?ezzo surface S (r which is the image of the pla, ne (g~)* under the m~p:

= (/~o; ...;/~0) has a singular point s (the blow-down of the fixed line) which lies on [Ka @ gl. The projection from s onto ~6= 0, maps S (6) onto the Veronese surface, and if we further project onto [Ks]* = {/~ =/~5 = #~ = 0} we get Q. The equation of the projectivization of Tw,,g,~' is:

o ~ ~,~) det ~#4 ~tl /x3 =0 5 #3 #2 which is the cone with vertex s over the chordal variety of the Veronese surface.

BIBLIOGRAPHY

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