A Note on Projectivized Tangent Cone Quadrics of a Prym Canonical Curve
aa r X i v : . [ m a t h . AG ] M a y A NOTE ON PROJECTIVIZED TANGENT CONEQUADRICS OF RANK ≤ IN THE IDEAL OF APRYM-CANONICAL CURVE
ALI BAJRAVANI
Abstract.
Throughout the paper, among other results, we givein theorem 3.1 and proposition 3.2 a partial analogue of theorem 1.1for projectivized tangent cone quadrics of rank equal or less than 4,for Prymians. During the lines of the paper it would be seen thatfor an un-ramified double covering of a general smooth tetragonalcurve X induced by a line bundle η on X with η = 0, the Prym-canonical model of X is projectively normal in P ( H ( K X · η )).Then we consider a genus g = 7, tetragonal curve C which isbirationally isomorphic to a plane sextic curve X with ordinarysingularities. As byproduct of theorem 3.1 and proposition 3.2,we show that the stable projectivized tangent cone quadrics withrank equal or less than 4 of an un-ramified double covering of C ,generate the space of quadrics in P ( H ( K C · η )) containing K C · η -model of C , where η is a line bundle on C with η = 0, obtainedin section 4. Keywords:
Clifford Index; Projectivized Tangent Cone; Prym-Canonical Curve; Prym variety; Tetragonal Curve.
MSC(2010): Introduction
For an ´ e tale double covering π : ˜ X → X of smooth curves, it is nat-urally associated a principally polarized abelian variety the so calledPrym variety of π , which is denoted by P ( π ), whose principal polar-ization is induced twice by Θ ˜ X , the theta divisor of ˜ X . While thisP.P.A.V. enjoys from some interesting properties analogous to Jaco-bians, it behaves differently in some another properties. Usually inmost cases these differences lead to a rich geometry which provideswide areas of research. For example, although the theory of Prym vari-eties is old enough and has been studied variously by many well knownmathematicians since decades ago, but surprisingly an analogue of thewell known Riemann singularity theorem for Prymians has been given relatively lately, by R. Smith and R. Varley in [13] and its completeanalogue has given recently by S. C. Martin in [12].Another useful and nice package in the land of Jacobians of canonicalcurves, is the well known theorem 1.1, proved by Andreotti-Mayer in[2] and by G. Kempf in [10]. Theorem 1.1.
Let X be a smooth projective curve of genus g on analgebraically closed field of characteristic zero and | D | = g g − a com-plete linear series of degree g − and dimension on X . Consider thecorresponding double point of Θ X : O X ( D ) = O X ( K X − D ) ∈ Θ sing . Then the projectivized tangent cone to Θ X at O X ( D ) is a quadric ofrank at least or equal to containing the canonical model of X whichcan be described as the union of the linear span of divisors in | D | = g g − . Moreover the quadric is of rank precisely when | D | = | K X | .Conversely a quadric Q of rank less than or equal to , through X isa tangent cone to Θ X if one of its rulings cuts out a complete linearseries of degree g − and dimension on X . Although it is completely known in the literature that the projec-tivized tangent cone at a double point a of Prym-Theta divisor of gen-eral Prym-Canonical curves is a quadric of rank 6 rather than rank 4,but it might be interesting to know: • How is the effect of linear subspaces of a Prym quadric tangentcone on C , when the quadric is of rank 4 containing C ?Equivalently we look for an analogue of theorem 1.1 for Prymians.The genus g = 7 case, the first case where the singular locus of prymtheta divisor is nonempty, is the first case that has to be dealt with. Wesee that a projectivized tangent cone quadric of rank equal or less than4 at a stable singularity of Prym-theta divisor of an ´ e tale double cover-ing ˜ X → X , through the Prym-canonical model of X , imposes a linearseries of degree d such that d ∈ { g − , g − , g − } . If d ∈ { g − , g − } then for each g ≥ d = g − g = 7. A partial converse tothis result will be given in Proposition 3.2.Then we proceed to provide an evidence for the above question. Pre-cisely we give an example of a curve admitting projectivized tangentcone quadrics of rank equal or less than 4. We would like to give an ex-ample, on which not only a complete converse of theorem 3.1 is valid,but also its rank 4 Prym quadric tangent cones generate the spaceof quadrics containing Prym-canonical model of C . A curve which is NOTE ON PROJECTIVIZED TANGENT CONE QUADRICS ... 3 birationally equivalent to a plane sextic with three number of ordi-nary singularities, might seem a candidate for this aim. But becauseof technical reasons a curve of this type with three collinear ordinarysingularities is needed. Whereas the existence of a plane sextic curvewith three number of ordinary double points is a well known fact, theexistence of a plane sextic curve with three number of collinear ordi-nary singularities needs an actual proof. Such a proof will be given inTheorem 4.1.H. Lange and E. Sernesi in [11] have proved that any Prym-canonicalline bundle on a curve of Clifford index ≥ Preliminaries and Notations
A nontrivial line bundle η on an irreducible nonsingular projectivecurve X with η = 0 gives rise to a double covering π : ˜ X → X andvice versa. For a nontrivial line bundle η with η = 0 we denote by π η the map induced by η . The kernel of the norm map of π η denotedby Nm( π η ), which is a subset of J( ˜ X ), turns to be the union of twoirreducible isomorphic components one of them containing zero. Thecomponent containing zero, denoted by P ( π η ), is called the Prym vari-ety of the double covering π η and consists of line bundles ˜ H on ˜ X , with˜ H ∈
Ker(Nm) and h ( ˜ H ) is an even number. The other one which isdenoted by Z , consists of line bundles ˜ H on ˜ X , with ˜ H ∈
Ker(Nm)and h ( ˜ H ) is an odd number.The theta divisor Θ ˜ X induces a principal polarization on P ( π η ) andthe Prym variety P ( π η ) turns to be a principally polarized abelianvariety with principal polarization E ( π η ). In terms of dimensions ofglobal sections of the points in J( ˜ X ), the principally polarized abelianvariety P ( π η ) can be described as follows: P ( π η ) = { ˜ L ∈
J( ˜ X ) | Nm( ˜ L ) = K X , h ( ˜ L ) ≡ mod } . The singular locus of E ( π η ) has a similar description in these termstoo:Sing( E ( π η )) = { ˜ L ∈ P ( π η ) | h ( ˜ L ) ≥ }∪ { ˜ L ∈ P ( π η ) | h ( ˜ L ) = 2 , T ˜ L ( P ( π η )) ⊂ T C ˜ L (Θ ˜ X ) } ALI BAJRAVANI where
T C ˜ L (Θ ˜ X ) denotes the tangent cone of Θ ˜ X at ˜ L . The singularpoints of E ( π η ) with h ( ˜ L ) ≥ η is a nontrivial line bundle on X such that η = 0, then the linebundle K X · η is called a Prym-canonical line bundle on X , if K X · η isglobally generated and very ample, the irreducible (possibly singular)curve φ K X · η ( X ) is a Prym-canonical curve, where φ K X · η : X → P ( H ( K X · η ))is the morphism defined by global sections of K X · η . We will denotethe curve φ K X · η ( X ) by X η . A linear series g rd on X gives rise to a samelinear series on X η via φ K X · η and vice versa. In the absence of anyconfusion, we use a same symbol for both of these linear series on X or on X η . Theorem 2.1.
Let π : ˜ X → X be an ´ e tale double covering induced by aline bundle η such that η = 0 . Assume moreover that the line bundle K X · η is very ample and globally generated. Then the projectivizedtangent cone of E ( π η ) at a double point ˜ L is a quadric hypersurface in P ( H ( K X · η )) containing X η if and only if ˜ L is a stable singularitywith h ( ˜ L ) = 4 .Proof. Let ˜
L ∈
Sing( E ( π η )) be a double point of E ( π η ). Consider thatusing [12, Corollary 6 . . φ K X · η ( X ) ⊂ P T C ˜ L ( E ( π η )) if andonly if h ( ˜ L ) ≥
4. If h ( ˜ L ) > h ( ˜ L ) ≥
6. There-fore by Riemann-Kempf singularity theorem deg( P T C ˜ L (Θ ˜ X )) ≥ P T C ˜ L ( E ( π η )) = 2 P T C ˜ L (Θ ˜ X ) · P ( H ( K X · η )), the hyper-surface P T C ˜ L ( E ( π η )) would be of degree at least 3 and vice versa. (cid:3) Lemma 2.2. If ˜ p, ˜ q ∈ ˜ X and L ˜ p, ˜ q is the line in P ( H ( K ˜ X )) joining ˜ p to ˜ q then φ K X · η ( π (˜ p )) = p = L ˜ p, ˜ q ∩ P ( H ( K X · η )) .Proof. This is claimed and proved in [14, page 4954]. (cid:3) Rank ≤ Projectivized Tangent Cone Quadrics
Assume that F is a smooth projective curve of genus g with a veryample Prym-canonical line bundle K F · η . Theorem 3.1.
Assume that F is a smooth non-hyperelliptic projectivecurve of genus g with a very ample Prym-canonical line bundle K F · η .Assume moreover that π η : ˜ F → F is an ´ e tale double cover of F . Let Q be a quadric of rank equal or less than containing F η in P ( H ( K F · η )) . NOTE ON PROJECTIVIZED TANGENT CONE QUADRICS ... 5
Furthermore for ˜ L ∈
Sing( E ( π η )) , assume that Q is the projectivizedtangent cone of E ( π η ) at ˜ L . Then one of the rulings of Q cuts a g d on F η with g − ≤ d ≤ g − and O ( g d )) ⊗ L = K F , for some line bundle L on F which L is of degree , or . Additionally, the g d is completewhen d ∈ { g − , g − } . It is complete in the case d = g − as well,provided that g = 7 .Proof. Assume that π η : ˜ F → F is an ´ e tale double covering. If Q = Q ˜ L is a projectivized tangent cone of Sing( E ( π η )) at ˜ L which is a quadricof rank equal or less than 4 containing F η , then one of its rulings cuts a g d , on F η . For a divisor E ∈| g d | , using the geometric Riemann-Roththeorem and considering that the linear space < E > inside Q ˜ L is ofcodimension 2 in P ( H ( K F · η )), one obtains that h ( K F · η − E ) = 2.Since dim( < E > ) = g − d = deg( E ) ≥ g − < E > and consider that Λ = ˜Λ ∩ P ( H ( K F · η )) where ˜Λ = < ˜ E > for some ˜ E ∈ ˜ F g − such that ˜ L = O ( ˜ E ), see [13]. For p ∈ F η set-ting π − η ( p ) = { ˜ p, ˜ q } using Lemma 2.2, we have p ∈ Supp( E ) if and onlyif { ˜ p, ˜ q } ⊂ Supp( ˜ E ). This observation implies that d = deg( E ) ≤ g − g d assume that g d ⊂| D | for some divisor D in g d and set ˜ D = π ∗ η ( D ).Assume first that d = g −
1: then the equality h ( D · η ) = h ( K F · η − D ) together with the geometric Riemann-Roth theorem imply that h ( D · η ) = 2. The assumption d = g −
1, implies that 2 g d = K F η andone can see from this that π ∗ η ( g d ) ∈ Sing( E ( π η )). In fact by Lemma 2.2,for each divisor Γ ∈| g d | the divisor ˜Γ = π ∗ η (Γ) is a divisor associatedto a global section of ˜ L , so one has π ∗ η ( g d ) = ˜ L , in this case. Therefore π ∗ η ( g d ) ∈ Sing( E ( π η )).If h ( g d ) > h ( π ∗ η ( g d )) = h ( g d ) + h ( g d · η ) >
4. This impliesthat Q π ∗ η ( g d ) = ∪ ˜ D ∈| π ∗ η ( g d ) | < ˜ D > is a hypersurface at least of degree 6in P ( H ( K ˜ F )). Therefore the hypersurface Q = [( Q π ∗ η ( g d ) ) · P ( H ( K F · η ))] would be a hypersurface of degree at least three. This by equalityof the quadrics Q and Q ˜ L is absurd. This implies that the g g − iscomplete.If d = g − h ( D · η ) = 1 and there exist ˜ p, ˜ q ∈ ˜ F such that O ( ˜ D ) ⊗ O (˜ p + ˜ q ) ∈ Sing( E ( π η )). In fact as in the previous case for eachdivisor Γ ∈| g d | there are points ˜ p, ˜ q ∈ ˜ F such that ˜Γ = π ∗ η (Γ) + (˜ p + ˜ q )is a divisor associated to a global section of ˜ L . So one has | O ( ˜ D ) ⊗ O (˜ p + ˜ q ) | = | ˜ L |∈
Sing( E ( π η )) . ALI BAJRAVANI
Now the relations4 = h ( O ( ˜ D ) ⊗O (˜ p +˜ q )) ≥ h ( D )+ h ( D · η )+ h ( O (˜ p +˜ q )) = h ( D )+1+1imply that h ( D ) = 2 = h ( g d ).Finally if d = g − g = 7, if h ( D ) > F has a g r with r ≥
2. This by Clifford’s theorem and non-hyper ellipticity of F is acontradiction.Consider moreover that since h ( K F − O ( g d )) ≥
1, the line bundle L := K F − O ( g d )) is a line bundle satisfying 2 O ( g d )) ⊗ L = K F . (cid:3) Proposition 3.2.
Assume that F and the assumptions about it areas in Theorem 3.1. Let Q be a quadric of rank equal or less than containing F η such that one of its rulings cuts a complete g d on F η with g − ≤ d ≤ g − and O ( g d ) ⊗ L = K F , for some line bundle L which L is of degree or on F . Then Q is a projectivized tangentcone of Sing( E ( π η )) .Proof. Assume that Q ∈| I F η (2) | is of rank equal or less than 4 suchthat one of its rulings cuts a complete g d with g − ≤ d ≤ g − O ( g d ) ⊗ L = K F . If L = O F ( p + p + · · · + p t ) and π ∗ η ( L ) = O ˜ F (¯ p + ¯ p + · · · + ¯ p t + ¯ q + ¯ q + · · · + ¯ q t ), where ¯ p i and ¯ q i are conjugate,then for a sub divisor ˜ D of ˜ D = ¯ p + ¯ p + · · · + ¯ p t + ¯ q + ¯ q + · · · + ¯ q t whichis of degree deg( ˜ D ) and no two points of its support are conjugate,setting 12 ( π ∗ η L ) := O ˜ F ( ˜ D ) , ˜ L = π ∗ η ( O ( g d )) ⊗
12 ( π ∗ η L ) . one has Nm( ˜ L ) = K F . This reads to say that ˜ L ∈
Ker(Nm) = P ( π η ) ∪ Z where Z is the isomorphic copy of P ( π η ) which we alreadyintroduced in backgrounds.Consider the relations: h ( ˜ L ) = h ( π ∗ η ( O ( g d )) + ( π ∗ η L )) ≥ h ( π ∗ η ( O ( g d ))) + h ( ( π ∗ η L ))= h ( O ( g d )) + h ( O ( g d ) · η ) + h ( ( π ∗ η L )) . If d = g − h ( O ( g d )) = h ( O ( g d ) · η ) = 2 and L has to be equal to O F . Therefore h ( ˜ L ) = 4 and so ˜ L ∈
Sing( E ( π η )). Moreover Q ˜ L = Q and this implies that Q is a projectivized tangent cone in this case.In the case of d = g − h ( O ( g d ) · η ) = 1 which implies thatdim( | π ∗ η ( O ( g d )) | ) = h ( π ∗ η ( O ( g d ))) − h ( O ( g d )) + h ( O ( g d ) · η ) − . NOTE ON PROJECTIVIZED TANGENT CONE QUADRICS ... 7
This reads to say that taking a global section σ of π ∗ η ( O ( g d )) and con-sidering its associated divisor, ˜ D , one has dim( | ˜ D | ) = 2. From this itcan be seen that taking a global section γ of ˜ L and considering its as-sociated divisor ˜ B , there exists a divisor ˜ E ∈| ˜ B | such that h ( ˜ E ) = 4.In fact for a point p in the support of a divisor associated to a globalsection of the line bundle ( π ∗ η L ), assume that ˜ D ∈| ˜ D | is a divisorsuch that p ∈ Supp( ˜ D ). Now set ¯ D := ˜ D + p + q = ¯ M + 2 p + q forsome divisor ¯ M on ˜ F and some point q ∈ ˜ F such that p + q ∈ ( π ∗ η L ).Consider that O ( ¯ D ) ∈| ˜ L | anddim( < ¯ D > ) = dim( < ¯ M + p + q > )= dim( < ¯ M + p > ) + 1 = 2 g − . This equivalently reads to say that h ( ¯ D ) = 4 which implies that h ( ˜ L ) = 4. Therefore ˜ L ∈
Sing( E ( π η )) which finishes the proof. (cid:3) Let X be an irreducible plane sextic curve with ¯ x , ¯ y and ¯ z as itsnodes or double points. Assume moreover that ¯ x , ¯ y and ¯ z are collinear.Let i : C → X be its normalization. Notice that by genus formulafor plane curves, X , and consequently C , is of genus 7. Now on thecurve C consider the linear series | H − ∆ | for which H = i ∗ ( O X (1)),∆ = x + x + y + y + z + z with i − (¯ x ) = { x , x } , i − (¯ y ) = { y , y } and i − (¯ z ) = { z , z } . Notice that K C = 3 H − ∆ and deg(5 H − ∆) = 24.Now take a divisor D ∈ C (12) such that 2 D belongs to | H − ∆ | and set η := 2 H − D . Trivially η = 4 H − D ∼ H − (5 H − ∆) =∆ − H ∼ O where the last equality holds because the points ¯ x , ¯ y and ¯ z are collinear. Notice moreover that the lines passing through one of thesingularities of the curve X define a base point free g on X and i ∗ ( g )is a base point free g on C . This implies that C is an irreducible,nonsingular, tetragonal curve of genus 7 with three number of basepoint free g ’s.Next we verify the existence of a curve C admitting a D with men-tioned properties: Theorem 4.1.
There exists a curve C admitting a divisor D withthe mentioned properties and admitting a globally generated and veryample prym-canonical line bundle K C · η .Proof. Let Q and Q be quadrics in P tangent to each other exactlyin one point. Bezout’s theorem implies that they cut each other in two ALI BAJRAVANI extra points ¯ x and ¯ y with multiplicity one. Consider the line l passingthrough ¯ x and ¯ y . Since the tangent variety of Q , (resp. Q ) fills up allthe surface P , an arbitrary point ¯ z ∈ l lies on at least a tangent lineof Q , (resp. Q ). Therefore for an arbitrary point ¯ z on l there are acouple of lines L and L passing through ¯ z such that L is tangent to Q and L is tangent to Q . Then with this assignments, the reduciblecurve X defined by the polynomial h = Q Q L L is a curve of degreesix which has three collinear ordinary singularities.Denote by t the point where Q and Q are tangent to each other.A computation shows that there are infinitely many quadrics through¯ x , ¯ y and t such that each of these quadrics has the same tangent lineat the point t . In fact, quadrics in P passing through the points p = (1 : 0 : 0), q = (0 : 1 : 0) and r = (0 : 0 : 1), are given by b x x + b x x + b x x = 0. These quadrics have the line b x + b x =0 as their tangent line at the point p . These imply that for fixed d , d the infinitely many quadrics b x x + b x x + b x x = 0 passe through p , q , r and are tangent to each other at the point p .Choose a couple of quadrics ¯ Q , ¯ Q passing through ¯ x , ¯ y , t , tangent toeach other at t and distinct with Q and Q respectively. As in thesextic h , there are lines ¯ L and ¯ L distinct from L and L respectively,passing through ¯ z and are tangent to ¯ Q and ¯ Q respectively. Again thecurve k = ¯ Q ¯ Q ¯ L ¯ L is a reducible plane sextic having three collinearpoints ¯ x , ¯ y and ¯ z as its ordinary singularities. Now since h and k hasno common irreducible component, Bertini’s theorem implies that X ,a general member of the pencil generated by h and k , is an irreducibleplane sextic having three collinear points as its ordinary singularities.Choosing the normalization of X gives the desired curve C .To show existence of a D with desirable properties, take a planequintic T with three number of nodes p , p , p and passing throughthree distinct prescribed collinear points ¯ x , ¯ y and ¯ z . Choose nine extrapoints p , · · · , p on T . Notice that passing through points p , · · · , p ,being tangent to T at the points p , · · · , p and having three collinearpoints ¯ x , ¯ y , ¯ z as only singularities, impose at most 24 conditions on thespace of plane sextics. Since the space of plane sextics is of dimension27, there are plane sextics X , passing through the points p , · · · , p ,being tangent to T at the points p , · · · , p and having three collinearpoints ¯ x , ¯ y , ¯ z as only singularities. On such a curve X , setting X := p + · · · + p one obtains the desired D .It can be seen easily that any Prym-canonical line bundle on a non-hyperelliptic curve is globally generated. See [11, Lemma 2 . C , considerthat the prym-canonical line bundle K C · η with η = 2 H − D is very NOTE ON PROJECTIVIZED TANGENT CONE QUADRICS ... 9 ample. In fact the proof of Lemma 4.2 implies that in lack of veryampleness of K C · η , there would exist two another points z, w ∈ X such that x + y ≁ z + w , x + 2 y ∼ z + 2 w ∈ g . This means that taking L , the line passing through singularities of X ,there exists a line ´ L other than L that is tangent to X in two points α and β distinct with x, y and z . But this is impossible for a generalcurve of type X . (cid:3) More than what we saw in Theorem 4.1 and at least as an indepen-dent interest, one can prove that any Prym-canonical line bundle on ageneral tetragonal curve is very ample and the Prym-canonical modelof this line bundle is projectively normal:Assume for a moment that the line bundle K X · η is very ample. Then X η , the Prym-Canonical model of X in P ( H ( K X · η )), is 2-normal,namely the map H ( P ( H ( K X · η )) , O P ( H ( K X · η )) (2)) → H ( X η , O X η (2))is surjective. In fact by [9], a curve X is n -regular if and only if, it is( n − O X ( n −
2) is non-special Thereforeto prove 2-normality of X η , it is enough to prove its 3-regularity, whichmeans that its sheaf of ideals, I X η , is 3-regular. Moreover for n ≥ → I X η → O P ( H ( K X · η )) → O X η → , it is enough to prove that the sheaf O X η is 2-regular. Trivially H i ( X η , O X η (2 − i )) = 0 for i ≥ H ( X η , O X η (1)) = 0. To see this, consider the isomorphisms H ( X η , O X η (1)) ∼ = H ( X, K X · η ) ∼ = ( H ( X, K X − K X · η )) ∨ = ( H ( X, η )) ∨ together with H ( X, η ) = 0. These finish, 2-normality of X η .The discussion just has been done, proves projective normality ofcurves X with Cliff( X ) ≥
3, in which case any prym-canonical linebundle K X · η is very ample by [11]. In the case Cliff( X ) = 2, itcan happen that the line bundle K X · η is not very ample for specialtetragonal curves X . But it can be proved that for a general tethragonalcurve X , any prym-canonical line bundle K X · η is very ample. In fact,an equality h ( K X · η ( − x − y )) = h ( K X · η ) − x, y on X , implies that there exist another points z, w ∈ X such that x + y ≁ z + w , x + 2 y ∼ z + 2 w ∈ g . This by [6] is absurd for a general tetragonal curve.Summarizing we have proved:
Theorem 4.2.
Assume that X is a general smooth tetragonal curveof genus g and ˜ X → X an etale double covering of X induced by σ ∈ Pic( X ) with σ = 0 . Then X σ , the Prym-canonical model of X in P ( H ( K X · σ )) , is projectively normal in P ( H ( K X · σ )) . Projectivized Tangent Cone Quadrics of C generatethe Space of H ( I C η (2))Everywhere in this paper by C we mean the tetragonal curve ob-tained in Theorem 4.1. Moreover by η we mean the 2-torsion linebundle obtained there in the rest of the paper. Theorem 5.1.
Let C and η be the curve and the line bundle obtainedin Theorem 4.1. Assume moreover that the double covering induced by η is an ´ e tale. Then a quadric Q ∈ I C η (2) of rank equal or less than is a prjectivized tangent cone if and only if one of its rulings cuts acomplete g d with d ∈ { g − , g − , g − } and O ( g d ) ⊗ L = K C , forsome line bundle L on C which is of degree , or .Proof. If a quadric Q ∈ I C η (2) is a projectivized tangent cone of E ( π η ),then since g ( C ) = 7 one of its rulings cuts the desired complete linearseries, by Theorem 3.1.Conversely if one of the rulings of a quadric Q ∈ I C η (2) of rank equal orless than 4 cuts a complete g d with prescribed conditions, then Proposi-tion 3.2 implies that Q is a projectivized tangent cone of E ( π η ) providedthat d ∈ { g − , g − } . If d = g −
3, then one obtains a g on C . ByMartens-Mumford’s theorem there are only finitely many g ’s on C . Infact the pencils of lines through ¯ x through ¯ y or through ¯ z cut out three g ’s on C and one can see that a g on C is one of these pencils. Butit is easy to see that the rulings generated by divisors in these g ’s areat most of dimension 2 and therefore these rulings can not sweep outa quadric. In fact for D ∈| H − ( x + x ) | one has:dim( < D > ) = 5 − h ( K C · η − g )= 5 − h (3 H − ∆ + 2 H − D − ( H − ( x + x )))= 5 − h ( D − H + x + x ) ≤ − h ( D − H ) . Since C is non-hyperelliptic, the Clifford’s theorem asserts that h ( D − H ) ≤
3. Consider now that multiplying by H gives the following exactsequence: 0 → H ( D − H ) → H ( D ) → H ( H ) NOTE ON PROJECTIVIZED TANGENT CONE QUADRICS ... 11 which implies that h ( D ) − h ( D − H ) ≤ h ( H ) = 3. Therefore onehas h ( D − H ) ≥
3. Summarizing one has h ( D − H ) = 3 and sodim( < D > ) ≤ − h ( D − H ) = 2.These imply that d can not be equal to g − Q generated by divisors in g d have to sweep out thequadric itself. (cid:3) Now, in order to give an application of Theorem 5.1, we describe W and W on C : Example 5.2. (i) g ’s on C : By Mumford-Martens theorem and considering that C is not hyperel-liptic nor trigonal and nor a smooth plane quintic, one has dim( W ) ≤ − − p ∈ X − { ¯ x, ¯ y, ¯ z } , the lines passing through p cut out a pencil of degree 5 on X as well as do the quadrics through¯ x, ¯ y, ¯ z and p . These pencils give rise to pencils of the same kind on C via pulling them back to C by the normalization map, i . Considermoreover that the only way for a one dimensional sub vector space V ,of quadrics in P to cut a g on X is that each member of V has topass through ¯ x, ¯ y, ¯ z and p . The cubiques in P can not cut a g on X . Generally picking 6 d −
11 points p , p , ..., p d − fixed on X , thehypersurfaces of degree d in P passing through ¯ x, ¯ y, ¯ z and the chosen6 d −
11 points p , p , ..., p d − will cut a g r on X . For d ≥ r ≥ d ≥ g .Therefore dim( W ) = 1 and g ’s are cut on X by lines or quadrics of P . Now these pencils pulled back via i , are the only g ’s on C .If g = H − p for some p ∈ C then K C · η − g = D − H + p .Consider that h ( D ) = 6 and by proof of Proposition 3.2, one has h ( D − H ) = 3. Therefore h ( D − H − p ) = 3 if p belongs to thebase locus of | D − H | , and h ( D − H − p ) = 2 otherwise. Takingthe exact sequence0 → H ( D − H − p ) → H ( D − H ) → H ( H ) → h ( K C · η − g ) = 3 if p belongs to the base locusof | D − H | and h ( K C · η − g ) = 2 otherwise. These imply thatfor each divisor D ∈| g | one has dim( < D > ) = 2 if p belongs to thebase locus of | D − H | , and dim( < D > ) = 3 otherwise. Moreoverconsider that since by 3 H − ∆ ∼ H , one has K C = 2 g ⊗ O (2 p ). NowProposition 3.2 implies that the ruled hypersurface ∪ ´ D ∈ g < ´ D > isa prym projectivized tangent cone provided that p does not belong tothe base locus of | D − H | . In the case g = 2 H − x − y − z − p we have K C = 2 g ⊗ O (2 p ) and K C · η − g = 3 H − ∆ + 2 H − D − (2 H − x − y − z − p )= 4 H − ∆ − D + p ∼ D − H + p. Therefore the situation is the same as in the case g = H − p . (ii) g ’s on C : Again by Mumford-Martens theorem and consideringthat C is not hyperelliptic nor trigonal and nor a smooth plane quintic,one has dim( W ) ≤ − − P cut a g on C . Foreach p, q ∈ X − { ¯ x, ¯ y, ¯ z } the quadrics through p, q and through two ofthe points ¯ x, ¯ y, ¯ z cut a g on X . Again these pencils are pulled back to g ’s on C via the normalization map.Now similarly as in (i) , the only way for a one dimensional sub-vectorspace V , of quadrics in P to cut a g on X is that each member of V has to pass through two points p, q and through two of the points¯ x, ¯ y, ¯ z . A computation similar for g ’s case shows that these are theonly g ’s on X . Any g on C will be obtained by pulling back a g on X via i . If g = 2 H − x − y − p − q then K C · η − g = 3 H − ∆ + 2 H − D − (2 H − x − y − p − q ) ∼ D − H + 2¯ x + 2¯ y + p + q ∼ D − H + p + q − z. Therefore one has h ( K C · η − g ) = h ( D − H + p + q − z )= h ( D − H − z ) = h ( D − H ) − . where the last equality is valid because 2¯ z is not contained in the baselocus of | D − H | . These computations imply that for each D ∈| g | one has dim( < D > ) = 4 and the union of linear spaces < D > ⊂ P ,when D varies in g , fill up all the space P and therefore the linebundle g can not give a prym projectivized tangent cone. (iii) g , g ’s on C : The curve C can’t admit any g and dim( W ) ≤ P cut a g on X . The quadrics passing throughthe points ¯ x, ¯ y and ¯ z cut a g . This is nothing but the g cut by thelines in P . Again any g on C will be obtained by pulling back a g on X via i , the normalization map.As a byproduct of the computations just have been done, Theorem 3.1and Proposition 3.2, one has the following: Theorem 5.3.
The space of quadrics containing the K C · η -model of C is generated by Prym projectivized tangent cones at double points of E ( π η ) . Precisely the set of quadrics { Q g | g ∈ W } NOTE ON PROJECTIVIZED TANGENT CONE QUADRICS ... 13 which consists of a subset of projectivized tangent quadrics of rank equalor less than , generate the space of quadrics containing C η in P .Proof. There exists a map Φ defined byΦ : W → P ( I ( C )) ∼ = P Φ( g ) = Q g := ∪ D ∈ g < D > Consider that Φ is an embedding and by our computations, its imageis contained in the locus of projectivized tangent cone quadrics inside P ( I ( C )). Moreover consider again by example 5.2 that W consistsof two copies both are birational to C itself. Therefore the image ofthe map Φ is non-degenerate. These imply that the linear span ofprojectivized tangent cones is P ( I ( C )). (cid:3) Remark . Based on Debarre’s work in [7], the Prym-Torelli map P : R g → A g − is generically injective. This fails in the non-generic locus’ because ofwell known reasons. In fact Donagi’s tetragonal construction as well asgeneralized tetragonal construction introduced in [8] imply that, for anetale double covering ¯ Y → Y , of a tetragonal (generalized tetragonal)curve Y , there exist two another un-ramified double coverings havingPrymians isomorphic to that of ¯ Y → Y . This implies non-injectivityof the Prym-Torelli map in the locus of double coverings of tetragonal(generalized tetragonal) curves in R g .On the other hand as it has been noticed by H. Lange and E. Sernesifor the injectivity of the Torelli map in [11], an effective strategy todeal with the injectivity of the Prym-Torelli map seems to consist oftwo main steps. The first step is to show that for a given unramifieddouble cover π : ¯ X → X , the projectivized tangent cones at doublepoints of the Prym-theta divisor of the Prym variety generate the spaceof quadrics through the Prym-canonical model of the double covering.This step has been done by Debarre in [7] for curves varying in an opensubset of R g , as we already noticed.The second main step consists in proving that the quadrics throughthe Prym-canonical model of π : ¯ X → X in the projective space of thePrym-canonical differential forms, cut the Prym-canonical model. Thisstep has been also proved not only for general Prym-canonical curvesby Debarre in [7], but also for general tetragonal curves of genus atleast 11 by him in [6]. Meanwhile, H. Lange and E. Sernesi have donethis step for unramified double coverings of curves X with Cliff( X ) ≥ Consider now that Debarre’s work in [6], together with non-injectivityof Prym-Torelli map in tetragonal(generalized tetragonal) locus, implythat the projectivized tangent cones at double points of the Prym-thetadivisor of the Prym variety of a general tetragonal curve of genus atleast 11, or that of a generalized tetragonal curve, do not generate thespace of quadrics through the Prym-canonical model of an un-ramifieddouble coverings of such a curve.These however won’t give any information about validity or invalidityof the first step for an arbitrary tetragonal curve of genus g ≤
11, asit is concluded from our work that the first step remains valid for anetale double cover of the curve C . This as well proves that the quadricsthrough the Prym-canonical model of an arbitrary etale double coveringof the curve C , does not cut its Prym-canonical model. Acknowledgements:
Professor E. Sernesi read parts of an earlyversion of this manuscript and suggested some corrections. He more-over answered my various questions patiently. I am grateful to himand I express my hearty thanks to him. The curve C in the paper, hasbeen addressed to me by professor A. Verra and professor G. Farkaswhen I was visiting the Rome Tre University on fall of 2011. I expressmy deep gratitude for professor A. Verra and professor G. Farkas forthis and another hints that I received from them. References [1] E. Arbarello, M. Cornalba, Ph. Griffiths, J. Harris; Geometry of AlgebraicCurves. I Grundlehren 267(1985), Springer.[2] A. Andreotti, A. Mayer; On Period relations for Abelian Integrals on Alge-braic Curves. Ann. Scoula Norm. Sup. Pisa 21(1967) 189-238.[3] E. Arbarello, E. Sernesi; Petri’s Approach to the study of Ideal associatedto a Special Divisor, Inventions Math. 49, 99-119(1978).[4] E. Arbarello, J. Harris; Canonical Curves and Quadrics of rank 4, Compositiomathematica. 43 p.145-179(1981).[5] Ch. Birkenhake, H. Lange; Complex Abelian Varieties (Seconed Edition). IGrundlehren der mathematischen Wissenschaften, Vol. 302, Springer(2004).[6] O. Debarre; Sur les vari´ e t´ e s de Prym des courbes t´ e tragonales, Ann. Scient.´ E c. Norm. Sup. 4e serie, t.21 (1988), 545-559.[7] O. Debarre; Sur les probleme de Torelli pour les vari´ e t´ e s de Prym, Amer. J.Math. 111(1989), 193-212.[8] E. Izadi, H. Lange; Counter Examples of high Clifford Indices for Prym-Torelli, J. A. Geometry,[9] L. Gruson, R. Lazarsfeld, C. Peskine; On a theorem of Castelnouvo and theequations defining space curves. Invent. Math. 72(1983) 491-506.[10] G. Kempf; On the Geometry of a Theorem of Riemann. Ann. of Math.98(1973) 178-185. NOTE ON PROJECTIVIZED TANGENT CONE QUADRICS ... 15 [11] H. Lange, E. Sernesi; Quadrics containing a Prym-Canonical Curve, J. Al-gebraic Geometry 5(1996), 387-399.[12] S.C. Martin; Singularities of the Prym theta divisor, Annales of Mathemat-ics, Vol. 170, No. 1, (2009), 163-204.[13] R. Smith, R. Varley; A Riemann Singularities Theorem for Prym ThetaDivisors, with applications, Pacific Journal of Mathematics, volume 201,No. 2, 479-509, December 2001.[14] R. Smith, R. Varley; The Curve of ”Prym Canonical” Gauss Divisors ona Prym Theta Divisor, Transactions of the A. M. S. V.353, N.12, pages4949-4962(2001).
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