aa r X i v : . [ m a t h . AG ] F e b Prym curves witha vanishing theta-null
CARLOS MAESTRO PER ´EZ
Abstract
If the theta-null divisor Θ null is moved to the Prym moduli space throughthe diagram S + g → M g ← R g , it splits into two irreducible components P +null and P − null . Using test curve techniques, we compute the expressionof the rational divisor classes of P +null and P − null in terms of the generatingclasses of Pic( R g ) Q . When looking at the moduli space S + g of even spin curves of genus g , there isa natural, geometric divisor that quickly takes center stage, namely the divisorof curves with a vanishing theta-null, or theta-null divisor :Θ null = { ( C, θ ) ∈ S + g / h ( C, θ ) ≥ } ⊂ S + g In this setting, the theta-null divisor has been thoroughly studied, and its classhas been computed by [Far10]. The same can be said about its pushforward to M g , which was similarly described at an even earlier date in [TiB88]. However,its interaction with the moduli space R g of Prym pairs of genus g , by means ofthe diagramΘ null ⊂ S + g π + " " ❊❊❊❊❊ R gπ R | | ②②②②② ⊃ P null = ( π R ) ∗ ( π + ) ∗ (Θ null ) M g remains largely unexplored. In this new setting, the divisor P null splits into twoirreducible components P +null = { ( C, η ) ∈ R g / ∃ θ ∈ S + g ( C ) with θ ⊗ η ∈ Θ null ( C ) } ⊂ R g P − null = { ( C, η ) ∈ R g / ∃ θ ∈ S − g ( C ) with θ ⊗ η ∈ Θ null ( C ) } ⊂ R g which we call even and odd Prym-null divisors . The main result of our work isthe computation of the classes of P +null and P − null in Pic( R g ) Q with standard testcurve techniques, culminating in theorem 3.12. We obtain: P +null ≡ g − (cid:18) (2 g − + 1) λ − (cid:16) g − δ ′ + (2 g − + 1) δ ram0 (cid:17) − X (cid:16) (2 i − − g − i − δ i + (2 i − g − i − − δ g − i ++ (2 g − − i − − g − i − + 1) δ i : g − i (cid:17)(cid:19) P − null ≡ g − (cid:18) g − λ − (cid:16) g − δ ′′ + 2 g − δ ′ + (2 g − − δ ram0 (cid:17) − X (cid:16) i − (2 g − i − δ i + (2 i −
1) 2 g − i − δ g − i ++ (2 g − − i − − g − i − ) δ i : g − i (cid:17)(cid:19) In order to arrive at these expressions, we require some groundwork, which weestablish in the remainder of this section. Subsection 1.1 offers a brief reminderon the compactifications of R g and S g , and their corresponding boundaries aredescribed in subsection 1.2. Subsection 1.3 introduces the different collections oftest curves in R g that play a role in the proof of theorem 3.12. Once the properbackground has been set up, the second section deals with the construction ofthe Prym-null divisors (subsection 2.1) and their connection with the potentialparity change induced on theta characteristics by a Prym root (subsection 2.2for the smooth case, and 2.3 for the irreducible nodal one). The final section isdevoted to the test curve computation, which is carried out in subsections 3.1,3.2 and 3.3 for reducible nodal curves, curves with elliptic tails and irreduciblenodal curves respectively, and whose conclusion leads to subsection 3.4, where thedesired expansions of the even and odd Prym-null classes are given and appliedto additional families of Prym curves. Acknowledgements.
The author would like to thank Prof. Farkas for bringingthis problem to his attention, as well as the Berlin Mathematical School and theHumboldt-Universit¨at zu Berlin for their financial support. Special thanks go toAndrei Bud for his helpful discussions regarding remark 1.14.
We work over C , and take C to be a smooth, integral curve of genus g . .1 Stable Prym and spin curves Definition 1.1. A Prym root of C is a nontrivial square root of O C , that is, aline bundle η ≇ O C of degree zero equipped with an isomorphism η ⊗ ∼ = O C . A Prym pair is a pair (
C, η ) such that η is a Prym root of C . The set of Prymroots of C is denoted by R g ( C ) ֒ → Pic ( C ) − {O C } . The moduli space of Prympairs of genus g is denoted by R g . Definition 1.2. A theta characteristic of C is a square root of ω C , that is, aline bundle θ of degree g − θ ⊗ ∼ = ω C . A spincurve is a pair ( C, θ ) such that θ is a theta characteristic of C . The set of thetacharacteristics of C is denoted by S g ( C ) ֒ → Pic g − ( C ). The moduli space of spincurves of genus g is denoted by S g = S + g ⊔ S − g .We now want to consider stable versions of these notions. To that end, recallthe following standard defitions. Definition 1.3.
Let X be a complete, connected, nodal curve. We say that X is stable (resp. semistable ) if every smooth rational component of X meets theother components of X in at least 3 points (resp. at least 2 points). Definition 1.4.
Let E be an irreducible component of a semistable curve X .Then E is said to be exceptional if it is smooth, rational, and meets the othercomponents in exactly 2 points. Definition 1.5.
Let X be a semistable curve. We say that X is quasistable ifany two distinct exceptional components are disjoint. In turn, the stable model of a quasistable curve X is the stable curve st( X ) obtained by contracting eachexceptional component to a point. Definition 1.6. A stable Prym curve is a triplet ( X, η, β ) where:(i) X is a quasistable curve (of genus g ).(ii) η ∈ Pic ( X ) is a nontrivial line bundle of total degree 0 on X such that η | E = O E (1) for every exceptional component E of X .(iii) β : η ⊗ → O X is a sheaf homomorphism such that the restriction β | A isgenerically non-zero for every non-exceptional component A of X .Similarly, a stable even spin curve is a triplet ( X, θ, α ) where:(i) X is a quasistable curve (of genus g ).(ii) θ ∈ Pic g − ( X ) is a line bundle of total degree g − X with h ( X, θ )even, and θ | E = O E (1) for every exceptional component E of X .(iii) α : η ⊗ → ω X is a sheaf homomorphism such that the restriction α | A isgenerically non-zero for every non-exceptional component A of X .For a stable odd spin curve , simply take h ( X, θ ) odd.
Definition 1.7.
Let S be a scheme. A family of stable Prym curves over thebase S , or a stable Prym curve over S , is a triplet ( f : X → S, η, β ) such that:(i) f : X → S is a quasistable (genus g ) curve over S .(ii) η ∈ Pic ( X ) is a line bundle on X .(iii) β : η ⊗ → O X is a sheaf homomorphism.(iv) The restriction of ( f : X → S, η, β ) to any fiber f − ( s ) = X s gives riseto a stable Prym curve ( X s , η s , β s ).An isomorphism ( X → S, η, β ) ∼ = ( X ′ → S, η ′ , β ′ ) is a pair ( ϕ, ψ ) where:(i) ϕ : X ∼ = X ′ is an isomorphism over S .(ii) ψ : ϕ ∗ ( η ′ ) ∼ = η is a sheaf isomorphism such that ϕ ∗ ( β ′ ) = β ◦ ψ ⊗ .With minimal changes, we could likewise define families of stable spin curves .The resulting moduli problems all admit proper moduli spaces, namely R g , S + g and S − g , which respectively compactify R g , S + g and S − g . Further details onthese can be found in [Cor89], for the compactification of S g , and [BCF04], forthe Cornalba-inspired compactification of R g . It is also important to highlight[Bea77], where a different but earlier compactification of R g was built throughthe use of admissible double covers of stable curves.Observe that stabilising preserves genus, and consider the natural maps π R : R g → M g , ( X, η, β ) st( X ) π S : S g → M g , ( X, θ, α ) st( X )where st( X ) is the stable model of X . These maps are finite and ramified overthe boundary, and extend the finite, unramified covers R g → M g , S g → M g . In order to study the boundary of R g , we can take advantage of the map π R : R g → M g , ( X, η, β ) st( X )which turns the decomposition in irreducible components ∂ M g = ∆ ∪ ∆ ∪ . . . ∪ ∆ ⌊ g/ ⌋ into a building block for the corresponding decomposition of ∂ R g . .2 Boundary divisors Y ∈ ∆ i is of the form:( i > Y = C ∪ p ∼ q D with ( C, p ) ∈ M i, , ( D, q ) ∈ M g − i, ( i = 0) Y = B pq with ( B, p, q ) ∈ M g − , where B pq denotes the irreducible 1-nodal curve obtained from B by gluing thepoints p and q . Let us describe the fibers π − R ( Y ) for each i . Example 1.8 ( i > . Let (
X, η, β ) ∈ R g with st( X ) = Y = C ∪ p ∼ q D . Theexistence of β prevents X from having exceptional components, i.e. X = st( X ) = Y = C ∪ p ∼ q D, β : η ⊗ ∼ = O Y = ( O C , O D )Then η is a nontrivial element of J ( C ) ⊕ J ( D ), and we have three irreduciblecomponents over ∆ i , characterized by their general point ( X, η, β ):(∆ n i ) Condition: η = ( η C , O D ) with η C ∈ R i ( C ).Notation: ∆ n i ⊂ R g (for nontrivial on i ), or traditionally ∆ i .Degree: deg(∆ n i | ∆ i ) = 2 i − t i ) Condition: η = ( O C , η D ) with η D ∈ R g − i ( D ).Notation: ∆ t i ⊂ R g (for trivial on i ), or traditionally ∆ g − i .Degree: deg(∆ t i | ∆ i ) = 2 g − i ) − p i ) Condition: η = ( η C , η D ) with η C ∈ R i ( C ), η D ∈ R g − i ( D ).Notation: ∆ p i ⊂ R g (for Prym ), or traditionally ∆ i : g − i .Degree: deg(∆ p i | ∆ i ) = (2 i − g − i ) − i ⊂ M g can be written as π ∗R (∆ i ) = ∆ n i + ∆ t i + ∆ p i and, in terms of divisor classes, we have relations π ∗R ( δ i ) = δ n i + δ t i + δ p i for 1 ≤ i ≤ ⌊ g/ ⌋ and δ x i = O R g (∆ x i ) ∈ Pic( R g ), x ∈ { t , n , p } . Observe thatdeg(∆ n i | ∆ i ) + deg(∆ t i | ∆ i ) + deg(∆ p i | ∆ i ) = 2 g − π R )as expected. Example 1.9 ( i = 0) . Let (
X, η, β ) ∈ R g with st( X ) = Y = B pq . There aretwo possibilities for X , depending on whether it contains or not an exceptional component. If it does not, i.e. X = st( X ) = Y = B pq , β : η ⊗ ∼ = O Y then the normalization ν : B → B pq induces an exact sequence0 −→ Z −→ J ( B pq ) ν ∗ −→ J ( B ) −→ , η B = ν ∗ η ∈ J ( B )and the potential triviality of η B = ν ∗ η determines two irreducible components:(∆ t0 ) Condition: η B = O B , hence η ∈ ( ν ∗ ) − ( η B ) − {O Y } unique.Notation: ∆ t0 ⊂ R g (for trivial ), or traditionally ∆ ′′ .Degree: deg(∆ t0 | ∆ ) = 1.(∆ p0 ) Condition: η B ∈ R g − ( B ), hence η ∈ ( ν ∗ ) − ( η B ) ∼ = Z .Notation: ∆ p0 ⊂ R g (for Prym ), or traditionally ∆ ′ .Degree: deg(∆ p0 | ∆ ) = 2 (2 g − − X has an exceptional component E , then we can projectit onto Y as a sort of “exceptional blow-up”, i.e. there is a map X = B ∪ p ∼ , q ∼∞ E −→ st( X ) = Y = B pq induced by ν : B → B pq , E z = ν ( p ) = ν ( q ). Then we have e X = X − E ∼ = B, β : η ⊗ B ∼ = O B ( − p − q )for η B = η | B ∈ Pic( B ), and Mayer-Vietoris yields an exact sequence0 −→ C ∗ −→ Pic( X ) ξ −→ Pic( B ) ⊕ Pic( E ) −→ , ξ ( η ) = ( η B , O E (1))This way, we obtain one last irreducible component:(∆ b0 ) Condition: η B ∈ p O B ( − p − q ).Notation: ∆ b0 ⊂ R g (for blown-up ), or traditionally ∆ ram0 .Degree: deg(∆ b0 | ∆ ) = 2 g − .Due to the appearance of an exceptional component over the node z ∈ B pq , thedivisor ∆ b0 is in fact the ramification divisor of π R : R g → M g . The pullback of∆ ⊂ M g can accordingly be written as π ∗R (∆ ) = ∆ t0 + ∆ p0 + 2 ∆ b0 and, in terms of divisor classes, we have the relation π ∗R ( δ ) = δ t0 + δ p0 + 2 δ b0 .2 Boundary divisors δ x0 = O R g (∆ x0 ) ∈ Pic( R g ), x ∈ { t , p , b } . Observe thatdeg(∆ t0 | ∆ ) + deg(∆ p0 | ∆ ) + 2 deg(∆ b0 | ∆ ) = 2 g − π R )as expected. Remark 1.10.
In example 1.9, note that deg(∆ b0 | ∆ ) is finite because, for η B ∈ p O B ( − p − q ) fixed, any two line bundles λ, µ ∈ ξ − ( η B , O E (1)) ∼ = C ∗ even if non-isomorphic as bundles, induce triplets( X, λ, β λ ) ∼ = ( X, µ, β µ ) ∈ R g that are always isomorphic as stable Prym curves; see [BCF04] Lemma 2.We can now repeat the process for S g , or rather its irreducible components S + g , S − g . Recall the projection π S : S g → M g , ( X, θ, α ) st( X ) = Y whose fibers π − S ( Y ) we describe for Y ∈ ∆ i general, 0 ≤ i ≤ ⌊ g/ ⌋ . Example 1.11 ( i > . Let (
X, θ, α ) ∈ S g with st( X ) = Y = C ∪ p ∼ q D . Theexistence of α forces X to have an exceptional component, i.e. there is a map X = C ∪ p ∼ E ∪ q ∼∞ D −→ st( X ) = Y = C ∪ p ∼ q D induced by E z = [ p ] = [ q ], and we get e X = X − E ∼ = C ⊔ D, α : ( θ C , θ D ) ⊗ ∼ = ω X | e X ( − p − q ) = ( ω C , ω D )for ( θ C , θ D ) = θ | e X ∈ Pic( C ) ⊕ Pic( D ). Therefore, θ is determined by a pair( θ C , θ D ) ∈ S i ( C ) ⊕ S g − i ( D ) , θ = ( θ C , O E (1) , θ D ) ∈ Pic( X )In particular, notice that the (even, odd) parity of θ is subject to the (identical,alternating) character of the parities of θ C and θ D , since we have the relation h ( X, θ ) = h ( C, θ C ) + h ( D, θ D )by Mayer-Vietoris. As a result, out of the four irreducible components that areobtained over each ∆ i , two lie in S + g and two lie in S − g . The even ones are: (∆ + i ) Condition: θ C ∈ S + i ( C ), θ D ∈ S + g − i ( D ).Notation: ∆ + i = ∆ + g − i ⊂ S + g (for even on i ), or traditionally A + i .Degree: deg(∆ + i | ∆ i ) = 2 g − (2 i + 1)(2 g − i + 1).(∆ − i ) Condition: θ C ∈ S − i ( C ), θ D ∈ S − g − i ( D ).Notation: ∆ − i = ∆ − g − i ⊂ S + g (for odd on i ), or traditionally B + i .Degree: deg(∆ − i | ∆ i ) = 2 g − (2 i − g − i − + i ) Condition: θ C ∈ S + i ( C ), θ D ∈ S − g − i ( D ).Notation: ∆ + i = ∆ − g − i ⊂ S − g (for even on i ), or traditionally A − i .Degree: deg(∆ + i | ∆ i ) = 2 g − (2 i + 1)(2 g − i − − i ) Condition: θ C ∈ S − i ( C ), θ D ∈ S + g − i ( D ).Notation: ∆ − i = ∆ + g − i ⊂ S − g (for odd on i ), or traditionally B − i .Degree: deg(∆ − i | ∆ i ) = 2 g − (2 i − g − i + 1).Observe that a factor of 2 has to be considered in the computation∆ + i ⊂ S + g , deg(∆ + i | ∆ i ) = 2 · S + i ( C ) · S + g − i ( D )= 2 · i − (2 i + 1) · g − i − (2 g − i + 1)= 2 · g − (2 i + 1)(2 g − i + 1)to account for the nontrivial automorphism of ( X, θ, α ) that arises from scalingby − S + g → M g ,this factor is not present. Consequently, the pullback of ∆ i ⊂ M g (resp. ⊂ M g )by π + : S + g → M g (resp. π + : S + g → M g ) can be written as π ∗ + (∆ i ) = 2 ∆ + i + 2 ∆ − i (resp. π ∗ + (∆ i ) = ∆ + i + ∆ − i )and, in terms of divisor classes, we have relations π ∗ + ( δ i ) = 2 δ + i + 2 δ − i (resp. π ∗ + [∆ i ] = [∆ + i ] + [∆ − i ])for 1 ≤ i ≤ ⌊ g/ ⌋ and δ x i = O S + g (∆ x i ) ∈ Pic( S + g ), x ∈ { + , −} . The same analysisworks for S − g and S − g . Example 1.12 ( i = 0) . Let (
X, θ, α ) ∈ S g with st( X ) = Y = B pq . There areagain two possibilities for X . If it has no exceptional components, i.e. X = st( X ) = Y = B pq , α : θ ⊗ ∼ = ω Y .2 Boundary divisors ν : B → B pq induces a double cover ν ∗ : p ω Y −→ p ω B ( p + q ) , ( ν ∗ θ ) ⊗ ∼ = ν ∗ ω Y ∼ = ω B ( p + q )so that θ is determined by a square root θ B ∈ p ω B ( p + q ) and a choice on howto glue its fibers θ B | p and θ B | q . Only two such gluings are possible, one making h ( X, θ ) even and the other one making it odd. We describe the component ∆ n0 obtained in this way only for S + g , as its S − g counterpart is very similar.(∆ n0 ) Condition: θ B ∈ p ω B ( p + q ) with even gluing.Notation: ∆ n0 ⊂ S + g (for not blown-up ), or traditionally A +0 .Degree: deg(∆ n0 | ∆ ) = 2 g − .On the other hand, if X has an exceptional component E , then X = B ∪ p ∼ , q ∼∞ E −→ st( X ) = Y = B pq , e X = X − E ∼ = B and we have θ B = θ | B ∈ S g − ( B ), since α gives rise to an isomorphism α : θ ⊗ B ∼ = ω X | B ( − p − q ) ∼ = ν ∗ ω Y ( − p − q ) ∼ = ω B Moreover, recall the exact sequence0 −→ C ∗ −→ Pic( X ) ξ −→ Pic( B ) ⊕ Pic( E ) −→ , ξ ( θ ) = ( θ B , O E (1))and note that h ( X, θ ) = h ( B, θ B ), again by Mayer-Vietoris. In conclusion, weget the remaining irreducible component of ∂ S + g , and similarly for ∂ S − g , as:(∆ b0 ) Condition: θ B ∈ S + g − ( B ).Notation: ∆ b0 ⊂ S + g (for blown-up ), or traditionally B +0 .Degree: deg(∆ b0 | ∆ ) = 2 g − (2 g − + 1).The pullback of ∆ ⊂ M g by π + : S + g → M g can be written as π ∗ + (∆ ) = ∆ n0 + 2 ∆ b0 and, in terms of divisor classes, we have the relation π ∗ + ( δ ) = δ n0 + 2 δ b0 for δ x0 = O S + g (∆ x0 ) ∈ Pic( S + g ), x ∈ { n , b } . Finally, the divisor∆ b0 + P (∆ + i + ∆ − i ) (resp. ∆ b0 )is the ramification divisor of π + : S + g → M g (resp. π + : S + g → M g ).0 Examples 1.8 and 1.9 provide us with a collection of boundary classes of R g ,while examples 1.11 and 1.12 follow suit with S + g (and S − g ): δ t0 , δ p0 , δ b0 , δ t i , δ n i , δ p i ∈ Pic( R g ) , ≤ i ≤ ⌊ g/ ⌋ δ n0 , δ b0 , δ + i , δ − i ∈ Pic( S + g ) , ≤ i ≤ ⌊ g/ ⌋ For g ≥
5, we then getPic( R g ) Q = λ Q ⊕ δ t0 Q ⊕ δ p0 Q ⊕ δ b0 Q ⊕ ⌊ g/ ⌋ M i =1 ( δ t i Q ⊕ δ n i Q ⊕ δ p i Q )and similarlyPic( S + g ) Q = λ Q ⊕ δ n0 Q ⊕ δ b0 Q ⊕ ⌊ g/ ⌋ M i =1 ( δ + i Q ⊕ δ − i Q )where λ denotes the pullback of λ ∈ Pic( M g ) to R g and S + g , respectively. Remark 1.13.
As the Hodge bundle construction used to build λ ∈ Pic( M g )commutes with base change, the class λ in R g or S + g can likewise be defined bymeans of the Hodge bundle associated to each of these spaces. Remark 1.14.
For the decompositions of Pic( R g ) Q and Pic( S + g ) Q to hold, it isenough to see that Pic( R g ) Q and Pic( S + g ) Q are infinite cyclic. In the case of R g ,we have finite maps M g (2) → R g → M g , ( C, η , . . . , η g ) ( C, η ) C where M g (2) is the moduli of curves with a level 2 structure, that is, a basis ofthe 2-torsion of their Jacobian. As a result, we get injective pullback mapsPic( M g ) Q ֒ → Pic( R g ) Q ֒ → Pic( M g (2)) Q Since Putman’s work [Put12a, Put12b] shows that Pic( M g (2)) Q ∼ = Q for g ≥ R g ) Q ∼ = Q in this range. The corresponding result for S + g isdue to Harer [Har93] for g ≥
9, and Putman [Put12a] for g ≥ .3 Test curves Let us take the most basic families of test curves on M g and examine waysof lifting them to R g . In the following examples, we denote π : R g → M g instead of π R , as we will not work with spin curves here. However, descriptionsof common test curves on S g can be found in [Far10] or [FV14]. Example 1.15 (reducible nodal curves) . For each integer 2 ≤ i ≤ g −
1, we fixgeneral curves C ∈ M i and ( D, q ) ∈ M g − i, and consider the test curve C i = ( C × C ) ∪ ∆ C ∼ C ×{ q } ( C × D ) −→ C corresponding to the family of reducible nodal curves C i ≡ { C ∪ y ∼ q D } y ∈ C ⊂ ∆ i ⊂ M g Using the standard test curve techniques of [HM98] Chapter 3, we can see thatthe intersection numbers of C i with the generators of Pic( M g ) Q given earlier inthe section are described by the table: λ δ i δ ( j = i ) C i − i η C ∈ R i ( C ), η D ∈ R g − i ( D ) and lift C i to test curves F i , G i , H i → C , as follows: F i ≡ { ( C ∪ y ∼ q D, ( η C , O D )) } y ∈ C ⊂ ∆ n i ⊂ R g G i ≡ { ( C ∪ y ∼ q D, ( O C , η D )) } y ∈ C ⊂ ∆ t i ⊂ R g H i ≡ { ( C ∪ y ∼ q D, ( η C , η D )) } y ∈ C ⊂ ∆ p i ⊂ R g Observe that π ∗ ( F i ) = π ∗ ( G i ) = π ∗ ( H i ) = C i . Then F i · δ n i = F i · π ∗ δ i = C i · δ i = 2 − iG i · δ t i = G i · π ∗ δ i = C i · δ i = 2 − iH i · δ p i = H i · π ∗ δ i = C i · δ i = 2 − i and all other intersection numbers are 0, which is collected in the table: λ δ t0 δ p0 δ b0 δ n i δ t i δ p i δ ( j = i ) F i − i G i − i H i − i g = 2 i , where we have F i · δ n i = G i · δ n i = 2 − i .Let us move on to the standard degree 12 pencil of elliptic tails in M g . Example 1.16 (elliptic tails) . We fix a general curve (
C, p ) ∈ M g − , and ageneral pencil f : Bl ( P ) → P of plane cubics, with fibers { E λ = f − ( λ ) } λ ∈ P ⊂ M together with a section σ : P → Bl ( P ) induced by one of the basepoints. Wemay then glue the curve ( C, p ) to the pencil f along σ , thus producing a pencilof stable curves C = ( C × P ) ∪ { p }× P ∼ σ ( P ) Bl ( P ) −→ P which corresponds to C ≡ { C ∪ p ∼ σ ( λ ) E λ } λ ∈ P ⊂ ∆ ⊂ M g As in the previous example, [HM98] shows that the intersection numbers of thepencil C with the generators of Pic( M g ) Q are given by the table: λ δ δ δ ( j ≥ C − η C ∈ R g − ( C ), then the degree 3 branched covering γ : R , → M , allows us to lift C to test curves F , G , H , as follows: F ≡ { ( C ∪ p ∼ σ ( λ ) E λ , ( η C , O E λ )) } λ ∈ P ⊂ ∆ t1 ⊂ R g G ≡ { ( C ∪ p ∼ σ ( λ ) E λ , ( O C , η E λ )) / η E λ ∈ γ − ( E λ ) } λ ∈ P ⊂ ∆ n1 ⊂ R g H ≡ { ( C ∪ p ∼ σ ( λ ) E λ , ( η C , η E λ )) / η E λ ∈ γ − ( E λ ) } λ ∈ P ⊂ ∆ p1 ⊂ R g .3 Test curves π ∗ ( F ) = C and π ∗ ( G ) = π ∗ ( H ) = 3 C , so in particular F · δ t1 = F · π ∗ δ = C · δ = − G · δ n1 = G · π ∗ δ = 3 C · δ = − H · δ p1 = H · π ∗ δ = 3 C · δ = − λ ∞ ∈ P that correspond to singular fibers of C andblowing up the node of the rational component E λ ∞ ∈ ∆ , we see that, for F ,the pullback of η λ ∞ = ( η C , O E λ ∞ ) is ( η C , O P ), which is nontrivial. As discussedin example 1.9, this implies that F , λ ∞ ∈ ∆ p0 , hence F · δ p0 = F · π ∗ δ = C · δ = 12Furthermore, the covering γ : R , → M , is branched over E λ ∞ , and thus thefiber γ − ( E λ ∞ ) consists of two elements: one lying in the ramification divisor of γ , which we denote by η b E λ ∞ , and one outside, which we denote by η t E λ ∞ . Thenthe pullback of ( O C , η t E λ ∞ ) is ( O C , O P ), that is, ( O C , η t E λ ∞ ) ∈ ∆ t0 , and we get G · δ t0 = C · δ = 12 G · δ b0 = C · δ = 12Finally, the pair ( η C , η t E λ ∞ ) pulls back to the nontrivial pair ( η C , O P ), and so itbelongs to ∆ p0 , yielding H · δ p0 = C · δ = 12 H · δ b0 = C · δ = 12All other intersection numbers are 0, except for F · λ = 1, G · λ = H · λ = 3.In summary, we obtain a table: λ δ t0 δ p0 δ b0 δ n1 δ t1 δ p1 δ ( j ≥ F − G − H − π ∗ ( G ) · δ = G · π ∗ δ = G · ( δ t0 + 2 δ b0 ) π ∗ ( H ) · δ = H · π ∗ δ = H · ( δ p0 + 2 δ b0 )both hold. Example 1.17 (irreducible nodal curves) . In keeping with the notation used in4 example 1.9, we fix a general curve (
B, p ) ∈ M g − , and consider the test curveobtained by gluing p to a varying point y ∈ B , namely Y = Bl ( p,p ) ( B × B ) / (∆ B ∼ B × { p } ) −→ B This corresponds to a family
Y ≡ { B py } y ∈ B ⊂ ∆ ⊂ M g where B py is an irreducible nodal curve for y = p and B pp is a copy of B with a pigtail attached to p , in the sense of [HM98] Section 3.C. Again, we can readilysee that the intersection table of Y with the generators of Pic( M g ) Q is: λ δ δ δ ( j ≥ Y − g Y by the map ∆ t0 → ∆ , we lift it to a test curve Y such that: Y ≡ { ( B py , η t y ) / η t y ∈ ∆ t0 ( B py ) } y ∈ B ⊂ ∆ t0 ⊂ R g Since deg(∆ t0 | ∆ ) = 1, we have π ∗ ( Y ) = Y , hence Y · δ t0 = Y · π ∗ δ = Y · δ = 2 − g In addition, the special fiber η t p lies in ∆ n1 , as it pulls back to the trivial bundle( O B , O P ) on the normalization B × P of B pp , and thus is trivial over B . Thenthe last non-zero intersection number standing is Y · δ n1 = Y · π ∗ δ = Y · δ = 1and we get a table: λ δ t0 δ p0 δ b0 δ n1 δ t1 δ p1 δ ( j ≥ Y − g p0 → ∆ or ∆ b0 → ∆ . In this section, we define the even and odd Prym-null divisors and study howa theta characteristic changes parity when tensored by a Prym root. .1 The Prym-null divisor and its irreducible components P null and its irreducible components Let C be a smooth, integral curve of genus g . Definition 2.1.
An even theta characteristic θ on C with h ( C, θ ) = 0 (that is,with h ( C, θ ) ≥ h ( C, θ ) ≡ vanishing theta-null .The terminology here may seem confusing, as vanishing theta-nulls are eventheta characteristics with non-vanishing global sections. This is justified by theclassical theory of theta functions, whose Thetanullwert vanishes only when theassociated even theta characteristic is a vanishing theta-null; see [Bea13].The locus of curves with a vanishing theta-null, namelyΘ null = { ( C, θ ) ∈ S + g / h ( C, θ ) ≥ } = S + g ∩ W g − , g gives rise to the theta-null divisor Θ null on S + g , as well as its closure Θ null in S + g .This divisor plays an important role in the study of the geometry of S + g , due toits effective nature and geometric characterization: for example, a computationof the class of Θ null allows [Far10] to prove that S + g is of general type for g ≥ g = 8.The theta-null divisor can be pushed forward by π + : S + g → M g to obtain M null g = { C ∈ M g / ∃ θ ∈ S + g ( C ) with h ( C, θ ) ≥ } ⊂ M g whose closure M null g in M g is described by [TiB88]. In turn, pulling back M null g by π R : R g → M g results in a divisor P null = { ( C, η ) ∈ R g / ∃ θ ∈ S + g ( C ) with h ( C, θ ) ≥ } ⊂ R g Note that the line bundle θ ⊗ η is again a theta characteristic, different from θ ,which may therefore be even or odd. Moreover, [TiB87] shows that the projectionΘ null → M null g is generically finite of degree 1, hence we can build a rational map P null → S g = S + g ⊔ S − g , ( C, η ) ( C, θ ⊗ η )where θ ∈ Θ null ( C ). Then, with the temporary notation¯ θ = θ ⊗ η ∈ S g ( C ) , θ = ¯ θ ⊗ η ∈ Θ null ( C )we may rewrite the defining condition of P null as P null = { ( C, η ) ∈ R g / ∃ ¯ θ ∈ S g ( C ) with ¯ θ ⊗ η ∈ Θ null ( C ) } ⊂ R g and deduce that the parity of ¯ θ = θ ⊗ η yields a decomposition P null = P +null ⊔ P − null Dropping the bar for the sake of simplicity, we get the following:
Definition 2.2.
We refer to the divisor P null on R g as the Prym-null divisor .Accordingly, its irreducible components P +null and P − null , namely P +null = { ( C, η ) ∈ R g / ∃ θ ∈ S + g ( C ) with θ ⊗ η ∈ Θ null ( C ) } ⊂ R g P − null = { ( C, η ) ∈ R g / ∃ θ ∈ S − g ( C ) with θ ⊗ η ∈ Θ null ( C ) } ⊂ R g with P null = P +null + P − null , are called the even and odd Prym-null divisors .Since the Prym-null divisors are natural, geometric divisors on R g , our goalis to compute the class of their closures P +null , P − null in R g . Such a computationwould continue the work of [TiB88] and [Far10], where the classes of M null g andΘ null are respectively expressed in terms of the generating classes of Pic( M g ) Q and Pic( S + g ) Q . In particular, write µ null g = O M g ( M null g ) ∈ Pic( M g ) , ϑ null = O S + g (Θ null ) ∈ Pic( S + g ) . for the aforementioned classes, and consider the notation λ, δ i , δ x i introduced inexamples 1.11 and 1.12. Then [TiB88] and [Far10] provide formulas µ null g = 2 g − (cid:16) (2 g + 1) λ − g − δ − ⌊ g/ ⌋ X i =1 (2 i − g − i − δ i (cid:17) ϑ null = 14 λ − δ n0 − ⌊ g/ ⌋ X i =1 δ − i the latter of which implies the former, as the class [ M null g ] can also be obtainedby pushing forward the class [Θ null ] by the coarse moduli map S + g → M g .Let us write the classes of P null , P +null and P − null as ̺ null = O R g ( P null ) ∈ Pic( R g ) , ̺ null = ̺ +null + ̺ − null ̺ +null = O R g ( P +null ) ∈ Pic( R g ) ̺ − null = O R g ( P − null ) ∈ Pic( R g )and recall the notation δ t0 , δ p0 , δ b0 , δ n i , δ t i , δ p i from examples 1.8 and 1.9. The sum .1 The Prym-null divisor and its irreducible components ̺ null can be directly computed as the pullback of µ null g by the map π R : R g → M g , π ∗R ( λ ) = λπ ∗R ( δ ) = δ t0 + δ p0 + 2 δ b0 π ∗R ( δ i ) = δ n i + δ t i + δ p i with 1 ≤ i < g/
2, and moreover π ∗R ( δ g/ ) = δ n g/ + δ p g/ for even g . Proposition 2.3.
The class of P null in Pic( R g ) Q is given by ̺ null = 2 g − (cid:16) (2 g + 1) λ − g − ( δ t0 + δ p0 + 2 δ b0 ) − k X i =1 (2 i − g − i − δ n i + δ t i + δ p i ) − ψ ( g ) · (2 g/ − ( δ n g/ + δ p g/ ) (cid:17) where the upper bound k and the parity-checking function ψ ( g ) , defined as k = ⌈ g/ ⌉ − ( ⌊ g/ ⌋ if g odd ⌊ g/ ⌋ − g even ψ ( g ) = 1 + ( − g ( g odd1 if g even account for the slight variation in pullback that occurs when g = 2 i . Proof.
Follows from π ∗R ( µ null g ) = ̺ null and the formulas above. Remark 2.4.
Once the classes ̺ +null and ̺ − null are computed, proposition 2.3 offersa quick double-check of their accuracy, by virtue of ̺ +null + ̺ − null = ̺ null .8 Remark 2.5.
With the notation of proposition 2.3, we may write ̺ +null = λ + · λ − (cid:16) δ t , +0 · δ t0 + δ p , +0 · δ p0 + δ b , +0 · δ b0 (cid:17) − k X i =1 ( δ n , + i · δ n i + δ t , + i · δ t i + δ p , + i · δ p i ) − ψ ( g ) · ( δ n , + g/ · δ n g/ + δ p , + g/ · δ p g/ ) ̺ − null = λ − · λ − (cid:16) δ t , − · δ t0 + δ p , − · δ p0 + δ b , − · δ b0 (cid:17) − k X i =1 ( δ n , − i · δ n i + δ t , − i · δ t i + δ p , − i · δ p i ) − ψ ( g ) · ( δ n , − g/ · δ n g/ + δ p , − g/ · δ p g/ )and subsequently aim our efforts at determining the rational coefficients λ + , δ t , +0 , δ p , +0 , δ b , +0 , δ n , + i , δ t , + i , δ p , + i ∈ Q (resp. − )for 1 ≤ i ≤ ⌊ g/ ⌋ . To that end, the assortment of test curves introduced earlierwill prove to be most useful.According to definition 2.2, the essential distinction between P +null and P − null isthe parity of the theta characteristic θ ∈ S g ( C ), whereas its associated tensoredversion θ ⊗ η ∈ Θ null ( C ) is always even. This leads towards the question of howthe parity of a theta characteristic changes when it is tensored by a Prym root. Given a Prym pair (
C, η ) of genus g , we can consider a map S g ( C ) −→ S g ( C ) , θ θ ⊗ η and wonder how the parity of θ is affected by it. Definition 2.6.
Let (
C, η ) be a Prym pair of genus g . Consider the subsets S + , + η ( C ) = { θ ∈ S + g ( C ) / θ ⊗ η ∈ S + g ( C ) } ⊂ S + g ( C ) S + , − η ( C ) = { θ ∈ S + g ( C ) / θ ⊗ η ∈ S − g ( C ) } ⊂ S + g ( C ) S − , + η ( C ) = { θ ∈ S − g ( C ) / θ ⊗ η ∈ S + g ( C ) } ⊂ S − g ( C ) S − , − η ( C ) = { θ ∈ S − g ( C ) / θ ⊗ η ∈ S − g ( C ) } ⊂ S − g ( C ) .2 Parity change: smooth case S g ( C ) decomposes as a disjoint union. Remark 2.7.
Note that S + , − η ( C ) ∼ = S − , + η ( C ), θ θ ⊗ η . This leaves us withthree distinct sets that we want to study.For any smooth, integral, genus g curve C , the group J ( C ) acts on S g ( C ) bymeans of the map J ( C ) × S g ( C ) −→ S g ( C ) , ( η, θ ) θ ⊗ η whose associated difference map can be written asdiff : S g ( C ) × S g ( C ) −→ J ( C ) , ( θ , θ ) θ ⊗ θ − If we remove the diagonal ∆ = diff − ( O X ), we get a mapdiff = : S g ( C ) × S g ( C ) − ∆ −→ R g ( C )whose fibers, of order 2 g , reflect how many ways there are of writing a Prymroot η as a difference of theta characteristics θ ⊗ θ − , that is, with θ = θ ⊗ η .Since we aim to keep track of the parity of θ = θ and θ ⊗ η = θ , we just needto consider the restrictionsdiff + : S + g ( C ) × S + g ( C ) − ∆ −→ R g ( C )diff − : S − g ( C ) × S − g ( C ) − ∆ −→ R g ( C )diff ± : S + g ( C ) × S − g ( C ) −→ R g ( C )Recalling that R g ( C ) = 2 g − , S + g ( C ) = 2 g − (2 g + 1) S − g ( C ) = 2 g − (2 g − Lemma 2.8.
With the previous notation, it holds that − ( η ) = 2 g − (2 g − + 1) − − ( η ) = 2 g − (2 g − − − ± ( η ) = 2 g − for any Prym pair ( C, η ) of genus g . These numbers follow from the computation − ( η ) = S + g ( C ) · ( S + g ( C ) − R g ( C ) = 2 g − (2 g − + 1) − − ( η ) = S − g ( C ) · ( S − g ( C ) − R g ( C ) = 2 g − (2 g − − − ± ( η ) = S + g ( C ) · S − g ( C ) R g ( C ) = 2 g − which depends only on the genus g of the curve. Definition 2.9.
We denoteN + g = − ( η ) = 2 g − (2 g − + 1)N − g = − − ( η ) = 2 g − (2 g − − ± g = − ± ( η ) = 2 g − for any positive integer g ∈ Z + . Proposition 2.10.
Let ( C, η ) be a Prym pair of genus g . Under the map S g ( C ) → S g ( C ) , θ θ ⊗ η that is, when tensoring by η , there are: (i) N + g = 2 g − (2 g − + 1) even theta characteristics on C that remain even. (ii) N − g = 2 g − (2 g − − odd theta characteristics on C that remain odd. (iii) N ± g = 2 g − even theta characteristics on C that become odd. (iv) N ± g = 2 g − odd theta characteristics on C that become even.In particular, S g ( C ) = N + g + N − g + 2 N ± g = 2 g . Proof.
As suggested above, we have S + , + η ( C ) = { θ ∈ S + g ( C ) / θ ⊗ η ∈ S + g ( C ) } = { ( θ , θ ) ∈ S + g ( C ) × S + g ( C ) / θ = θ ⊗ η ∈ S + g ( C ) } = { ( θ , θ ) ∈ S + g ( C ) × S + g ( C ) − ∆ / θ ⊗ θ − = η } = − ( η ) = N + g and similarly S − , − η ( C ) = N − g and S + , − η ( C ) = S − , + η ( C ) = N ± g . .3 Parity change: irreducible nodal case Let (
B, p, q ) ∈ M g − , and take the irreducible nodal curve X = B pq ∈ M g obtained from B by gluing the points p and q , with normalization ν : B → B pq .The dualizing bundle ω X is the subbundle of ν ∗ ( ω B ( p + q )) fulfilling the residuecondition, that is, such that the following diagram commutes: ω X (cid:9) res p | | ②②②②②②②② res q " " ❊❊❊❊❊❊❊❊ κ ( p ) ∼ = − / / κ ( q )In this particular case, we actually have H ( X, ω X ) = H ( B, ω B ( p + q )), since h ( ω B ( p + q )) = 2 g − − ( g −
1) + 1 = g = h ( ω X )by Riemann-Roch and duality. As mentioned in example 1.12, a spin curve( X, θ, α ) ∈ S g , α : θ ⊗ ∼ = ω X , ( ν ∗ θ ) ⊗ ∼ = ω B ( p + q )is given by a root θ B ∈ p ω B ( p + q ) and a suitable gluing ϕ : θ B | p ∼ = θ B | q , whichby the above discussion is bound to a condition ϕ ⊗ ≡ − ω B ( p + q ) | p ϕ ⊗ ∼ = / / res p ∼ = (cid:15) (cid:15) (cid:9) ω B ( p + q ) | q res q ∼ = (cid:15) (cid:15) κ ( p ) ∼ = − / / κ ( q )Specifically, consider the canonical isomorphism ψ induced by the diagram θ B | pψ ∼ = (cid:15) (cid:15) & & ▼▼▼▼▼▼ / / θ B ( − p − q ) / / θ B (cid:8) ♣♣♣♣♣♣ & & ◆◆◆◆◆◆ θ B | q rrrrrr where θ B | p and θ B | q are expressed as cokernels of the same map. Let us give anexplicit description of ψ . On the one hand, Riemann-Roch and duality yield h ( θ B ) − h ( θ B ( − p − q )) = g − − ( g −
1) + 1 = 12 so we can write θ B | p = h σ ( p ) i and θ B | q = h σ ( q ) i for any section σ ∈ H ( B, θ B ) − H ( B, θ B ( − p − q ))and see that ψ is, by definition, the isomorphism ψ : θ B | p ∼ = θ B | q , σ ( p ) σ ( q )On the other hand, we have σ ⊗ ∈ H ( B, ω B ( p + q )) = H ( X, ω X ), henceres p ( σ ⊗ ) + res q ( σ ⊗ ) = 0and it is clear that ψ ⊗ ≡ −
1, in the sense of: ω B ( p + q ) | p ψ ⊗ ∼ = / / res p ∼ = (cid:15) (cid:15) (cid:9) ω B ( p + q ) | q res q ∼ = (cid:15) (cid:15) σ ⊗ ( p ) ✤ / / ❴ (cid:15) (cid:15) (cid:9) σ ⊗ ( q ) ❴ (cid:15) (cid:15) κ ( p ) ∼ = − / / κ ( q ) res p ( σ ⊗ ) ✤ / / res q ( σ ⊗ )If we also consider the opposite isomorphism − ψ : θ B | p ∼ = θ B | q , σ ( p )
7→ − σ ( q ) , ( − ψ ) ⊗ ≡ − ψ and − ψ are the only possible ways of gluing p and q to make θ B into asquare root of ω X . The resulting bundles on X , which we denote by( θ B , ψ ) , ( θ B , − ψ ) ∈ p ω X are thus the two elements in the fiber ( ν ∗ ) − ( θ B ) of the double cover ν ∗ : p ω X −→ p ω B ( p + q )Furthermore, observe that the 1-dimensional space of sections h σ i ⊂ H ( B, θ B )is preserved under the gluing ψ , but lost under the gluing − ψ . As a result, thedimension of the glued global sections is given by h ( X, ( θ B , ψ )) = h ( B, θ B ) h ( X, ( θ B , − ψ )) = h ( B, θ B ) − θ B , ψ ) and ( θ B , − ψ ) always have different parity.Finally, let η t be the single Prym root of X lying in the divisor ∆ t0 ⊂ R g , asdefined in example 1.9. In other words, we have η t = O X , ν ∗ η t = O B η t permutes the elements of ( ν ∗ ) − ( θ B ), since( θ B , ψ ) ⊗ η t = ( θ B , ψ ) , ν ∗ (( θ B , ψ ) ⊗ η t ) = θ B and similarly for ( θ B , − ψ ). This corresponds to a change in parity: Proposition 2.11.
With the notation above, let ( X, θ, α ) ∈ S + g be a generalpoint of ∆ n0 . Then tensoring by ( X, η t , β ) ∈ ∆ t0 ⊂ R g induces a new spin curve ( X, θ ⊗ η t , α ⊗ β ) in ∆ n0 ⊂ S − g , of opposite parity (resp. S − g , S + g ). In this section, we use test curve techniques to determine all the coefficientsin the rational expansions of the Prym-null classes.
Recall the test curves from example 1.15, that is F i ≡ { ( C ∪ y ∼ q D, ( η C , O D )) } y ∈ C ⊂ ∆ n i ⊂ R g G i ≡ { ( C ∪ y ∼ q D, ( O C , η D )) } y ∈ C ⊂ ∆ t i ⊂ R g H i ≡ { ( C ∪ y ∼ q D, ( η C , η D )) } y ∈ C ⊂ ∆ p i ⊂ R g with C ∈ M i , ( D, q ) ∈ M g − i, general,and η C ∈ R i ( C ) , η D ∈ R g − i ( D ) arbitrary.If g = 2 i , their intersection table is: λ δ t0 δ p0 δ b0 δ n i δ t i δ p i δ ( j = i ) F i − i G i − i H i − i g = 2 i , we have F i · δ n i = G i · δ n i = 2 − i instead.We want to determine F i ∩ P +null . Remark 3.1.
If a stable Prym curve F i, y = ( C ∪ y ∼ q D, ( η C , O D )) ∈ F i lies in P +null , then it can be expressed as the limit of a smooth family in P +null , inthe following sense. First, let us write( X y , η y ) = ( C ∪ y ∼ E ∪ q ∼∞ D, ( η C , O E , O D ))st( X y , η y ) = ( C ∪ y ∼ q D, ( η C , O D )) = F i, y ∈ P +null to account for the exceptional component E ∼ = P which appears when workingwith stable spin structures on C ∪ y ∼ q D . Then there exist families f : X →
Spec( R ) = { ξ, p } of quasistable curves,(st( f ) , η, β ) ∈ R g of stable Prym curves, and( f, θ, α ) ∈ S + g of stable (even) spin curves,such that:(i) X is a smooth surface.(ii) R is a discrete valuation ring with maximal ideal m , whose spectrum iscomposed of a special point p ≡ m and a generic point ξ ≡ (0).(iii) On the special fiber X = f − ( p ), it holds that( X , η | X ) = ( X y , η y ) , st( X , η | X ) = F i, y ∈ P +null (iv) On the generic fiber X ξ = f − ( ξ ) = st( f ) − ( ξ ), it holds that( X ξ , η ξ ) ∈ P +null , ( X ξ , θ ξ ⊗ η ξ ) ∈ Θ null or equivalently ( θ ξ ⊗ η ξ ) ⊗ ≃ ω X ξ and h ( X ξ , θ ξ ⊗ η ξ ) ≥ ≡ θ | X = θ + y = ( θ C , O E (1) , θ D ) ∈ S + g ( X y )then it is clear that θ C and θ D must have the same parity. In addition, since C and D are general, the dimension of the global sections of θ C and θ D is at mostone, and thus we get h ( C, θ C ) = h ( D, θ D ) ∈ { , } Observe that if F i, y were to lie in P − null , then θ | X = θ − y would be odd instead ofeven and these theta characteristics would have opposite parity.As described in [EH86] Section 2, the data given in remark 3.1 produces a .1 Over reducible nodal curves g g − on C ∪ y ∼ q D , namely ℓ = (cid:16) ℓ C = ( L C , V C ) , ℓ D = ( L D , V D ) (cid:17) ∈ G g − ( C ) × G g − ( D )where the line bundles L C and L D are obtained by looking at the equality θ | X ⊗ η | X = θ + y ⊗ η y = ( θ C ⊗ η C , O E (1) , θ D )and twisting with y and q to adjust the degrees to g −
1, so that ( L C = θ C ⊗ η C (( g − i ) y ) L D = θ D ( iq )Since θ ξ ⊗ η ξ ∈ Θ null ( X ξ ) is even and parity is constant in families, we get h ( θ C ⊗ η C ) + h ( θ D ) = h ( θ + y ⊗ η y ) ≡ h ( θ ξ ⊗ η ξ ) ≡ θ C ⊗ η C and θ D must have the same parity, and the dimension oftheir global sections is again either 0 or 1 due to generality. This results in twodistinct possibilities for the P +null setting, and two more for the P − null one: h ( θ C ) h ( θ D ) h ( θ C ⊗ η C ) P +null (F , + , (F , + , P − null (F , − , (F , − , Remark 3.2.
Given a linear series (
L, V ) ∈ G rd ( C ) on a smooth curve C and apoint p ∈ C , we can find an ordered basis V = h s , . . . , s r i ⊂ H ( C, L )such that, if we write a i ( p ) = ord p ( s i ) for all i ∈ { , . . . , r } , then a ( p ) < . . . < a r ( p )is the vanishing sequence of ( L, V ) at p . Taking any b ∈ Z + and observing that V ( − bp ) is the subspace of sections s ∈ V such that ord p ( s ) ≥ b , we can extract6 a basis of V ( − bp ) out of h s , . . . , s r i , namely V ( − bp ) = h s j , . . . , s r i ⊂ H ( C, L ( − bp ))where the index j ∈ { , . . . , r + 1 } is determined by the inequalities a j ( p ) = ord p ( s j ) ≥ b, a j − ( p ) = ord p ( s j − ) < b whenever they make sense. Finally, the fact that there are ( r + 1) − j elementsin such a basis leads to the useful relationdim V ( − bp ) = r + 1 − j ⇔ a j − ( p ) < b ≤ a j ( p )which we will systematically use in the subsequent discussion. For example, wecan apply it to L C = θ C ⊗ η C (( g − i ) y ) and deduce that h ( θ C ⊗ η C ) = h ( L C ( − ( g − i ) y ) = 2 − j ⇔ a ℓ C j − ( y ) < g − i ≤ a ℓ C j ( y )with j ∈ { , } depending on the parity of θ C ⊗ η C .Let us start by analysing the two possibilities related to the even Prym-nulldivisor, labelled as in the table above. Possibility (F , + , . In this case, we have h ( θ C ⊗ η C ) = 0 ⇒ a ℓ C ( y ) < a ℓ C ( y ) < g − i ⇒ a ℓ C ( y ) ≤ g − i − h ( θ D ) = 0 ⇒ a ℓ D ( q ) < i ⇒ a ℓ D ( q ) ≤ i − g g − compatibility conditions g − ≤ a ℓ C ( y ) + a ℓ D ( q ) ≤ g − Possibility (F , + , . In this case, we have h ( θ C ⊗ η C ) = 1 ⇒ a ℓ C ( y ) < g − i ≤ a ℓ C ( y ) h ( θ D ) = 1 ⇒ a ℓ D ( q ) < i ≤ a ℓ D ( q )On the one hand, ( D, q ) is general, so we may assume that q / ∈ supp( θ D ). Then .1 Over reducible nodal curves ℓ D at q can be bounded further: h ( θ D ( − q )) = h ( θ D ) − ⇒ a ℓ D ( q ) < i + 1 h ( θ D ( q )) = h ( θ D ) = 1 ⇒ a ℓ D ( q ) < i − ≤ a ℓ D ( q )We thus get a ℓ D ( q ) = i and, by the limit g g − condition, a ℓ C ( y ) = g − i −
1. Onthe other hand, C is general, so supp( θ C ⊗ η C ) consists of i − a ℓ C ( y ), namelydiv( θ C ⊗ η C ) (cid:3) y ⇒ div( s ) (cid:3) ( g − i + 2) y ∀ s ∈ H ( L C ) ⇒ a ℓ C ( y ) ≤ g − i + 1which together with the condition a ℓ D ( q ) + a ℓ C ( y ) ≥ g − a ℓ D ( q ) = i − a ℓ C ( y ) = g − i + 1. In turn, this means that y ∈ supp( θ C ⊗ η C ), and that ℓ is a refined limit g g − of the form ( ℓ C = | θ C ⊗ η C ( y ) | + ( g − i − y ∈ G g − ( C ) ℓ D = | θ D (2 q ) | + ( i − q ∈ G g − ( D )with vanishing sequences ( g − i − , g − i + 1) and ( i − , i ).In conclusion, for each pair of theta characteristics θ C , θ D fulfilling (F , + , θ C ∈ S − i ( C ), θ D ∈ S − g − i ( D ) and θ C ⊗ η C ∈ S − i ( C ), then every y ∈ supp( θ C ⊗ η C ) yields a limit g g − as above, and these limit linear series arethe only ones contributing to the intersection F i ∩ P +null . Consequently, we needto count such pairs of theta characteristics.Fortunately, we already have all the necessary tools to do this. Lemma 3.3.
For all i ∈ { , . . . , g − } , it holds that F i · P +null = 2 g − (2 i − − g − i − i − Proof.
In light of the previous considerations, we may split the count into threeparts. Specifically, we want to compute the number of:(i)
Theta characteristics θ C ∈ S − i ( C ) such that θ C ⊗ η C ∈ S − i ( C ) . According to proposition 2.10, this is N − i = 2 i − (2 i − − Theta characteristics θ D ∈ S − g − i ( D ) . This is S − g − i ( D ) = 2 g − i − (2 g − i − Once θ C is fixed, points y in the support of θ C ⊗ η C . Since θ C ⊗ η C ∈ S i ( C ), there are deg( θ C ⊗ η C ) = i − Altogether, we obtain F i · P +null = ( ( θ C , θ D , y ) ∈ S − i ( C ) × S − g − i ( D ) × C /θ C ⊗ η C ∈ S − i ( C ) , y ∈ supp( θ C ⊗ η C ) ) = N − i · S − g − i ( D ) · ( i − i − (2 i − − · g − i − (2 g − i − · ( i − g − (2 i − − g − i − i − F i ∩ P − null we follow the same argument, with the onlydifference being that θ C and θ D now have opposite parity (remark 3.1). Sincethis brings about minimal variations, we merely outline the situation and carryout the corresponding count. There are again two possibilities to tackle. Possibility (F , − , . Similar contradiction to that of (F , + , Possibility (F , − , . As with its even counterpart, we are able to build a limitlinear series contributing to F i · P − null whenever y ∈ supp( θ C ⊗ η C ). In this case,however, we have θ C ∈ S + i ( C ).We thus get F i · P − null = ( ( θ C , θ D , y ) ∈ S + i ( C ) × S − g − i ( D ) × C /θ C ⊗ η C ∈ S − i ( C ) , y ∈ supp( θ C ⊗ η C ) ) = N ± i · S − g − i ( D ) · ( i − i − · g − i − (2 g − i − · ( i − g + i − (2 g − i − i − F i and each ofthe Prym-null divisors works with G i and H i as well. Nonetheless, we still needto carefully track the small changes that happen along the way.Let us briefly do this. If a stable Prym curve G i, y = ( C ∪ y ∼ q D, ( O C , η D )) ∈ G i lies in P +null (resp. P − null ), we can produce a limit g g − on C ∪ y ∼ q D such that ( L C = θ C (( g − i ) y ) L D = θ D ⊗ η D ( iq ) .1 Over reducible nodal curves h ( θ C ) + h ( θ D ⊗ η D ) ≡ θ C and θ D have the same parity(resp. opposite parity). Then θ C and θ D ⊗ η D have the same parity and we getthe following possibilities: h ( θ C ) h ( θ D ) h ( θ D ⊗ η D ) P +null (G , + ,
0) : contradiction1 1 1 (G , + ,
1) : y ∈ supp( θ C ) P − null (G , − ,
0) : contradiction1 0 1 (G , − ,
1) : y ∈ supp( θ C )With the only contribution of (G , + ,
1) and (G , − ,
1) to their respective in-tersections, we obtain G i · P +null = ( ( θ C , θ D , y ) ∈ S − i ( C ) × S − g − i ( D ) × C /θ D ⊗ η D ∈ S − g − i ( D ) , y ∈ supp( θ C ) ) = S − i ( C ) · N − g − i · ( i − i − (2 i − · g − i − (2 g − i − − · ( i − g − (2 i − g − i − − i − G i · P − null = ( ( θ C , θ D , y ) ∈ S − i ( C ) × S + g − i ( D ) × C /θ D ⊗ η D ∈ S − g − i ( D ) , y ∈ supp( θ C ) ) = S − i ( C ) · N ± g − i · ( i − i − (2 i − · g − i − · ( i − g − i − (2 i − i − Remark 3.4.
The contradiction in (G , + ,
0) and (G , − ,
0) is again g − ≤ a ℓ C ( y ) + a ℓ D ( q ) ≤ g − g g − on our reducible nodal curve, so inthe future we will refrain from detailing it any further.Finally, if a stable Prym curve H i, y = ( C ∪ y ∼ q D, ( η C , η D )) ∈ H i lies in P +null (resp. P − null ), we can produce a limit g g − on C ∪ y ∼ q D such that ( L C = θ C ⊗ η C (( g − i ) y ) L D = θ D ⊗ η D ( iq )with h ( θ C ⊗ η C ) + h ( θ D ⊗ η D ) ≡ θ C and θ D have the sameparity (resp. opposite parity). Then θ C ⊗ η C and θ D ⊗ η D have the same parityand we get the following possibilities: h ( θ C ) h ( θ D ) h ( θ C ⊗ η C ) h ( θ D ⊗ η D ) P +null contradiction1 1 0 0 contradiction0 0 1 1 y ∈ supp( θ C ⊗ η C )1 1 1 1 y ∈ supp( θ C ⊗ η C ) P − null contradiction1 0 0 0 contradiction0 1 1 1 y ∈ supp( θ C ⊗ η C )1 0 1 1 y ∈ supp( θ C ⊗ η C )The count has now grown in complexity, but not by much. We have H i · P +null = ( θ C , θ D , y ) ∈ ( S + i ( C ) × S + g − i ( D ) × C ) ∪∪ ( S − i ( C ) × S − g − i ( D ) × C ) /θ C ⊗ η C ∈ S − i ( C ) , θ D ⊗ η D ∈ S − g − i ( D ) ,y ∈ supp( θ C ⊗ η C ) = (N ± i N ± g − i + N − i N − g − i ) · ( i − i − · g − i − + 2 i − (2 i − − · g − i − (2 g − i − − · ( i − g − (2 g − − i − − g − i − + 1)( i − H i · P − null = ( θ C , θ D , y ) ∈ ( S + i ( C ) × S − g − i ( D ) × C ) ∪∪ ( S − i ( C ) × S + g − i ( D ) × C ) /θ C ⊗ η C ∈ S − i ( C ) , θ D ⊗ η D ∈ S − g − i ( D ) ,y ∈ supp( θ C ⊗ η C ) = (N ± i N − g − i + N − i N ± g − i ) · ( i − i − · g − i − (2 g − i − −
1) + 2 i − (2 i − − · g − i − ) · ( i − g − (2 g − − i − − g − i − )( i − .1 Over reducible nodal curves Lemma 3.5.
For all i ∈ { , . . . , g − } , we have intersection numbers P +null P − null F i (2 i − − g − i − r i − (2 g − i − rG i (2 i − g − i − − r (2 i −
1) 2 g − i − rH i (2 g − − i − − g − i − + 1) r (2 g − − i − − g − i − ) r where r = 2 g − ( i −
1) = − g − (2 − i ) . Remark 3.6.
A quick computation shows that these numbers pass the checksuggested by remark 2.4. Indeed, we can easily see that F i · P +null + F i · P − null = 2 g − (2 i − g − i − i −
1) = F i · P null G i · P +null + G i · P − null = 2 g − (2 i − g − i − i −
1) = G i · P null H i · P +null + H i · P − null = 2 g − (2 i − g − i − i −
1) = H i · P null where we have used example 1.15 and proposition 2.3.If we combine lemma 3.5 with the intersection table of example 1.15, we canderive the first batch of coefficients from the formulas in remark 2.5. Proposition 3.7.
Fix integers g ≥ and i ∈ { , . . . , ⌊ g/ ⌋} . The generatingclasses δ n i , δ t i , δ p i ∈ Pic( R g ) Q have coefficients δ n , + i = 2 g − (2 i − − g − i − ̺ +null δ t , + i = 2 g − (2 i − g − i − − δ p , + i = 2 g − (2 g − − i − − g − i − + 1) δ n , − i = 2 g − i − (2 g − i − ̺ − null δ t , − i = 2 g − (2 i −
1) 2 g − i − δ p , − i = 2 g − (2 g − − i − − g − i − ) in the rational expansions of the Prym-null classes in genus g . Proof.
Every family of test curves generates a linear relation between the coeffi-cients of each expansion. Due to the simplicity of the intersection tables of F i , G i and H i with the generators of Pic( R g ) Q , their corresponding linear relationsdirectly determine one coefficient (sometimes the same one, if two relations arelinearly dependent). For the sake of brevity, we shall describe this computation2 simply in the F i case, as all others are analogous. To begin with, we havedeg ̺ +null ( F i ) = F i · P +null = − g − (2 i − − g − i − − i )by lemma 3.5. Furthermore, remark 2.5 and example 1.15 show thatdeg ̺ +null ( F i ) = − δ n , + i deg δ n i ( F i ) = − δ n , + i (2 − i )with the convention δ n , + i = δ t , + g − i for all i . Since only one coefficient survives, wecan immediately extract it from the resulting equation: F i δ n , + i = 2 g − (2 i − − g − i − ≤ i ≤ ⌊ g/ ⌋ , each coefficient can be similarly computed by means of: F i or G g − i ( δ n , + i = 2 g − (2 i − − g − i − δ n , − i = 2 g − i − (2 g − i − G i or F g − i ( δ t , + i = 2 g − (2 i − g − i − − δ t , − i = 2 g − (2 i −
1) 2 g − i − H i or H g − i ( δ p , + i = 2 g − (2 g − − i − − g − i − + 1) δ p , − i = 2 g − (2 g − − i − − g − i − )For i = 1, we do not have families F , G and H . Nevertheless, we can use: G g − ( δ n , +1 = 0 δ n , − = 2 g − (2 g − − F g − ( δ t , +1 = 2 g − (2 g − − δ t , − = 2 g − g − H g − ( δ p , +1 = 2 g − g − δ p , − = 2 g − (2 g − − i = 1 as well. Remark 3.8.
Observe that δ n , +1 = 0 is a consequence of the fact that genus 1curves do not have nontrivial odd theta characteristics, hence N − = 0. .2 Over curves with elliptic tails Recall the test curves from example 1.16, that is F ≡ { ( C ∪ p ∼ σ ( λ ) E λ , ( η C , O E λ )) } λ ∈ P ⊂ ∆ t1 ⊂ R g G ≡ { ( C ∪ p ∼ σ ( λ ) E λ , ( O C , η E λ )) / η E λ ∈ γ − ( E λ ) } λ ∈ P ⊂ ∆ n1 ⊂ R g H ≡ { ( C ∪ p ∼ σ ( λ ) E λ , ( η C , η E λ )) / η E λ ∈ γ − ( E λ ) } λ ∈ P ⊂ ∆ p1 ⊂ R g with ( C, p ) ∈ M g − , general, { E λ } λ ∈ P ⊂ M general pencil of plane cubics with basepoint σ , η C ∈ R g − ( C ) arbitrary,and γ : R , → M , forgetful degree 3 branched covering.Our goal is to expand their intersection table to include the Prym-null divisors,as we did in lemma 3.5 for the previous families of test curves. In this case, itturns out that the new intersection numbers are much less imposing: λ δ t0 δ p0 δ b0 δ n1 δ t1 δ p1 δ ( j ≥ P +null P − null F − G − H − F ∩ P +null (resp. P − null ). If a stable Prym curve F , λ = ( C ∪ p ∼ e E λ , ( η C , O E λ )) ∈ F lies in P +null (resp. P − null ), we can produce a limit g g − on C ∪ p ∼ e E λ such that ( L C = θ C ⊗ η C ( p ) L E λ = θ E λ (( g − e )with h ( θ C ⊗ η C ) + h ( θ E λ ) ≡ θ C and θ E λ have the same parity(resp. opposite parity). Then θ C ⊗ η C and θ E λ have the same parity and we getthe following possibilities: h ( θ C ) h ( θ E λ ) h ( θ C ⊗ η C ) P +null contradiction1 1 1 (F , + , P − null contradiction0 1 1 (F , − , Note that all four theta characteristics of a genus 1 curve are of degree zero,hence the nontrivial ones have no global sections other than zero. In particular,the dimension of the global sections of θ E λ is given by h ( θ E λ ) = ( θ E λ = O E λ λ ∈ P , meaning that the table above is comprehensive. Since half of thescenarios are already covered by remark 3.4, we just need to look at (F , + , , − ,
1) to conclude. This can be done in one fell swoop.
Possibilities (F , + , , − , . In both of these cases, we have h ( θ C ⊗ η C ) = 1 ⇒ a ℓ C ( p ) < ≤ a ℓ C ( p ) h ( θ E λ ) = 1 ⇒ a ℓ Eλ ( e ) < g − ≤ a ℓ Eλ ( e )Now ( C, p ) is general, so we may assume that p / ∈ supp( θ C ⊗ η C ). Therefore h ( θ C ⊗ η C ( p )) = h ( θ C ⊗ η C ) = 1 ⇒ a ℓ C ( p ) < ≤ a ℓ C ( p )Furthermore, E λ is a genus 1 curve, so deg( θ E λ ) = 0. As a result, we get h ( θ E λ ( − e )) = 0 ⇒ a ℓ Eλ ( e ) < g that is, a ℓ Eλ ( e ) = g −
1, which contradicts the limit g g − condition g − ≤ a ℓ C ( p ) + a ℓ Eλ ( e ) < g − p / ∈ supp( θ C ⊗ η C ) we may also deduce h ( θ C ⊗ η C ( − p )) = h ( θ C ⊗ η C ) − ⇒ a ℓ C ( p ) < a ℓ C ( p ) = 1 and, via the limit g g − condition, a ℓ Eλ ( e ) = g −
2. Thisshould prevent θ E λ from being trivial, due to the implication h ( θ E λ ( e )) = h ( O E λ ( e )) = 1 ⇒ a ℓ Eλ ( e ) < g − ≤ a ℓ Eλ ( e )However, the triviality of θ E λ is ensured by h ( θ E λ ) = 1 and deg( θ E λ ) = 0.)As every possibility leads to a contradiction, the intersections F ∩ P +null and F ∩ P − null are both empty, and thus F · P +null = F · P − null = 0.Next we study G ∩ P +null (resp. P − null ). If a stable Prym curve G , λ = ( C ∪ p ∼ e E λ , ( O C , η E λ )) ∈ G .2 Over curves with elliptic tails P +null (resp. P − null ), we can produce a limit g g − on C ∪ p ∼ e E λ such that ( L C = θ C ( p ) L E λ = θ E λ ⊗ η E λ (( g − e )with h ( θ C ) + h ( θ E λ ⊗ η E λ ) ≡ θ C and θ E λ have the same parity(resp. opposite parity). Then θ C and θ E λ ⊗ η E λ have the same parity and we getthe following possibilities: h ( θ C ) h ( θ E λ ) h ( θ E λ ⊗ η E λ ) P +null contradiction1 1 1 (G , + , P − null contradiction1 0 1 (G , − , F counterpart. Possibility (G , + , . We may repeat (F , + , E λ is a genus 1 curve, it holds that h ( θ E λ ) = h ( θ E λ ⊗ η E λ ) = 1deg( θ E λ ) = deg( θ E λ ⊗ η E λ ) = 0 ) ⇒ ( θ E λ = θ E λ ⊗ η E λ = O E λ ⇒⇒ η E λ = O E λ (!!)in direct contradiction with the nontriviality of a Prym root. Possibility (G , − , . Same contradiction as in (F , − , G · P +null = G · P − null = 0.Finally, let us consider H ∩ P +null (resp. P − null ). If a stable Prym curve H , λ = ( C ∪ p ∼ e E λ , ( η C , η E λ )) ∈ H lies in P +null (resp. P − null ), we can produce a limit g g − on C ∪ p ∼ e E λ such that ( L C = θ C ⊗ η C ( p ) L E λ = θ E λ ⊗ η E λ (( g − e )with h ( θ C ⊗ η C ) + h ( θ E λ ⊗ η E λ ) ≡ θ C and θ E λ have the sameparity (resp. opposite parity). Then θ C ⊗ η C and θ E λ ⊗ η E λ have the same parity6 and we get the following possibilities: h ( θ C ) h ( θ E λ ) h ( θ C ⊗ η C ) h ( θ E λ ⊗ η E λ ) P +null contradiction1 1 0 0 contradiction0 0 1 1 (H , + , , (H , + , , P − null contradiction1 0 0 0 contradiction0 1 1 1 (H , − , , (H , − , , Possibilities (H , + , , , − , , . Same contradiction as in (F , + , Possibilities (H , + , , , − , , . Same contradiction as in (G , + , H · P +null = H · P − null = 0 as well. Remark 3.9.
The above computations, in combination with example 1.16 andproposition 2.3, show that F · P +null + F · P − null = 0 = F · P null (resp. G , H ), as expected by remark 2.4. Lemma 3.10.
In the setting of proposition 3.7, the families F , G and H provide three linearly independent linear relations F ( λ + − δ p , +0 = − g − (2 g − − λ − − δ p , − = − g − G ( λ + − δ t , +0 − δ b , +0 = 0 λ − − δ t , − − δ b , − = − g − (2 g − − H ( λ + − δ p , +0 − δ b , +0 = − g − λ + − δ p , − − δ b , − = − g − (2 g − − between the coefficients of λ, δ t0 , δ p0 , δ b0 ∈ Pic( R g ) Q in each expansion. Proof.
Follows from combining proposition 3.7 with the intersection of F andthe Prym-null divisors (resp. G , H ). .3 Over irreducible nodal curves Recall the test curve from example 1.17, that is Y ≡ { ( B py , η t y ) / η t y ∈ ∆ t0 ( B py ) } y ∈ B ⊂ ∆ t0 ⊂ R g with ( B, p ) ∈ M g − , general, B py = B/ { y ∼ p } irreducible nodal curve for y = p ,and B pp copy of B with a pigtail attached to p .As we will see below, its extended intersection table is: λ δ t0 δ p0 δ b0 δ n1 δ t1 δ p1 δ ( j ≥ P +null P − null Y − g g − (2 g − ( g −
3) + 1)The reason that the Prym-null intersection gravitates entirely towards the oddside is the parity change explored in proposition 2.11. With the help of both thisresult and the notation used to prove it, we can easily determine Y ∩ P +null and Y ∩ P − null .On the one hand, if a stable Prym curve Y , y = ( B py , η t y ) ∈ Y lies in the even Prym-null divisor P +null , then by definition there exists a stablevanishing theta-null θ y ∈ Θ null ⊂ S + g over B py such that θ y ⊗ η t y ∈ S + g is even.Since proposition 2.11 shows that θ y and θ y ⊗ η t y always have opposite parity, nosuch vanishing theta-null can exist, and therefore Y · P +null = 0.On the other hand, if a stable Prym curve Y , y = ( B py , η t y ) ∈ Y lies in the odd Prym-null divisor P − null instead, then by definition there exists astable vanishing theta-null θ y ∈ Θ null ⊂ S + g over B py such that θ y ⊗ η t y ∈ S − g isodd. Proposition 2.11 now makes redundant the second part of the condition, sowe just need to count how many B py admit a vanishing theta-null. This can bedone by taking advantage of the formula for the theta-null class ϑ null = 14 λ − δ n0 − ⌊ g/ ⌋ X i =1 δ − i introduced in section 2 and originally given by [Far10]. If we consider the test8 curve Y n0 obtained as the pullback of { B py } y ∈ B by the divisor ∆ n0 ⊂ S + g , i.e. Y n0 ≡ { ( B py , θ y ) / θ y ∈ ∆ n0 ( B py ) } y ∈ B ⊂ ∆ n0 ⊂ S + g then the previous discussion identifies the intersection Y ∩ P − null with the inter-section Y n0 ∩ Θ null . The latter can easily be derived from the theta-null formulaand the intersection table of Y n0 with the generators of Pic( S + g ) Q , which is: λ δ n0 δ b0 δ +1 δ − δ ( j ≥ Y n0 g − (1 − g ) 0 2 g − (2 g − + 1) 2 g − (2 g − −
1) 0Indeed, example 1.17 and deg(∆ n0 | ∆ ) = 2 g − yield all coefficients except for the δ +1 , δ − ones, while over the special point ( B pp , θ p ) we can see that θ p = ( θ B , ( O P , ϕ )) ∈ ∆ n0 ( B pp )for some θ B ∈ S g − ( B ) and ϕ ∈ { ψ, − ψ } such that h ( θ B ) ≡ h ( O P , ϕ ) mod 2.Since by construction of ψ and − ψ we have h ( O P , ϕ ) = ( ⇔ ϕ = ψ ⇔ ϕ = − ψ it follows that ϕ is determined by the parity of θ B and thus Y n0 · δ +1 = S + g − ( B ) = 2 g − (2 g − + 1) Y n0 · δ − = S − g − ( B ) = 2 g − (2 g − − Y · P − null = Y n0 · Θ null = − − Y n0 · δ n0 − − Y n0 · δ − = − g − (1 − g ) − g − (2 g − − g − (2 g − ( g −
3) + 1)as indicated earlier.
Proposition 3.11.
In the setting of proposition 3.7, the generating classes λ, δ t0 , δ p0 , δ b0 ∈ .4 Class expansion and application to other families R g ) Q have coefficients ̺ +null λ + = 2 g − (2 g − + 1) = 2 g − (2 g − + 1) δ t , +0 = 0 = 0 δ p , +0 = 2 g − = 2 g − − g − δ b , +0 = 2 g − (2 g − + 1) = 2 g − − (2 g − + 1) ̺ − null λ − = 2 g − = 2 g − g − δ t , − = 2 g − = 2 g − − g − δ p , − = 2 g − = 2 g − − g − δ b , − = 2 g − (2 g − −
1) = 2 g − − (2 g − − in the rational expansions of the Prym-null classes in genus g . Proof.
Since the δ n1 coefficients have already been computed (proposition 3.7), itis straightforward to check that the linear relation provided by the family Y ineach case directly determines the corresponding δ t0 coefficient: Y ( δ t , +0 = 0 δ t , − = 2 g − Plugging these into lemma 3.10, we obtain the following linear systems: −
12 01 0 − − − · λ + δ p , +0 δ b , +0 = − g − (2 g − − − g − −
12 01 0 − − − · λ − δ p , − δ b , − = − g − g − − g − (2 g − − The two sets of solutions are precisely the expressions stated above.
For the first time, all of the rational coefficients introduced in remark 2.5 areknown to us, by virtue of propositions 3.7 and 3.11. As a result, we are finallyin a position to express the rational classes of P +null and P − null in terms of thegenerating classes of Pic( R g ) Q , which was our main goal.0 Theorem 3.12.
For g ≥ , the classes ̺ +null , ̺ − null ∈ Pic( R g ) Q are given by ̺ +null = 2 g − (cid:18) (2 g − + 1) λ − (cid:16) g − δ p0 + (2 g − + 1) δ b0 (cid:17) − k X i =1 (cid:16) (2 i − − g − i − δ n i + (2 i − g − i − − δ t i ++ (2 g − − i − − g − i − + 1) δ p i (cid:17) − ψ ( g ) · (cid:16) (2 g/ − − g/ − δ n g/ + (2 g − − g/ + 1) δ p g/ (cid:17)(cid:19) ̺ − null = 2 g − (cid:18) g − λ − (cid:16) g − δ t0 + 2 g − δ p0 + (2 g − − δ b0 (cid:17) − k X i =1 (cid:16) i − (2 g − i − δ n i + (2 i −
1) 2 g − i − δ t i ++ (2 g − − i − − g − i − ) δ p i (cid:17) − ψ ( g ) · (cid:16) g/ − (2 g/ − δ n g/ + (2 g − − g/ ) δ p g/ (cid:17)(cid:19) where the upper bound k and the parity-checking function ψ ( g ) , defined as k = ⌈ g/ ⌉ − ( ⌊ g/ ⌋ if g odd ⌊ g/ ⌋ − g even ψ ( g ) = 1 + ( − g ( g odd1 if g even account for the slight variation that occurs when g = 2 i . Example 3.13 (quartic tails) . We fix a general curve (
C, p ) ∈ M g − , and ageneral pencil γ : Bl ( P ) → P of plane quartics, with fibers { Q λ = γ − ( λ ) } λ ∈ P ⊂ M together with a section ζ : P → Bl ( P ) induced by one of the basepoints. Wemay then glue the curve ( C, p ) to the pencil γ along ζ , thus producing a pencilof stable curves Q = ( C × P ) ∪ { p }× P ∼ ζ ( P ) Bl ( P ) −→ P .4 Class expansion and application to other families Q ≡ { C ∪ p ∼ ζ ( λ ) Q λ } λ ∈ P ⊂ ∆ ⊂ M g Standard techniques show its intersection table to be: λ δ δ δ δ δ ( j ≥ Q − η C ∈ R g − ( C ) and lift Q to a test curve R , as follows: R ≡ { ( C ∪ p ∼ ζ ( λ ) Q λ , ( η C , O Q λ )) } λ ∈ P ⊂ ∆ t3 ⊂ R g Observe that π ∗ ( R ) = Q . Then R · λ = Q · λ = 3 and R · δ t3 = Q · δ = − λ ∞ ∈ P corresponding to singular quartics of γ andblow up the node of the component Q λ ∞ ∈ ∆ , we can see that the pullback of η λ ∞ = ( η C , O Q λ ∞ ) is ( η C , O P ), which is nontrivial. Hence R λ ∞ ∈ ∆ p0 and R · δ p0 = Q · δ = 27All other intersection numbers are 0, so we get a table: λ δ t0 δ p0 δ b0 δ n3 δ t3 δ p3 δ ( j =0 , R − R and the family F from example 1.16. Remark 3.14.
Applying theorem 3.12 to example 3.13, we get R · P +null = 3 λ + − δ p , +0 + δ t , +3 = 2 g − (2 g − − R · P − null = 3 λ − − δ p , − + δ t , − = 2 g − These intersection numbers may in fact be interpreted, as the limit linear seriestechniques introduced in earlier cases are also quite useful here.Again, we want to consider R ∩ P +null (resp. P − null ). If a stable Prym curve R λ = ( C ∪ p ∼ z Q λ , ( η C , O Q λ )) ∈ R lies in P +null (resp. P − null ), we can produce a limit g g − on C ∪ p ∼ z Q λ such that ( L C = θ C ⊗ η C (3 p ) L Q λ = θ Q λ (( g − z )with h ( θ C ⊗ η C ) + h ( θ Q λ ) ≡ θ C and θ Q λ have the same parity(resp. opposite parity). Then θ C ⊗ η C and θ Q λ have the same parity and we getthe following possibilities: h ( θ C ) h ( θ Q λ ) h ( θ C ⊗ η C ) P +null contradiction1 1 1 (R , + , P − null contradiction0 1 1 (R , − , X ∈ M , its canonical embeddingrealises it as a plane quartic Q ֒ → P , with the canonical series manifesting asthe restriction of the hyperplane series to the curve.Take a theta characteristic θ on X , with θ ⊗ ∼ = ω X ∈ W ( X ). Then we havedeg( θ ) = 2, and θ is of type g r on X whenever h ( θ ) = r + 1 >
0. But X is nothyperelliptic, so it does not admit any g and thus r ≤ ⇒ h ( θ ) = r + 1 ≤ X have h ( θ ) = 0 and the 28 oddones have h ( θ ) = 1. In particular, X has no vanishing theta-nulls.Let θ be odd. Then | θ | = { D } with 2 D ∼ K X , that is, D = x + y and2 D = 2 x + 2 y = H ∩ Q for some hyperplane H ֒ → P . If moreover x = y (for example, for X general),we get a one-to-one correspondence between odd theta characteristics of X andbitangents to its canonical model Q . Note that, if X is special enough for Q tohave any hyperflexes, then the tangent lines at such points must be included inthe correspondence too. Possibilities (R , + , , − , . In both of these cases, we have h ( θ C ⊗ η C ) = 1 ⇒ a ℓ C ( p ) < ≤ a ℓ C ( p ) h ( θ Q λ ) = 1 ⇒ a ℓ Qλ ( z ) < g − ≤ a ℓ Qλ ( z ) .4 Class expansion and application to other families C, p ) is general, so we may assume that p / ∈ supp( θ C ⊗ η C ). Therefore h ( θ C ⊗ η C ( − p )) = h ( θ C ⊗ η C ) − ⇒ a ℓ C ( p ) < h ( θ C ⊗ η C ( p )) = h ( θ C ⊗ η C ) = 1 ⇒ a ℓ C ( p ) < ≤ a ℓ C ( p )Hence a ℓ C ( p ) = 3 and, via the limit g g − condition, a ℓ Qλ ( z ) = g −
4. Moreover,we may assume that the basepoint z is not a hyperflex of Q λ , as the pencil γ isalso general. Consequently, supp( θ Q λ ) does not consist of z twice, that is,div( θ Q λ ) = 2 z ⇒ a ℓ Qλ ( z ) ≤ g − a ℓ C ( p ) + a ℓ Qλ ( z ) ≥ g − a ℓ C ( p ) = 1and a ℓ Qλ ( z ) = g −
2. In turn, this means that z ∈ supp( θ Q λ ), and that ℓ is arefined limit g g − of the form ℓ C = | θ C ⊗ η C (2 p ) | + p ∈ G g − ( C ) ℓ Q λ = | θ Q λ ( z ) | + ( g − z ∈ G g − ( Q λ )with vanishing sequences (1 ,
3) and ( g − , g − Q λ , θ Q λ ) consisting of a plane quartic Q λ of γ equipped with an odd theta characteristic θ Q λ such that z = ζ ( λ ) ∈ supp( θ Q λ ),then every θ C ∈ S − g − ( C ) with θ C ⊗ η C ∈ S − g − ( C ) yields a limit g g − as above,and these limit linear series are the only ones contributing to the intersection R ∩ P +null (resp. θ C ∈ S + g − ( C ) with θ C ⊗ η C ∈ S − g − ( C ), R ∩ P − null ). The naturalquestion then arises as to how many such pairs ( Q λ , θ Q λ ) there are.We have discussed that odd theta characteristics θ Q λ of the plane curve Q λ correspond to bitangents to the quartic. Under this identification, the condition z = ζ ( λ ) ∈ supp( θ Q λ ) corresponds to the bitangent having the basepoint z asone of its contact points. In particular, for each Q λ we get only one candidate:the tangent line T z ( Q λ ) ⊂ P , which will intersect Q λ in two additional points.If we find out for how many values of λ these two points coincide, we will havefound the pairs ( Q λ , θ Q λ ) ≡ ( Q λ , T z ( Q λ )) we are trying to count.Now, we can study the pencil γ by taking two general polynomials F ( x ) = P i + j + k =4 a ijk x i x j x k , G ( x ) = P i + j + k =4 b ijk x i x j x k ∈ C [ x , x , x ] and considering the family { Q λ } described by H ( x , λ ) = λ F ( x ) + λ G ( x ) = 0,with basepoints { ( x , λ ) / F ( x ) = G ( x ) = 0 } ∋ ζ ( λ ) = z . By a suitable change ofcoordinates, we may assume z = (1 : 0 : 0) ∈ P . If we write H ( x , λ ) as H ( x , λ ) = H λ ( x ) = P i + j + k =4 c ijk ( λ ) x i x j x k ∈ C [ x , x , x ] with c ijk ( λ ) = λ a ijk + λ b ijk ∈ C [ λ ] , this means that c ( λ ) = 0.Moreover, the tangent line T z ( Q λ ) is given by ∂H λ ( x ) ∂x ( z ) x + ∂H λ ( x ) ∂x ( z ) x + ∂H λ ( x ) ∂x ( z ) x = c ( λ ) x + c ( λ ) x = 0Switching to coordinates u, v on T z ( Q λ ) = P u,v , that is, x = u , x = − c ( λ ) v , x = c ( λ ) v we see that z = { v = 0 } = (1 : 0) ∈ P u,v and that the intersection Q λ ∩ T z ( Q λ )is given by H λ ( u, v ) = P i + j + k =4 ( − j c ( λ ) j c ( λ ) k c ijk ( λ ) u i v j + k = 0Since the intersection contains z twice, this polynomial has no v , v terms: j + k = 0 ⇒ i = 4 c ( λ ) = 0 j + k = 1 ⇒ i = 3 − c ( λ ) c ( λ ) + c ( λ ) c ( λ ) = 0Factoring out v , we get a quadric q λ ( u, v ) = P ≤ i ≤ j + k =4 − i ( − j c ( λ ) j c ( λ ) k c ijk ( λ ) u i v − i = c uu ( λ ) u + c uv ( λ ) uv + c vv ( λ ) v whose roots correspond to the two additional points lying in Q λ ∩ T z ( Q λ ). Thedegree of each summand ( − j c ( λ ) j c ( λ ) k c ijk ( λ ) is j + k + 1, so we have c uu ( λ ) = P j + k =2 ( − j c ( λ ) j c ( λ ) k c jk ( λ ) ∈ C [ λ , λ ] c uv ( λ ) = P j + k =3 ( − j c ( λ ) j c ( λ ) k c jk ( λ ) ∈ C [ λ , λ ] c vv ( λ ) = P j + k =4 ( − j c ( λ ) j c ( λ ) k c jk ( λ ) ∈ C [ λ , λ ] Finally, the values of λ for which the roots of q λ ( u, v ) coincide are determined bythe roots of the discriminant∆( λ ) = ∆( q λ ( u, v )) = c uv ( λ ) − c uu ( λ ) c vv ( λ ) ∈ C [ λ , λ ] which is an octic polynomial. Therefore we obtain { ( Q λ , θ Q λ ) / z ∈ supp( θ Q λ ) } = { λ ∈ P / ∆( λ ) = 0 } = 8 = 2 and we can finally observe the appearance of the intersection numbers provided .4 Class expansion and application to other families { ( Q λ , θ Q λ ) / z ∈ supp( θ Q λ ) } ·· { θ C ∈ S − g − ( C ) / θ C ⊗ η C ∈ S − g − ( C ) } = 2 · N − g − = 2 g − (2 g − − { ( Q λ , θ Q λ ) / z ∈ supp( θ Q λ ) } ·· { θ C ∈ S + g − ( C ) / θ C ⊗ η C ∈ S − g − ( C ) } = 2 · N ± g − = 2 g − In particular, this indicates the lack of contribution from singular fibers.
Example 3.15 (more irreducible nodal curves) . If we recall the family
Y ≡ { B py } y ∈ B ⊂ ∆ ⊂ M g from example 1.17, which was lifted to a test curve Y ⊂ ∆ t0 ⊂ R g , then thereare two more standard lifts Z and T in R g , which arise when Y is pulled backby the maps ∆ p0 → ∆ and ∆ b0 → ∆ respectively: Z ≡ { ( B py , η p y ) / η p y ∈ ∆ p0 ( B py ) } y ∈ B ⊂ ∆ p0 ⊂ R g T ≡ { ( B ∪ p ∼ , y ∼∞ E, η b y ) / η b y ∈ ∆ b0 ( B py ) } y ∈ B ⊂ ∆ b0 ⊂ R g If we set k = R g − ( B ) = 2 g − −
1, we can see that their intersection table is: λ δ t0 δ p0 δ b0 δ n1 δ t1 δ p1 δ ( j ≥ Y − g Z k (1 − g ) 0 0 k k T g − (1 − g ) 1 0 k p0 | ∆ ) = 2 k and deg(∆ b0 | ∆ ) = 2 g − = k + 1. Remark 3.16.
Applying theorem 3.12 to example 3.15, it follows that Z · P +null = (2 g − − µ = R g − ( B ) · µZ · P − null = (2 g − − µ = R g − ( B ) · µT · P +null = 2 g − (2 g − + 1) µ = S + g − ( B ) · µT · P − null = 2 g − (2 g − − µ = S − g − ( B ) · µ with the factor µ = Y n0 · Θ null = 2 g − (2 g − ( g −
3) + 1) indicating the number ofnodal curves B py in Y that admit a vanishing theta-null θ y ∈ Θ null ( B py ), whichwe computed in the argument preceding preposition 3.11. Once more, it may beinteresting to provide an interpretation of these results.6 References
According to example 1.9, any Prym root η B ∈ R g − ( B ) gives rise to twoelements η p , + y , η p , − y ∈ ∆ p0 ( B py ), depending on which of the two possible gluings η B | p ∼ = η B | y is chosen. In particular, for each pair ( B py , θ y ) ∈ Θ null , tensoring θ y by either η p , + y or η p , − y produces stable spin curves of opposite parity, so that( B py , η p , + y ) ∈ Z ∩ P +null ( B py , η p , − y ) ∈ Z ∩ P − null which explains the emergence of the factors k = { η B ∈ R g − ( B ) } µ = { ( B py , θ y ) ∈ Θ null } in the intersection numbers Z · P +null and Z · P − null .Similarly, for each pair ( B py , θ y ) ∈ Θ null , the root θ y | B ∈ p ω B ( p + q ) can besubtracted from any theta characteristic θ B ∈ S g − ( B ) = √ ω B so as to create aroot η B ∈ p O B ( − p − q ). This in turn yields a unique stable Prym curve( X, η b y ) = ( B ∪ p ∼ , y ∼∞ E, η b y ) ∈ T ∩ P null such that η b y restricts to ( η B , O E (1)) on Pic( B ) ⊕ Pic( E ). Furthermore, ( X, η b y )lies in P +null (resp. P − null ) whenever θ B is even (resp. odd) by construction of thePrym-null divisors, which brings to light the connection between S + g − ( B ) , S − g − ( B ) , µ and the intersection numbers T · P +null and T · P − null . References [BCF04] Edoardo Ballico, Cinzia Casagrande, and Claudio Fontanari. Moduliof Prym curves.
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