Very stable and wobbly loci for elliptic curves

As a warm-up exercise we first focus on $\mathbb{P}^{1}$. If $t\geq 0$ there does not exist any $t$-very stable bundle over $\mathbb{P}^{1}$. It is enough to check the statement for $r=2$. Let $E\cong\mathcal{O}(d_{1})\oplus\mathcal{O}(d_{2})$. Then either $d_{1}-d_{2}\geq 0$ or $d_{2}-d_{1}\geq 0$. Furthermore, without loss of any generality, we can take $d_{1}-d_{2}+t\geq 0$. Define

Here, $\phi^{\prime}$ is a nonzero section of a line bundle, appearing as a global component of $\phi$. This $\phi$ is a non-zero nilpotent Higgs field.

The following results are well known, but we outline their proofs.

The crux of Atiyah’s work is the exploration of indecomposable bundles and their multiplicative structures on elliptic curves. Tu organizes necessary information about the moduli of stable and semistable bundles out of Atiyah’s construction of indecomposable bundles. Moreover, they computed the respective cohomology groups of such moduli space of bundles. We collect a list of useful results (with some changes in language) from both papers ([2] and [25]) in service to the readers.

A point $A$ on an elliptic curve $X$ gives rise to a unique line bundle $\mathcal{O}(A)\to X$ so that every nonzero section of $\mathcal{O}(A)$ vanishes at $A$. This is an isomorphism of complex manifolds (between the elliptic curve and its Picard variety of degree $1$), usually called the Abel-Jacobi map. There is an isomorphism $\mbox{Pic}^{1}X\cong\mbox{Pic}^{0}X$ by $L\mapsto L\otimes\mathcal{O}(A)^{-1}$. Commonly the elements of a symmetric product of a curve are well defined divisors of the degree same as the power of the symmetric product. We will use the description of $\mbox{Sym}^{h}(X)$ as the collection of effective divisors interchangeably with the classical definition. More generally, there is a holomorphic Abel-Jacobi map $\alpha:\mbox{Sym}^{h}(X)\to\mbox{Pic}^{h}X$ assigning a divisor of $h$ points on $X$ to its corresponding line bundle of degree $h$. The tangent and the cotangent (canonical) bundles on $X$ are isomorphic to the trivial line bundle $X\times\mathbb{C}$.

1. (1)

   (The Uniqueness Theorem) Let $E\in\mathcal{E}(r,d)$ with $d\geq 0$. Then (i) $\dim H^{0}(E)=h^{0}(E)=d$ when $d>0$ and $0$ or $1$ when $d=0$, (ii) if $d<r$, there is a trivial subbundle $I_{s}=\bigoplus_{i=1}^{s}\mathcal{O}_{X}$ of $E$ while $E^{\prime}=E/I_{s}$ is indecomposable with $h^{0}(E^{\prime})=s$.

2. (2)

   (Existence Theorem) Let $E^{\prime}\in\mathcal{E}(r^{\prime},d)$ with $d\geq 0$ and if $d=0$ let $h^{0}(E^{\prime})\neq 0$. Then there exists a bundle $E\in\mathcal{E}(r,d)$ unique up to isomorphism, given by an extension

   $$0\to I_{s}\to E\to E^{\prime}\to 0$$

   where $r=r^{\prime}+s$ and $s=d$ when $d>0$ and $1$ when $d=0$.

3. (3)

   (Extensions of indecomposable bundles of degree $0$) There exists a vector bundle $F_{r}\in\mathcal{E}(r,0)$ with $h^{0}(F_{r})\neq 0$ that fits into an exact sequence

   $$0\to\mathcal{O}_{X}\to F_{r}\to F_{r-1}\to 0$$

   and a bundle $E\in\mathcal{E}(r,0)$ is of the form $E\cong F_{r}\otimes L$ for some $r$-torsion element $L\in\mbox{Pic}^{0}(X)$. The bundle $F_{r}$ is self dual i.e $F_{r}\cong F_{r}^{*}$ and $h^{0}(F_{r}\otimes L)\neq 0$ if and only if $h^{0}(L)\neq 0$.

4. (4)

   (Indecomposable bundles with nonzero degrees) Let $r,d$ be mutually prime integers. Then $E\in\mathcal{E}(r,d)$ is uniquely identified with its determinant $\det(E)\in\mbox{Pic}^{d}(X)$. Suppose that $A\in\mbox{Pic}^{1}(X)\cong X$ is a chosen base point. There is a unique bundle $E_{A}(r,d)\in\mathcal{E}(r,d)$ so that we are able to identify $E\in\mathcal{E}(r,d)$ as $E_{A}(r,d)\otimes L$. To be precise, $E_{A}(r,d)$ is the unique indecomposable bundle of rank $r$ and degree $d$ that admits determinant $\mathcal{O}(A)^{d}$. Moreover,

   $$E_{A}(r,d)^{*}\cong E_{A}(r,-d).$$

   Even if $r$ and $d$ admit the greatest common divisor $h$, a bundle $E_{A}(r,d)$ is uniquely defined by a bijective correspondence between $\mathcal{E}(h,0)$ and $\mathcal{E}(r,d)$. Related results are generalized with step by step reductions to the ‘mutually prime’ cases.

5. (5)

   (Multiplicative structures) If $r,s>1$ the bundle $F_{r}\otimes F_{s}$ decomposes as a finite direct sum of bundles $F_{i}$’s. If $r,d$ are mutually prime integers then

   $$E_{A}(r,d)\otimes F_{h}\cong E_{A}(rh,dh)$$

   is indecomposable for any integer $h$. For chosen mutually prime pairs of integers $r,d$ and $r^{\prime},d^{\prime}$ so that $r,r^{\prime}$ are mutually prime we have

   $$E_{A}(r,d)\otimes E_{A}(r^{\prime},d^{\prime})\cong E_{A}(rr^{\prime},rd^{\prime}+r^{\prime}d).$$

6. (6)

   (Semistability of indecomposable bundles) For a chosen base point $A$ in $\mbox{Pic}^{1}X$, the canonical indecomposable bundle $E_{A}(r,d)$ is semistable for any $r,d$. If $(r,d)=h>1$ and $r=r^{\prime}h$ and $d=d^{\prime}h$ then $E_{A}(r,d)$ is strictly semistable from the following exact sequence of bundles

   $$0\to E_{A}(r^{\prime},d^{\prime})\to E_{A}(r,d)\to E_{A}(r-r^{\prime},d-d^{\prime})\to 0.$$

   This confirms stability of $E$ precisely if $r$ and $d$ are mutually prime. In particular, $F_{r}$ is semistable for all $r\in\mathbb{N}$.

7. (7)

   (Decomposable semistable bundles) Set $h=(r,d)$ and $r=r^{\prime}h;~{}d=d^{\prime}h$. Every semistable bundle of rank $r$ and degree $d$ on $X$ is strongly equivalent to a bundle of the form

   $$E_{A}(r^{\prime},d^{\prime})\otimes\bigoplus_{i=1}^{h}M_{i},$$

   where $M_{i}\in\mbox{Pic}^{0}X$ with $M_{i}^{n^{\prime}}=\mathcal{O}_{X}$. There is an isomorphism $E_{A}(r^{\prime},d^{\prime})\otimes\oplus_{i=1}^{h}M_{i}\mapsto\oplus_{i=1}^{h}(M_{i}^{n^{\prime}}\otimes A)$.

8. (8)

   (Semistable bundles and symmetric products of an elliptic curve) Let the moduli space of (grading equivalent) semistable and the moduli space of (isomorphic) stable bundles of rank $r$ and degree $d$ on an elliptic curve $X$ be denoted with $\mathcal{M}_{X}^{ss}(r,d)$ and $\mathcal{M}_{X}^{s}(r,d)$ respectively. Suppose that $h$ is the greatest common divisor of $r$ and $d$. Then

   $$\mathcal{M}_{X}^{ss}(r,d)\cong\mbox{Sym}^{h}(X)$$

   and

   $$\mathcal{M}_{X}^{s}(r,d)\cong\begin{cases}\emptyset;~{}h>1\\ X;~{}h=1\end{cases}.$$

9. (9)

   There is a commutative diagram

   (2.3.1)

   In short we explain the commutative diagram as following. Choose an element $E$ in $\mbox{Sym}^{h}(X)$ and represent it in two ways, first as a bundle $E_{1}\oplus...\oplus E_{h}$ each with the same rank and the same degree and take its determinant $L$. Then consider the same element $E$ as a divisor $p_{1}+...+p_{h}$ and consider the line bundle $\otimes_{i=1}^{h}\mathcal{O}(p_{i})$. These two line bundles differ by a tensor product of a power of $\mathcal{O}(A)$ so that $\det(E_{i})=\mathcal{O}(A)^{\mbox{rank}(E_{i})}$.

10. (10)

    $\mbox{Pic}(\mathcal{M}_{X}^{ss}(r,d))\cong\mbox{Pic}(\mbox{Pic}(X))\oplus\mathbb{Z}$. Denote $\mathcal{M}_{X}^{ss}(r,L)$ the semistable bundles of a fixed determinant $L$. Then $\mathcal{M}_{X}^{ss}(r,L)\cong\mathbb{P}^{h-1}$. We derive it from the fact that a fiber of the Abel-Jacobi map is the projectivization of the linear system of effective divisors of $L\otimes\mathcal{O}(A)^{h-d}$.

We first consider $0\leq d<r$ and manage any other degree through adjusting via the division algorithm. We handle the indecomposable bundles at first then the case of the decomposable ones.

1. (1)

   Let $E\in\mathcal{E}(r,0)$. For a twist $\deg(L)<0$ we have $E$ is $L$-very stable due to the fact that $E$ is semistable. However a more elementary argument is available at our exposure, utilizing the fact that there exists an element $M\in\mbox{Pic}^{0}(X)$ so that $E\cong F_{r}\otimes M$. Decompose

   (3.1.1)

   $$F_{r}\otimes F_{r}=\bigoplus\limits_{i=1}^{n}F_{k_{i}}$$

   in to indecomposable components and observe that $H^{0}(F_{i}\otimes L)=0$ since $F_{r}\otimes L$ is indecomposable with a negative degree (cf. [3] Lemma 3.19) we arrive at $H^{0}(\mbox{End}(E)\otimes L)=0$. Hence $E$ is $L$-very stable.∎
   Next we choose a twist $L$ as $\deg(L)>0$ or $L=\mathcal{O}_{X}$ we apply Lemma 2.2.1 on the exact sequence

   (3.1.2)

   $$0\to M\to E\to F_{r-1}\otimes M\to 0$$

   whereas $H^{0}(\mbox{Hom}(F_{r-1}\otimes M,M\otimes L))=H^{0}(F_{r-1}\otimes L)\neq 0$. We conclude that $E$ is $L$-wobbly.∎

2. (2)

   Now we consider $0<d<r$. For $\deg(L)<0$ we have $E\in\mathcal{E}(r,d)$ is $L$-very stable once again as $E$ is semistable (Lemma 2.2.1).∎
   Choose a twist $L$ with $\deg(L)>0$. We focus on a line bundle $L$ so that $\deg(L)\geq 2$. We derive that $E$ is $L$-wobbly. Choose an exact sequence

   (3.1.3)

   $$0\to\mathcal{O}_{X}\to E\to E^{\prime}\to 0.$$

   We remark that $E^{\prime}$ may decompose in to bundles of smaller ranks. The inequality $0<d<r+(r-2)$ implies $\deg(L)\geq 2>\frac{d}{r-1}$. We obtain

   (3.1.4)

   $$\deg(E^{\prime*}\otimes L)=(r-1)\deg(L)-d>0.$$

   By 2.2.1 we have $E$ is $L$-wobbly via Riemann-Roch.∎
   Let $\deg(L)=1$. We choose the rank and the degree of $E$ so that $r>d+1$. The exactly same reasoning $E$ is $L$-wobbly. We sharpen our reasoning to investigate the $L$-very stable pairs in the case of $r=d+1$. At first we focus at $d=1$. Suppose that $E$ is a rank $2$ bundle of degree $1$ with determinant $\delta\in\mbox{Pic}^{1}(X)$. Due to its stability each line subbundle $\xi\subset E$ admits a degree $\leq 0$. In particular, there is an exact sequence of bundles

   (3.1.5)

   $$0\to\xi\to E\to\xi^{-1}\otimes\delta\to 0.$$

   If $E$ admits a line subbundle $\xi$ such that $\xi^{2}\otimes L=\delta$ then $E$ is $L$-wobbly. If such a line subbundle does not exist then $E$ is $L$-very stable due to 2.2.1. In general we use the exact sequence

   (3.1.6)

   $$0\to I_{d}\to E\to E^{\prime}\to 0$$

   where $E^{\prime}$ is a degree $1$ line bundle isomorphic to the determinant of $E$. If $E^{\prime}\cong L$ we can conclude $E$ is $L$-wobbly which confirms that the $L$-wobbly locus in $\mathcal{E}(d+1,d)$ is nonempty. We frame a relevant conjecture 3.1.2 in this context. We further choose the twist $L=\mathcal{O}_{X}=T^{*}X$. If $(r,d)=1$ then $E\in\mathcal{E}(r,d)$ is stable, thus simple and obviously very stable. However, if $(r,d)>1$ a bundle $E$ is not necessarily stable and $H^{0}(\mbox{End}E)$ may or may not admit a nonzero nilpotent Higgs field. We handle such bundles up to a grading equivalent polystable bundle in Remark 3.1.4.∎

   Conjecture 3.1.2.

   Consider the holomorphic one-to-one correspondence

   $$\det:\mathcal{E}(d+1,d)\cong X\to X\cong\operatorname{Pic}^{1}(X).$$

   The set of $L$-very stable locus is an open dense subset $\det^{-1}(\operatorname{Pic}^{1}(X)\backslash\{L\})=X\backslash\det^{-1}(L)$ for $d\geq 2$.

We summarize the whole discussion in form of the following theorem.

We dedicate this subsection to a rapid recollection of the results concerning the canonically wobbly loci on curves of higher genera. We furthermore discuss the merits and demerits of these results in case of an elliptic curve. The geometry of the canonically very stable and the canonically wobbly locus of bundles was formally introduced by Laumon [12] on the algebraic stack of rank $n$ bundles $\mbox{Fib}_{X,n}$, on a complex algebraic curve $X$. Laumon proved that the nilpotent cone in the cotangent bundle $T^{*}\mbox{Fib}_{X,n}$ is the support of a closed reduced Lagrangian submanifold of $T^{*}\mbox{Fib}_{X,n}$ of dimension $n^{2}(g_{X}-1)$ and later concluded that very stable bundles form an open dense subset in $\mbox{Fib}_{X,n}$. The very stable bundles live inside the open dense subset of slope-semistable bundles for $g_{X}>0$ (cf. [12] Proposition (3.5)) by Lemma 2.2.1 and Lemma 2.2.2. In the same article [12] Laumon referred to a result of Drinfeld stating that the collection of the wobbly bundles form a pure closed subset in $\mbox{Fib}_{X,n}$ of codimension $1$. Recently mathematicians ([18], [20], [21]) investigated in to topological properties of the moduli space of the very stable and the wobbly bundles within the moduli space of Higgs bundles on curves $X$ of genus $\geq 2$. At this stage, we briefly recollect features of the moduli space of semistable $L$-twisted pairs for a chosen line bundle $L$.

For the rest of the article, we restrict our focus to $L=K_{X}$ or $\deg(L)>2g_{X}-2$ on a curve $X$ of genus at least $1$. Nitsure ([17]) proved existence of a quasi-projective separated noetherian scheme $\mathcal{M}_{X}(r,d,L)$ of finite type over $\mathbb{C}$ parametrizing the collection of strongly equivalent semistable $L$-twisted pairs which contains an open subscheme $\mathcal{M}_{X}^{\prime}(r,d,L)$ parametrizing the collection of isomorphism classes of stable pairs. In particular, there is a smooth open subscheme $\mathcal{M}_{0}^{\prime}$ of complex dimension $\dim_{\mathbb{C}}\mathcal{M}_{0}^{\prime}=r^{2}\deg(L)+1+h^{1}(X,L)$ parametrizing the stable pairs which admits a stable underlying bundle. This scheme $\mathcal{M}_{0}^{\prime}$ is integral. In particular, $\mathcal{M}_{0}^{\prime}(K_{X})$ is the cotangent bundle $T^{*}\mathcal{M}_{X}^{s}(r,d)$ that admits hyperkähler structures. For a general twist the space of stable pairs $\mathcal{M}_{0}^{\prime}$ admits a Kähler structure. There is a proper morphism $H:\mathcal{M}_{X}(r,d,L)\to\oplus_{i=1}^{r}H^{0}(X,L^{i})$ evaluating the characteristic coefficients of pairs. In [21], Pauly and Peón-Nieto proved a chain of important criteria as a device to locate the very stable bundles inside $T^{*}\mathcal{M}_{X}^{s}(r,d)$ on a curve of genus $\geq 2$. The following result is a version of [21] Theorem 1.1 with minimal changes in the language and the notation.

The proof of the above statement engages deeply with the topology of the limit points coming from the usual $\mathbb{C}^{*}$-action on the moduli space $\mathcal{M}_{X}(r,d,K_{X})$. We recall, as a standard fact, that this moduli space is semiprojective i.e. the limit $\lim_{\lambda\to 0}(E,\lambda\cdot\phi)$ which exists in the projective completion of the moduli space lies inside the moduli space itself. To prove this specific assertion, they choose the path of showing a contradiction. They modify a rational map from a complex surface to $\mathcal{M}_{X}(r,d,K_{X})$ by resolving its indeterminacy locus and construct a morphism from a connected union of projective lines to the Zariski closure of $V$. This morphism assumes values $(E,0)$ and $(F,\psi)\in\bar{V}\backslash V$ at two distinct points and leads to the conclusion that $F=E$ to respect semistability of $F$. A favorable estimate of the dimensions of affine spaces plays an important role in this argument. Recall that a proper morphism between two affine schemes of the same dimension, in particular, the Hitchin base $\mathcal{B}$ and the vector space $H^{0}(\mbox{End}E\otimes K)$ of the same complex dimension $1+r^{2}(g_{X}-1)$, is quasi-finite. We can interpret the criterion in a slightly different way. Recall that there is a holomorphic projection morphism $\pi:\mathcal{M}_{0}^{\prime}\to\mathcal{M}_{X}^{s}(r,d)$ and $\pi^{-1}([E])$ is the vector space $V$ which is closed in $\mathcal{M}_{0}^{\prime}$ but may not be closed inside $\mathcal{M}_{X}(r,d,L)$. Theorem 3.2.1 confirms that on a curve of genus $>1$, a fiber $\pi^{-1}([E])$ is closed if and only if $E$ is canonically very stable.

The nilpotent cone $\mathcal{N}$ is the collection of semistable nilpotent Higgs bundles. It is formally defined as the fiber $H^{-1}(0)$. Due to properness $\mathcal{N}$ is a compact subset (possibly containing singular points) and it is not easy to explore the generally twisted nilpotent cone. For a stable bundle $E\to X$ we have $[(E,0)]\in H^{-1}(0)$, thus the stable bundles contribute to the nilpotent cone a dimensional estimate at least $r^{2}(g_{X}-1)+1$. Geometrically, a stable point $(E,0)$ lies at the bottom of the nilpoent cone and appears as a global minima of the Morse function of stable pairs $(E,\phi)\mapsto 2i\int_{X}\mbox{trace}(\phi\phi^{*})$. To study the geometry of the nilpotent cone, Mathematicians ([8], [20]) restricted their investigation to the determinant morphism on trace-free canonically twisted pairs of rank $2$ with a fixed determinant $\Lambda$. The main advantage here is that the very stable locus lives inside the ‘zero-determinant locus’ of the stable Higgs bundles as an open subset and simplifies the analysis. The definition that is due to [20]. It futher leads to the fact that the wobbly locus is a reducible divisor.

Works of Narasimhan and Ramanan [15], [16] laid the foundation of research of stable bundles on complex curves, particularly in the base case of rank $2$. In case of $g_{X}=2$, $\mathcal{M}_{X}^{s}(2,\delta)$ is isomophic to the intersection of a smooth pencil of quadrics in $\mathbb{CP}^{5}$. As a known fact we mention that any non-trivial extension of the following type (this exact sequence is mentioned in Lemma 2.2.1)

where $\xi$ is a line bundle of degree $0$, is stable and any two such extensions are isomorphic if and only if these are scalar multiples of each other. Moreover, there is a linear embedding $\mathbb{P}(H^{1}(\xi^{-2}\otimes\delta^{-1}))\to\mathcal{M}_{X}^{s}(2,\delta)$. Pal further restricted the wobbly locus inside the cotangent bundle $T^{*}\mathcal{M}_{X}^{s}(2,\delta)$ and proved (cf. [18] Theorem 1.3). that the space of wobbly vector bundles of rank $2$ with determinant $\delta$ is isomorphic to a surface of degree $32$. Recently Pal has completed his proof for an arbitrary rank of the fact that wobbly locus forms a closed subvariety of codimension $1$ inside $\mathcal{M}_{X}^{s}(r,\delta)$ generalizing his techniques of extensions of the very stable bundles from the rank $2$ case to an arbitrary rank (cf. [19] Theorem 1.1). Pal proved that a stable bundle $E\in\mathcal{M}_{X}^{s}(r,\delta)$ is wobbly if and only if it passes through a nonfree minimal rational curve.

The work of Pal and Pauly ([20]) is significant for their explicit computation of the Chern classes of the irreducible subdivisors of the wobbly divisor. Assuming the definition 3.2.2, we present a theorem of Pal and Pauly (cf. [20] Theorem 1.1) with minimal changes in the language and the notation. We denote $\lambda=\deg(\Lambda)$. Unfortunately, we do not find any major information about the topology of the irreducible components of the wobbly locus of an arbitrary rank on a curve of genus $\geq 2$.

We explain the divisors $\mathcal{W}_{k}$ briefly. A sublocus is defined

and denote by $\mathcal{W}_{k}$ the Zariski closure of $\mathcal{W}_{k}^{0}\subset\mathcal{M}_{X}^{s}(2,\Lambda)$. It is deduced that $\mathcal{W}=\cup_{k=1}^{g}\mathcal{W}_{k}$ for $\lambda=0$ and $\mathcal{W}=\cup_{k=1}^{g-1}\mathcal{W}_{k}$ for $\lambda=1$. There is a filtration of $\mathcal{W}_{k}$’s as per [20] Proposition 2.3 that $\bigcup_{k=1}^{\lceil{\frac{g_{X}-\lambda}{2}}\rceil-1}\mathcal{W}_{k}\subset\mathcal{W}_{\frac{g_{X}-\lambda}{2}}$. Further each of the divisors $\mathcal{W}_{k}$ represents an element in the Picard group of $\mathcal{M}_{X}^{s}(2,\Lambda)$. According to a result by Drezet and Narasimhan we are aware that the Picard group of $\mathcal{M}_{X}^{s}(2,\Lambda)$ is isomorphic to $\mathbb{Z}$. The following theorem is Theorem 1.3 [20] which mentions the representatives in the Picard group in terms of the first Chern classes.

The proof is technical. We mention a few important steps. The actual strategy follows from decomposing $\mathcal{N}_{0}$ as a union of images of $\mathcal{W}_{k}$’s under the aforementioned forgetful rational map. A set of subvarieties $Z_{k}\subset\mbox{Pic}^{1-k}(X)$ for $1\leq k\leq g_{X}-\lambda$ appear as the pre-images of the Brill-Noether Loci $W_{2g_{X}-2k-\lambda}$ under the holomorphic map $L(\in\mbox{Pic}^{1-k}(X))\mapsto K_{X}L^{2}\Lambda^{-1}(\in\mbox{Pic}^{2g_{X}-2k-\lambda}(X))$. The fundamental class $[Z_{k}]$ of $Z_{k}$ as a subvariety of $\mbox{Pic}^{1-k}(X)$ is computed as $\frac{2^{4k-2g_{X}+2}}{(2k-g_{X}+1)!}\cdot\Theta_{\mbox{Pic}^{1-k}(X)}^{2k-g_{X}+1}$ ([20] Lemma 4.1) wherein we use $\Theta_{\mbox{Pic}^{1-k}(X)}$ to denote a theta divisor of $\mbox{Pic}^{1-k}(X)$. Under a classifying map $f:S\to\mathcal{M}_{X}^{s}(2,\Lambda)$ the wobbly subloci $\mathcal{W}_{k}$ is identified as $f^{-1}(\mathcal{W}_{k})=\pi_{S}(\Delta_{k}\cup(S\times Z_{k}))$ where $\pi_{S}:S\times X\to S$ is the projection map. Here $\Delta_{k}$ is a subvariety of $S\times\mbox{Pic}^{1-k}(X)$ (cf. page 9 [20]). The fundamental class $[\Delta_{k}]$ of $\Delta_{k}$ is computed as $\frac{2^{2g_{X}-2k-2}}{(2g_{X}-2k-1)!}\Theta_{S}\otimes\Theta_{\mbox{Pic}^{1-k}(X)}^{2g_{X}-2k-1}$ given that $\Theta_{S}$ is the first Chern class of the pullback of an ample generator of $\mbox{Pic}(\mathcal{M}_{X}^{s}(2,\Lambda))$ under $f$. Finally, the fundamental class of $f^{-1}(\mathcal{W}_{k})$ is the tensor product of $[Z_{k}]$ with $[\Delta_{k}]$.