Hyperelliptic classes are rigid and extremal in genus two
Vance Blankers
math.AG










Chiral transport along magnetic domain walls in the quantum anomalous Hall effect
    D. Goldhaber-Gordon
    December 30, 2023
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Français






























§ INTRODUCTION




Every smooth curve of genus two admits a unique degree-two hyperelliptic map to ℙ^1. The Riemann-Hurwitz formula forces such a map to have six ramification points called Weierstrass points; each non-Weierstrass point p exists as part of a conjugate pair (p,p') such that the images of p and p' agree under the hyperelliptic map.

The locus of curves of genus two with ℓ marked Weierstrass points is codimension ℓ inside the moduli space _2,ℓ, and in <cit.> it is shown that the class of the closure of this locus is rigid and extremal in the cone of effective classes of codimension ℓ. Our main theorem extends their result to _2,ℓ,2m,n⊆_2,ℓ+2m+n, 
the locus of genus-two curves with ℓ marked Weierstrass points, m marked conjugate pairs, and n free marked points (see Definition <ref>).


For ℓ,m,n≥ 0, the class _2,ℓ,2m,n, if non-empty, is rigid and extremal in the cone of effective classes of codimension ℓ+m in _2,ℓ+2m+n.



In <cit.>, the authors show that the effective cone of codimension-two classes of _2,n has infinitely many extremal cycles for every n. Here we pursue a perpendicular conclusion: although in genus two ℓ≤ 6, the number of conjugate pairs and number of free marked points are unbounded, so that the classes _2,ℓ,2m,n form an infinite family of rigid and extremal cycles in arbitrarily-high codimension. Moreover, the induction technique used to prove the main result is genus-agnostic, pointing towards a natural extension of the main theorem to higher genus given a small handful of low-codimension cases.

When ℓ + m ≥ 3, our induction argument (Theorem <ref>) is a generalization of that used in <cit.> to include conjugate pairs and free points; it relies on pushing forward an effective decomposition of one hyperelliptic class onto other hyperelliptic classes and showing that the only term of the decomposition to survive all pushforwards is the original class itself. This process is straightforward when there are at least three codimension-one conditions available to forget; however, when ℓ + m = 2, and in particular when ℓ = 2 and m = 0, more care must be taken. The technique used in <cit.> to overcome this problematic subcase relies on an explicit expression for [_2,2,0,0] which becomes cumbersome when a non-zero number of free marked points are allowed. Although adding free marked points can be described via pullback, pullback does not preserve rigidity and extremality in general, so we introduce an intersection-theoretic calculation using tautological ω-classes to handle this case instead.

The base case of the induction (Theorem <ref>) is shown via a criterion (Lemma <ref>) given by  <cit.> for rigidity and extremality for divisors; it amounts to an additional pair of intersection calculations. We utilize the theory of moduli spaces of admissible covers to construct a suitable curve class for the latter intersection, a technique which generalizes that used in <cit.> for the class of _2,1,0,0.





 §.§ Structure of the paper.
 We begin in §<ref> with some background on _g,n and cones of effective cycles. This section also contains the important Lemma <ref> upon which Theorem <ref> depends. In §<ref>, we prove Theorem <ref>, which establishes the base case for the induction argument of our main result, Theorem <ref>. Finally, we conclude in §<ref> with a discussion of extending these techniques for g≥ 3 and possible connections to a CohFT-like structure.




 §.§ Acknowledgments.
 The author wishes to thank Nicola Tarasca, who was kind enough to review an early version of the proof of the main theorem and offer his advice. The author is also greatly indebted to Renzo Cavalieri for his direction and support.





§ PRELIMINARIES ON _G,N AND EFFECTIVE CYCLES





 §.§ Moduli spaces of curves, hyperelliptic curves, and admissible covers.
 We work throughout in _g,n, the moduli space of isomorphism classes of stable genus g curves with n (ordered) marked points. If 2g-2+n > 0 this space is a smooth Deligne-Mumford stack of dimension 3g-3+n. We denote points of _g,n by [C; p_1,…,p_n] with p_1,…,p_n∈ C smooth marked points. For fixed g, we may vary n to obtain a family of moduli spaces related by forgetful morphisms: for each 1≤ i ≤ n, the map π_p_i:_g,n→_g,n-1 forgets the ith marked point and stabilizes the curve if necessary. The maps ρ_p_i:_g,n→_g,{p_i} are the rememberful morphisms which are the composition of all possible forgetful morphisms other than π_p_i. 

Due to the complexity of the full Chow ring of _g,n, the tautological ring R^*(_g,n) is often considered instead <cit.> (for both rings we assume rational coefficients). Among other classes, this ring contains the classes of the boundary strata, as well as all ψ- and λ-classes. For 1≤ i ≤ n the class ψ_p_i is defined to be the first Chern class of the line bundle on _g,n whose fiber over a given isomorphism class of curves is the cotangent line bundle at the ith marked point of the curve; λ_1 is the first Chern class of the Hodge bundle. The tautological ring also includes pullbacks of all ψ- and λ-classes, including the ω-classes, sometimes called stable ψ-classes. The class ω_p_i is defined on _g,n for g,n≥ 1 as the pullback of ψ_p_i along ρ_p_i. Several other notable cycles are known to be tautological, including the hyperelliptic classes considered below (<cit.>).

Hyperelliptic curves are those which admit a degree-two map to ℙ^1. The Riemann-Hurwitz formula implies that a hyperelliptic curve of genus g contains 2g+2 Weierstrass points which ramify over the branch locus in ℙ^1. For a fixed genus, specifying the branch locus allows one to recover the complex structure of the hyperelliptic curve and hence the hyperelliptic map. Thus for g≥ 2, the codimension of the locus of hyperelliptic curves in _g,n is g-2. In this context, requiring that a marked point be Weierstrass (resp. two marked points be a conjugate pair) is a codimension-one condition for genus at least two.

We briefly use the theory of moduli spaces of admissible covers to construct a curve in _2,n in Theorem <ref>. These spaces are particularly nice compactifications of Hurwitz schemes. For a thorough introduction, the standard references are <cit.> and <cit.>. For a more hands-on approach in the same vein as our usage, see as well <cit.>.




 §.§ Notation.
 We use the following notation for boundary strata on _g,n
; all cycles classes are given as stack fundamental classes. For g≥ 1, the divisor class of the closure of the locus of irreducible nodal curves is denoted by δ_irr. By δ_h,P we mean the class of the divisor whose general element has one component of genus h attached to another component of genus g-h, with marked points P on the genus h component and marked points {p_1,…,p_n}\ P on the other. By convention δ_h,P = 0 for unstable choices of h and P.

Restrict now to the case of g=2. We use W_2,P to denote the codimension-two class of the stratum whose general element agrees with that of δ_2,P, with the additional requirement that the node be a Weierstrass point. We denote by γ_1,P the class of the closure of the locus of curves whose general element has a genus-one component containing the marked points P meeting in two points conjugate under a hyperelliptic map a rational component with marked points {p_1,…,p_n}\ P (see Figure <ref>).





The space Adm_2 0,t_1,…,t_6,u_1±,…,u_n± is the moduli space of degree-two admissible covers of genus two with marked ramification points (Weierstrass points) t_i and marked pairs of points (conjugate pairs) u_j+ and u_j-. This space comes with a finite map c to _0,{t_1,…,t_6,u_1,…,u_n} which forgets the cover and remembers only the base curve and its marked points, which are the images of the markings on the source. It comes also with a degree 2^n map s to _2,1+n which forgets the base curve and all u_j+ and t_i other than t_1 and remembers the (stabilization of the) cover.








 §.§ ω-class lemmas.
 The following two lemmas concerning basic properties of ω-classes prove useful in the last subcase of Theorem <ref>. The first is a unique feature of these classes, and the second is the ω-class version of the dilaton equation.



Let g≥ 1, n≥ 2, and P⊂{p_1,…,p_n} such that |P|≤ n-2. Then for any p_i,p_j∉P

    ω_p_i·δ_g,P = ω_p_j·δ_g,P

on _g,n.


This follows immediately from Lemma 1.9 in <cit.>.




Let g,n≥ 2. Then on _g,n,

    π_p_i*ω_p_j = 2g - 2

if i=j, and 0 otherwise.


Let P = {p_1,…,p_n}. When i=j, the pushforward reduces to the usual dilaton equation for ψ_p_i on _g,{p_i}. If π is the morphism which forgets all marked points, the diagram


[row sep=1cm, column sep=1cm] 
_g,n [d, "ρ_p_i"] [r, "π_p_i"]     _g,P\{p_i} [d, "π"] 

_g,{p_i} [r, "π_p_i"]     _g



commutes, so π_p_i*ω_p_i = π_p_i*ρ_p_i^*ψ_p_i = π^*π_p_i*ψ_p_i = (2g-2)1.

If i≠ j, then π_p_i*ω_p_j = π_p_i*π_p_i^*ω_p_j = 0.





 §.§ Cones and properties of effective classes.
 For a projective variety X, the sum of two effective codimension-d classes is again effective, as is any ℚ_+-multiple of the same. This gives a natural convex cone structure on the set of effective classes of codimension d inside the ℚ vector space of all codimension-d classes, called the effective cone of codimension-d classes and denoted ^d(X). The closure of this cone (in the usual ℝ^n topology) is ^d(X), the pseudo-effective cone of codimension-d classes. Given an effective class E in the Chow ring of X, an effective decomposition of E is an equality

    E = ∑_s=1^m a_sE_s

with a_s > 0 and E_s irreducible effective cycles on X for all s. The main properties we are interested in for classes in the pseudo-effective cone are rigidity and extremality.



Let E∈^d(X).

E is rigid if any effective cycle with class rE is supported on the support of E.

E is extremal if, for any effective decomposition of E, all E_s are proportional to E.



When d=1, elements of the cone correspond to divisor classes, and the study of ^1(_g,n) is fundamental in the theory of the birational geometry of these moduli spaces. For example, _0,n is known to fail to be a Mori dream space for n≥ 10 (first for n≥ 134 in <cit.>, then for n≥ 13 in <cit.>, and the most recent bound in <cit.>). For n≥ 3 in genus one, <cit.> show that _1,n is not a Mori dream space; the same statement is true for _2,n by <cit.>. In these and select other cases, the pseudo-effective cone of divisors has been shown to have infinitely many extremal cycles and thus is not rational polyhedral (<cit.>).

These results are possible due in large part to the following lemma, which plays an important role in Theorem <ref>. Here a moving curve 𝒞 in D is an irreducible effective curve 𝒞, the deformations of which cover a Zariski-dense subset of D.



Let D be an irreducible effective divisor in a projective variety X, and suppose that 𝒞 is a moving curve in D satisfying ∫_X[D]·[𝒞] < 0. Then [D] is rigid and extremal. □
 


Using Lemma <ref> to show a divisor D is rigid and extremal in fact shows more: if the lemma is satisfied, the boundary of the pseudo-effective cone is polyhedral at D. We do not rely on this fact, but see <cit.> for further discussion.


Lemma <ref> allows us to change a question about the pseudo-effective cone into one of intersection theory and provides a powerful tool in the study of divisor classes. Unfortunately, it fails to generalize to higher-codimension classes, where entirely different techniques are needed. Consequently, much less is known about ^d(_g,n) for d≥ 2. This paper is in part inspired by <cit.>, where the authors show that certain hyperelliptic classes of higher codimension are rigid and extremal in genus two. In <cit.>, the authors develop additional extremality criteria to show that in codimension-two there are infinitely many extremal cycles in _1,n for all n≥ 5 and in _2,n for all n≥ 2, as well as showing that two additional hyperelliptic classes of higher genus are extremal. These criteria cannot be used directly for the hyperelliptic classes we consider; this is illustrative of the difficulty of proving rigidity and extremality results for classes of codimension greater than one.




§ MAIN THEOREM



In this section we prove our main result, which culminates in Theorem <ref>. The proof proceeds via induction, with the base cases given in Theorem <ref>. We begin by defining hyperelliptic classes on _g,n.


Fix integers ℓ,m,n≥ 0. Denote by _g,ℓ,2m,n the closure of the locus of hyperelliptic curves in _g,ℓ+2m+n with marked Weierstrass points w_1,…,w_ℓ; pairs of marked points +_1,-_1,…,+_m,-_m with +_j and -_j conjugate under the hyperelliptic map; and free marked points p_1,…,p_n with no additional constraints. By hyperelliptic class, we mean a non-empty class equivalent to some [_g,ℓ,2m,n] in the Chow ring of _g,ℓ+2m+n.






Lemma <ref> allows us to establish the rigidity and extremality of the two divisor hyperelliptic classes for genus two, which together provide the base case for Theorem <ref>.




For n ≥ 0, the class of _2,0,2,n is rigid and extremal in ^1(_2,2+n) and the class of _2,1,0,n is rigid and extremal in ^1(_2,1+n).




Define a moving curve 𝒞 in _2,0,2,n by fixing a general genus-two curve C with n free marked points p_1,…,p_n and varying the conjugate pair (+,-). 

Since [_2,0,2,n] = π_p_n^*⋯π_p_1^*[_2,0,2,0], by the projection formula and the identity (see <cit.>) 

    [_2,0,2,0] = -λ + ψ_+ + ψ_- - 3δ_2,∅ - δ_1,∅,

we compute

    ∫__2,2+n[_2,0,2,n]· [𝒞]    = ∫__2,2[_2,0,2,0] ·π_p_1*⋯π_p_n*[𝒞] 
       = 0 + (4-2+6) + (4-2+6) - 3(6) - 0 
       = -2.

In particular, intersecting with λ is 0 by projection formula. Intersecting with either ψ-class can be seen as follows: pullback ψ_i from _2,1 to ψ_i - δ_2,∅, then use projection formula on ψ_i back to _2,1. This is just 2g-2, since ψ_i is the first Chern class of the cotangent bundle of C over i. The intersection with δ_2,∅ corresponds to the 2g+2 Weierstrass points. Finally, δ_1,∅ intersects trivially, since by fixing C we have only allowed rational tail degenerations. As _2,0,2,n is irreducible, it is rigid and extremal by Lemma <ref>.



We next apply Lemma <ref> by constructing a moving curve ℬ which intersects negatively with _2,1,0,n using the following diagram. Note that the image of s is precisely _2,1,0,n⊂_2,1+n.



[row sep=1cm, column sep=1cm] 
Adm_2 0,t_1,…,t_6,u_1±,…,u_n± [d, "c"] [r, "s"]     _2,1+n 

_0,{t_1,…,t_6,u_1,…,u_n} [d, "π_t_6"] 

_0,{t_1,…,t_5,u_1,…,u_n} 




Fix a generic point [x_n] in _0,{t_1,…,t_5,u_1,…,u_n} corresponding to a smooth marked curve and the point [b_n] in _0,{t_1,…,t_5,u_1,…,u_n} corresponding to a chain of ℙ^1s with n+3 components and marked points as shown in Figure <ref> (if n=0, t_4 and t_5 are on the final component; if n=1, t_5 and u_1 are on the final component; etc.), and define [𝒳_n] = s_*c^*π_t_6^*[x_n] and [ℬ_n] = s_*c^*π_t_6^*[b_n] (with an additional relabeling of t_1 to w_1 and u_j_- to p_j). Now 𝒳_n is a moving curve in _2,1,0,n, and the deformations of 𝒳_n are parametrized by _0,{t_1,…,t_5,u_1,…,u_n}.






Because the image of s is _2,1,0,n, the intersection [_2,1,0,n] · [𝒳_n] is not transverse, so we correct with minus the Euler class of the normal bundle of _2,1,0,n in _2,1+n restricted to 𝒳_n. Further, as all points in _0,{t_1,…,t_5,u_1,…,u_n} are equivalent, we may replace [𝒳_n] with [ℬ_n] in the intersection. In other words,

    ∫__2,1+n[_2,1,0,n] · [𝒳_n]    = ∫__2,1+n -π_p_n^*⋯π_p_1^*ψ_w_1·[𝒳_n] 
       = ∫__2,1+n -π_p_n^*⋯π_p_1^*ψ_w_1·[ℬ_n] 
       = ∫__2,1 -ψ_w_1·[ℬ_0].

By passing to the space of admissible covers (see, for example, <cit.>), this integral is seen to be a positive multiple (a power of two) of

    ∫__1,2 -ψ_w_1·[_1,2,0,0]    = ∫__1,2 -ψ_w_1· (3ψ_w_1) 
       = -1/8,

where we have used the fact that [_1,2,0,0] = 3ψ_w_1 <cit.>. Therefore, by Lemma <ref>, _2,1,0,n is rigid and extremal.


This establishes the base case for the inductive hypothesis in Theorem <ref>. The induction procedure differs fundamentally for the codimension-two classes, so we first prove the following short lemma to simplify the most complicated of those.



The class W_2,{p_1,…,p_n} is not proportional to [_2,2,0,n] on _2,2+n.


Let P = {p_1,…,p_n}. Note that in W_2,P the marked points w_1 and w_2 carry no special restrictions, and the class is of codimension two. Because the point w_1 is on a three-pointed rational component of the general element of W_2,P,

    ∫__2,2+n W_2,P·ψ_w_1^n+3   = ∫__0,3ψ_w_1^n+3 = 0.

However, using the equality

    [_2,2,0,0] = 6ψ_w_1ψ_w_2 - 3/2(ψ_w_1^2+ψ_w_2^2) - (ψ_w_1 + ψ_w_2)(21/10δ_1,{w_1} + 3/5δ_1,∅ + 1/20δ_irr)

established in <cit.> and Faber's Maple program <cit.>, we compute

    ∫__2,2+n[_2,2,0,n] ·ψ_w_1^n+3   = ∫__2,2+nπ_p_1^*⋯π_p_n^*[_2,2,0,0] ·ψ_w_1^n+3
       = ∫__2,2[_2,2,0,0] ·π_p_1*⋯π_p_n*ψ_w_1^n+3
       = ∫__2,2(6ψ_w_1ψ_w_2 - 3/2(ψ_w_1^2+ψ_w_2^2) 
        - (ψ_w_1 + ψ_w_2)(21/10δ_1,{w_1} + 3/5δ_1,∅ + 1/20δ_irr)) ·ψ_w_1^3 
       = 1/384,

so W_2,P is not a non-zero multiple of [_2,2,0,n].


We are now ready to prove our main result. The bulk of the effort is in establishing extremality, though the induction process does require rigidity at every step as well. Although we do not include it until the end, the reader is free to interpret the rigidity argument as being applied at each step of the induction.

The overall strategy of the extremality portion of the proof is as follows. Suppose [_2,ℓ,2m,n] is given an effective decomposition. We show (first for the classes of codimension at least three, then for those of codimension two) that any terms of this decomposition which survive pushforward by π_w_i or π_+_j must be proportional to the hyperelliptic class itself. Therefore we may write [_2,ℓ,2m,n] as an effective decomposition using only classes which vanish under pushforward by the forgetful morphisms; this is a contradiction, since the hyperelliptic class itself survives pushforward.



For ℓ,m,n≥ 0, the class _2,ℓ,2m,n, if non-empty, is rigid and extremal in ^ℓ+m(_2,ℓ+2m+n).




We induct on codimension; assume the claim holds when the class is codimension ℓ+m-1. Theorem <ref> is the base case, so we may further assume ℓ+m ≥ 2. Now, suppose that

    [_2,ℓ,2m,n] = ∑_s a_s[X_s] + ∑_t b_t[Y_t]

is an effective decomposition with [X_s] and [Y_t] irreducible codimension-(ℓ+m) effective cycles on _2,ℓ+2m+n, with [X_s] surviving pushforward by some π_w_i or π_+_j and [Y_t] vanishing under all such pushforwards, for each s and t.

Fix an [X_s] appearing in the right-hand side of (<ref>). If ℓ≠ 0, suppose without loss of generality (on the w_i) that π_w_1*[X_s] ≠ 0. Since 

    π_w_1*[_2,ℓ,2m,n] = (6-(ℓ-1))[_2,ℓ-1,2m,n]

is rigid and extremal by hypothesis, π_w_1*[X_s] is a positive multiple of the class of _2,ℓ-1,2m,n and X_s⊆ (π_w_1)^-1_2,ℓ-1,2m,n. By the commutativity of the following diagrams and the observation that hyperelliptic classes survive pushforward by all π_w_i and π_+_j, we have that π_w_i*[X_s] ≠ 0 and π_+_j*[X_s] ≠ 0 for all i and j.










[row sep=1cm, column sep=1cm] 
_2,ℓ,2m,n [d, "π_w_1"] [r, "π_+_j"]     _2,ℓ,2(m-1),n+1 [d, "π_w_1"]         _2,ℓ,2m,n [d, "π_w_1"] [r, "π_w_i"]     _2,ℓ-1,2m,n [d, "π_w_1"] 

_2,ℓ-1,2m,n [r, "π_+_j"]     _2,ℓ-1,2(m-1),n+1         _2,ℓ-1,2m,n [r, "π_w_i"]     _2,ℓ-2,2m,n




If ℓ = 0, suppose without loss of generality (on the +_j) that π_+_1*[X_s] ≠ 0. Then the same conclusion holds that [X_s] survives all pushforwards by π_+_j, since

    π_+_1*[_2,ℓ,2m,n] = [_2,ℓ,2(m-1),n+1]

is rigid and extremal by hypothesis, and π_+_1 commutes with π_+_j.

It follows that for any ℓ+m≥ 2

    X_s ⊆⋂_i,j((π_w_i)^-1_2,ℓ-1,2m,n∩ (π_+_j)^-1_2,ℓ,2(m-1),n+1).

We now have two cases. If ℓ+m ≥ 3, any ℓ + 2m - 1 non-free marked points in a general element of X_s are distinct Weierstrass or conjugate pair marked points, and hence all ℓ + 2m such non-free marked points in a general element of X_s are distinct Weierstrass or conjugate pair marked points. In this case we conclude that [X_s] is a positive multiple of [_2,ℓ,2m,n]. If ℓ+m = 2, we must analyze three subcases.



If ℓ = 0 and m = 2, then

    X_s⊆ (π_+_1)^-1_2,0,2,n+1∩ (π_+_2)^-1_2,0,2,n+1.

The modular interpretation of the intersection leaves three candidates for [X_s]: W_2,P or γ_1,P for some P containing neither conjugate pair, or [_2,0,4,n] itself. However, for the former two, we have W_2,P≠π_+_1(W_2,P) and γ_1,P≠π_+_1(γ_1,P) for all such P, contradicting our assumption that the class survived pushforward. Thus [X_s] is proportional to [_2,0,4,n].

If ℓ = 1 and m = 1, similar to the previous case, [X_s] could be [_2,1,2,n] or W_2,P or γ_1,P for some P containing neither the conjugate pair nor the Weierstrass point. However, if X_s is either of the latter cases, we have X_s ≠π_+_1(X_s), again contradicting our assumption about the non-vanishing of the pushforward, and so again [X_s] must be proportional to [_2,1,2,n].

If ℓ = 2 and m = 0, as before, [X_s] is either [_2,2,0,n] itself or W_2,P or γ_1,P for P = {p_1,…,p_n}. Now W_2,P = π_w_iW_2,P, so the argument given in the other subcases fails (though γ_1,P is still ruled out as before). Nevertheless, we claim that W_2,P cannot appear on the right-hand side of (<ref>) for _2,2,0,n; to show this we induct on the number of free marked points n. The base case of n=0 is established in <cit.>, so assume that _2,2,0,n-1 is rigid and extremal for some n≥ 1. Suppose for the sake of contradiction that

    [_2,2,0,n] = a_0W_2,P + ∑_s a_s[Z_s]

is an effective decomposition with each [Z_s] an irreducible codimension-two effective cycle on _2,2+n. Note that

    W_2,P = π_p_n^*W_2,P\{p_n} - W_2,P\{p_n}.

Multiply (<ref>) by  ω_p_n and push forward by π_p_n. On the left-hand side,

    π_p_n*(ω_p_n·[_2,2,0,n])    = π_p_n*(ω_p_n·π_p_n^*[_2,2,0,n-1]) 
       = π_p_n*(ω_p_n) ·[_2,2,0,n-1] 
       = 2 [_2,2,0,n-1],

having applied Lemma <ref>. Combining this with the right-hand side,

    2[_2,2,0,n-1]    = a_0π_p_n*(ω_p_n·π_p_n^*W_2,P\{p_n} - ω_p_n· W_2,P\{p_n}) + ∑_s a_sπ_p_n*(ω_p_n·[Z_s]) 
       = 2a_0W_2,P\{p_n} + π_p_n*(ω_p_n· W_2,P\{p_n}) + ∑_s a_sπ_p_n*(ω_p_n·[Z_s]).

The term π_p_n*(ω_p_n· W_2,P\{p_n}) vanishes by Lemma <ref>:

    π_p_n*(ω_p_n· W_2,P\{p_n-1})    = π_p_n*(ω_w_1· W_2,P\{p_n}) 
       = π_p_n*(π_p_n^*ω_w_1· W_2,P\{p_n}) 
       = ω_w_1·π_p_n* W_2,P\{p_n}
       = 0,

where w_1 is the Weierstrass singular point on the genus-two component of W_2,P\{p_n}. Altogether, we have

    2[_2,2,0,n-1]    = 2a_0W_2,P\{p_n} + ∑_s a_sπ_p_n*(ω_p_n·[Z_s]).




<cit.> establishes that ψ_p_n is semi-ample on _2,{p_n}, so ω_p_n is semi-ample, and hence this is an effective decomposition. By hypothesis, _2,2,0,n-1 is rigid and extremal, so W_2,P\{p_n} must be a non-zero multiple of [_2,2,0,n-1], which contradicts Lemma <ref>. Therefore W_2,P cannot appear as an [X_s] in (<ref>).

Thus for all cases of ℓ+m = 2 (and hence for all ℓ+m≥ 2), we conclude that each [X_s] in (<ref>) is a positive multiple of [_2,ℓ,2m,n]. Now subtract these [X_s] from (<ref>) and rescale, so that

    [_2,ℓ,2m,n] = ∑_t b_t[Y_t].

Recall that each [Y_t] is required to vanish under all π_w_i* and π_+_j*. But the pushforward of [_2,ℓ,2m,n] by any of these morphisms is non-zero, so there are no [Y_t] in (<ref>). Hence [_2,ℓ,2m,n] is extremal in ^ℓ+m(_2,ℓ+2m+n).

For rigidity, suppose that E:= r[_2,ℓ,2m,n] is effective. Since π_w_i*E = (6-(ℓ-1))r[_2,ℓ-1,2m,n] and π_+_j*E = r[_2,ℓ,2(m-1),n+1] are rigid and extremal for all i and j, we have that π_w_i*E is supported on _2,ℓ-1,2m,n and π_+_j*E is supported on _2,ℓ,2(m-1),n+1. This implies that E is supported on the intersection of (π_w_i)^-1[_2,ℓ-1,2m,n] and (π_+_j)^-1[_2,ℓ,2(m-1),n+1] for all i and j. Thus E is supported on _2,ℓ,2m,n, so [_2,ℓ,2m,n] is rigid.





§ HIGHER GENUS



The general form of the inductive argument in Theorem <ref> holds independent of genus for g≥ 2. However, for genus greater than one, the locus of hyperelliptic curves in _g is of codimension g-2, so that the base cases increase in codimension as g increases.  The challenge in showing the veracity of the claim for hyperelliptic classes in arbitrary genus is therefore wrapped up in establishing the base cases of codimension g-1 (corresponding to Theorem <ref>) and codimension g (corresponding to the three ℓ+m = 2 subcases in Theorem <ref>).

In particular, our proof of Theorem <ref> relies on the fact that _2,0,2,n and _2,1,0,n are divisors, and the subcase ℓ = 2 in Theorem <ref> depends on our ability to prove Lemma <ref>. This in turn requires the description of _2,2,0,0 given by <cit.>. More subtly, we require that ψ_p_n be semi-ample in _2,{p_n}, which is known to be false in genus greater than two in characteristic 0 <cit.>. In genus three, <cit.> show that the base case _3,1,0,0 is rigid and extremal, though it is unclear if their method will extend to _3,1,0,n. Moreover, little work has been done to establish the case of a single conjugate pair in genus three, and as the cycles move farther from divisorial classes, such analysis becomes increasingly more difficult.

One potential avenue to overcome these difficulties is suggested by work of Renzo Cavalieri and Nicola Tarasca <cit.>. They use an inductive process to describe hyperelliptic classes in terms of decorated graphs using the usual dual graph description of the tautological ring of _g,n. Such a formula for the three necessary base cases would allow for greatly simplified intersection-theoretic calculations, similar to those used in Theorem <ref> and Lemma <ref>. Though such a result would be insufficient to completely generalize our main theorem, it would be a promising start.

We also believe the observation that pushing forward and pulling back by forgetful morphisms moves hyperelliptic classes to (multiples of) hyperelliptic classes is a useful one. There is evidence that a more explicit connection between marked Weierstrass points, marked conjugate pairs, and the usual gluing morphisms between moduli spaces of marked curves exists as well, though concrete statements require a better understanding of higher genus hyperelliptic loci. Although it is known that hyperelliptic classes do not form a cohomological field theory over the full _g,n, a deeper study of the relationship between these classes and the natural morphisms among the moduli spaces may indicate a CohFT-like structure, which in turn would shed light on graph formulas or other additional properties.






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