The Integral Chow Ring of the Stack of Pointed Hyperelliptic Curves
Alberto Landi

	THE CHOW RING OF THE STACK OF POINTED HYPERELLIPTIC CURVES]THE INTEGRAL CHOW RING OF THE STACK OF POINTED HYPERELLIPTIC CURVES
	
	Scuola Normale Superiore, Piazza Cavalieri 7, 56126 Pisa, Italy
	alberto_landi@brown.edu
	
	
	
	
	
	
	
		We study the integral Chow ring of the stack ℋ_g,n parametrizing n-pointed smooth hyperelliptic curves of genus g. We compute the integral Chow ring of ℋ_g,n for n=1,2 completely, while for 3≤ n≤2g+2 we compute it up to the additive order of a single class in degree 2. We obtain partial results also for n=2g+3. In particular, taking g=2 and recalling that ℋ_2,n=ℳ_2,n, our results hold for CH^*(ℳ_2,n) for 1≤ n≤7.
	
	[
    ALBERTO LANDI
    April 29, 2024
==================



	

§ INTRODUCTION

	Since the seminal work of David Mumford <cit.>, several researchers have put their efforts in computing (both integral and rational) Chow rings of natural stacks of curves, as the stack ℳ_g,n of stable n-pointed curves of genus g, and its smooth counterpart ℳ_g,n.
	
	In this paper we are interested in the closed substack ℋ_g,n of ℳ_g,n classifying n-pointed smooth hyperelliptic curves of genus g.
	Notice that for g=2 we have ℳ_2,n=ℋ_g,2.
	
	Since a considerable number of papers have been written on the intersection theory of stacks, we recall only a few known results about ℋ_g,n and related stacks. In <cit.>, Eric Larson has computed the Chow ring of ℳ_2, while the Chow ring of ℳ_2 was previously computed by Angelo Vistoli in <cit.>. This last result has been generalized to ℋ_g by Dan Edidin and Damiano Fulghesu in <cit.> for the even g case, and by Andrea Di Lorenzo in <cit.> for the odd g case. The pointed case was first addressed by Michele Pernice in <cit.>, where he studied the n=1 case. However, the result is not completely correct, due to a mistake at the end of the proof; in this article we supplement his ideas to correct it. Finally, Dan Edidin and Zhengning Hu studied the locus of ℋ_g,n where all the sections have image in the Weierstrass divisor (in particular, 1≤ n≤2g+2), and they have computed the Chow rings of those loci. We will use their results in this paper. Samir Canning and Hannah Larson (<cit.>) have computed the rational Chow ring of ℋ_g,n for n≤2g+6, using a different description of the stack.
	
	We compute the integral Chow ring of ℋ_g,n for n=1,2 completely, while for 3≤ n≤2g+3 we find the generators and almost every relation. As a consequence, we recover <cit.> when n≤2g+3.
	
	All our results are based on the techniques of <cit.>, where a new description of ℋ_g,n is found and used to compute its Picard groups for all g and n.
	

 §.§ Statement of the results and strategy

	We work over a field of characteristic strictly greater then 2g, where g≥2 is the genus. The main result of this paper is the (almost) computation of the integral Chow ring of ℋ_g,n for 1≤ n≤2g+3. This ring has already been computed in the case n=1 in <cit.> by Pernice, but the result is not correct as stated; we correct it by extending his ideas. The presentation of CH^*(ℋ_g,1) with the most geometric meaning that we get is described in the following Theorem; see the first Section for the definitions of the various classes.
	
		We use the following notation. If we have a ring A and elements β_1,…,β_s∈ A, and p_1,…,p_n∈ℤ[x_1,…,x_s] are polynomials in s variables, we write
		
    A≃ℤ[β_1,…,β_s]/(p_1(β_1,…,β_s),…,p_r(β_1,…,β_s))

		to mean that there is a surjective homomorphism
		
    ℤ[x_1,…,x_s][r,twoheadrightarrow]     A

		sending x_i to α_i and with kernel generated by p_1,…,p_r.
		
		Moreover, we use the square brackets to denote the Chow class associated to the closed subscheme written between them.
	
	
		Let k>2g. Then,
		
    CH^*(ℋ_g,1)≃ℤ[ψ_1,[𝒲_g,n^1]]/((4g+2)((g+1)ψ_1-(g-1)[𝒲_g,1^1]),[𝒲_g,1^1]^2+ψ_1·[𝒲_g,1^1])
			
		regardless of the parity of g.
	
	In general, we show that CH^*(ℋ_g,n) is generated in degree 1 if 1≤ n≤2g+3.
	
		For all 1≤ n≤2g+3, CH^*(ℋ_g,n) is generated in degree 1.
	
	The main idea is to use an exact sequence
	
    ⊕_i<jCH^*(ℋ_g,n-1^i,j)[r]    CH^*(ℋ_g,n)[r]    CH^*(ℋ_g,n^far)[r]     0

	where ℋ_g,n-1^i,j is an open substack of ℋ_g,n-1, and ℋ_g,n^far is an open substack of ℋ_g,n, whose Chow ring is computable for 2≤ n≤2g+3 (not completely for n=2g+3). Using induction on n and the result for n=1, one gets that each piece on the left and right is generated in degree 1, and concludes that the same holds for the piece in the middle.
	
	Moreover, one can find numerous relations between Chow classes using the results in <cit.> on the Picard group of ℋ_g,n. All this can be used to compute the Chow ring in the case n=2.
	
		Let g>2g. Then, CH^*(ℋ_g,2) is generated by the classes of ψ_1, 𝒲_g,2^1 and 𝒵_g,2^1,2, and the subgroup of relations is generated by
		2pt
		
    (4g+2)((g+1)ψ_1-(g-1)[𝒲_g,2^1]),
     [𝒵_g,2^1,2]^2+ψ_1·[𝒵_g,2^1,2],  [𝒲_g,2^1]^2+ψ_1·[𝒲_g,2^1],  
    			[𝒵_g,2^1,2]·[𝒲_g,2^1],
    
    			(2g+1)(g+1)ψ_1^2+(2g+1)(3g-1)[𝒲_g,2^1]^2-(2g+1)(g+1)[𝒵_g,2^1,2]^2.

		
	Unfortunately, the relations found are not enough to determine the integral Chow ring in the case of 3≤ n≤2g+2. The only piece missing is the additive order of the degree 2 class [𝒵_g,n^1,2]^2. We summarize the results here.
	
		Let k>2g. Then, CH^*(ℋ_g,3) is generated by the classes of 𝒲_g,3^1, 𝒵_g,3^1,2, 𝒵_g,3^1,3 and 𝒵_g,3^2,3, and the ideal of relations is generated by
		2pt
		
    (8g+4)[𝒲_g,3^1]-(4g+2)(g+1)([𝒵_g,3^1,2]+[𝒵_g,3^1,3]-[𝒵_g,3^2,3]),
     [𝒵_g,3^1,2]·[𝒵_g,3^1,3],  [𝒵_g,3^1,2]·[𝒵_g,3^2,3],  [𝒵_g,3^1,3]·[𝒵_g,3^2,3],  
    			[𝒵_g,3^1,j]·[𝒲_g,3^1] for j=2,3,
    
    			[𝒲_g,3^1]·[𝒵_g,3^2,3]+(g+1)[𝒵_g,3^2,3]^2,  2[𝒲_g,3^1]^2-(g+1)[𝒵_g,3^2,3]^2,
    
    			(2g+1)(g+1)([𝒵_g,3^i,j]^2+[𝒵_g,3^i',j']^2) for i≠j,i'≠j',   b_g[𝒵_g,3^i,j]^2 for i≠j.

		where b_g is either (2g+1)(g+1) or (4g+2)(g+1).
	
	
		Let k>2g. Then, CH^*(ℋ_g,4) is generated by the classes of 𝒲_g,4^1, 𝒵_g,4^1,2, 𝒵_g,4^1,3, 𝒵_g,4^1,4 and 𝒵_g,4^2,3, and the ideal of relations is generated by
		2pt
		
    (8g+4)[𝒲_g,4^1]-(4g+2)(g+1)([𝒵_g,4^1,2]+[𝒵_g,4^1,3]-[𝒵_g,4^2,3]),
     [𝒵_g,4^1,i]·[𝒵_g,4^1,j] for 1<i<j,  [𝒵_g,4^1,j]·[𝒵_g,4^2,3] fpr j=2,3,  
    			[𝒵_g,4^1,j]·[𝒲_g,3^1] for j>1,
    
    			[𝒵_g,4^2,3]·[𝒲_g,3^1]+(g+1)[𝒵_g,4^2,3]^2,  [𝒵_g,4^2,3]^2=-[𝒵_g,4^2,3]·[𝒵_g,4^1,4]=[𝒵_g,4^1,4]^2,
    
    			2[𝒲_g,4^1]^2-(g+1)[𝒵_g,4^2,3]^2,  
    			(g+1)[𝒵_g,4^i,j]^3 for all i≠j,
    
    			(g+1)([𝒵_g,4^i,j]^2-[𝒵_g,4^i',j']^2) for all i≠j,i'≠j',   b_g[𝒵_g,4^i,j]^2 for all i≠j

		for some integer b_g=b'_g(g+1) such that b'_g divides 4g+2.
	
	
		Let k>2g and 5≤ n≤2g+2. Then, CH^*(ℋ_g,n) is generated by the classes of 𝒲_g,n^1, 𝒵_g,n^1,i for 1<i≤ n and 𝒵_g,n^2,3, and the ideal of relations is generated by
		2pt
		
    (8g+4)[𝒲_g,n^1]-(4g+2)(g+1)([𝒵_g,n^1,2]+[𝒵_g,n^1,3]-[𝒵_g,n^2,3]),
     [𝒵_g,n^1,i]·[𝒵_g,n^1,j] for 1<i<j,  [𝒵_g,n^1,j]·[𝒵_g,n^2,3] for j=2,3,  
    			[𝒵_g,n^1,j]·[𝒲_g,n^1] for j≠1,
    
    			[𝒵_g,n^2,3]·[𝒲_g,n^1]+(g+1)[𝒵_g,n^2,3]^2,  2[𝒲_g,n^1]^2-(g+1)[𝒵_g,n^2,3]^2,
    
    			[𝒵_g,n^2,3]^2=-[𝒵_g,n^2,3]·[𝒵_g,n^1,i]=[𝒵_g,n^1,j]^2 for all i>3 and j≠1,
    
    			b_g[𝒵_g,n^i,j]^2 for all i≠j

		where b_g is a positive integer dividing (4g+2)(g+1).
	
	Notice that we have some bounds on the order of [𝒵_g,n^2,3]^2, but we do not manage to find the order exactly. Notice that [𝒵_g,n^2,3]^2=-[𝒵_g,n^1,4]·[𝒵_g,n^2,3], for n≥4, so in that case we are dealing with a proper intersection. It is possible that the implementation of some techniques such as higher Chow groups can be used to compute it, but it does not seem immediate. This technique is complicated to use as the first higher Chow group of ℋ_g,n^far is non-zero.
	
	The case of 2g+3 is less well behaved. The Chow ring of ℋ_g,n is very different from those for n≤2g+2, since this stack is now a scheme, thus the Chow ring has to be 0 in degree greater than its dimension.
	
		Let k>2g and n=2g+3. Then, CH^*(ℋ_g,2g+3) is generated by the classes of  𝒲_g,2g+3^1, 𝒵_g,2g+3^1,i for 1<i≤2g+3 and 𝒵_g,2g+3^2,3, and the ideal of relations is generated by
		2pt
		
    (8g+4)[W_g,2g+3^1]-(4g+2)(g+1)([𝒵_g,2g+3^1,2]+[𝒵_g,2g+3^1,3]-[𝒵_g,2g+3^2,3]),
     [𝒵_g,2g+3^1,i]·[𝒵_g,2g+3^1,j] for 1<i<j,  [𝒵_g,2g+3^1,j]·[𝒵_g,2g+3^2,3] for j=2,3,
    
    			[𝒵_g,2g+3^1,j]·[𝒲_g,2g+3^1] for j≠1,  [𝒵_g,2g+3^2,3]·[𝒲_g,2g+3^1]+(g+1)[𝒵_g,2g+3^2,3]^2,
    
    			2[𝒲_g,2g+3^1]^2-(g+1)[𝒵_g,2g+3^2,3]^2,
    
    			[𝒵_g,2g+3^2,3]^2=-[𝒵_g,2g+3^2,3]·[𝒵_g,2g+3^1,i]=[𝒵_g,2g+3^1,j]^2 for all i>3 and j≠1,
    
    			b_g[𝒵_g,n^i,j]^2 for all i≠j,  [𝒲_g,2g+3^1]^c_g

		where b_g is a positive integer dividing (4g+2)(g+1), and 1<c_g≤2g+2.
	
	The only other thing not known in this case is the multiplicative order of [𝒲_g,2g+2^1]. This should be easier to be found than the additive order of [𝒵_g,2g+3^1,2]^2.
	

 §.§ Structure of the paper

	In the first Section we recall some results obtained in <cit.>, which are essential for the rest of the paper.
	In the second and third Section we compute the Chow rings of ℋ_g,1 and ℋ_g,2^far, respectively. Next, in Section 4, we show that for 1≤ n≤2g+3 the Chow ring of ℋ_g,n is generated in degree 1 and we compute some relations. These are enough to compute the Chow ring of ℋ_g,2, as shown in Section 5. In Section 6, we compute ℋ_g,n^far for 3≤ n≤2g+2, and we conclude the computation of ℋ_g,n for n≤2g+3, except for the additive order of the degree 2 torsion class [𝒵_g,n^1,2]^2, and the multiplicative order of [𝒲_g,2g+3^1] in the case n=2g+3.
	

 §.§ Acknowledgements

	I would like to thank Angelo Vistoli for the valuable time spent discussing ideas on this and related subjects. I am also grateful to Dan Edidin and Zhengning Hu for the useful discussions we had. I wish to warmly thank Michele Pernice for his support and taking the time to talk about this work.
	

§ PRELIMINARIES AND THE GEOMETRY OF ℋ_G,N

	We will use the description of ℋ_g,n obtained in <cit.>, where the author computed the Picard group of ℋ_g,n for all g and n. We recall the description and results here. We will always assume that the characteristic of the base field does not divide 2g+2.
	

 §.§ Description of ℋ_g,n

	
		Let
		
    𝒞_g[r,"f"]    𝒫_g[r,"π"]    ℋ_g

		be the universal family of n-hyperelliptic curves, ramifying over the `universal' Brauer-Severi stack 𝒫_g→ℋ_g, and let α be the universal hyperelliptic involution on 𝒞_g. Let 𝒞_g^n be the fiber product of n copies of 𝒞_g over ℋ_g, and Δ_i,j be the i,j-th diagonal. Notice that ℋ_g,n is the open substack of 𝒞_g^n given by the complement of the extended diagonal. For all i≠j, define 𝒵_g,n^i,j⊂ℋ_g,n as the restriction of α_i^-1(Δ_i,j) to ℋ_g,n, which is a closed substack of ℋ_g,n. Here, α_i:𝒞_g^n→𝒞_g^n is the involution which acts as α on the i-th component, and as the identity on the others.
		Finally, define ℋ_g,n^far as the complement of the 𝒵_g,n^i,j's in ℋ_g,n.
		
		Denote with
		
    𝒞_g,n[r,"f"]    𝒫_g,n[r,"π"]    ℋ_g,n

		the universal family of pointed hyperelliptic curves over ℋ_g,n, which ramifies over 𝒫_g,n.
		Consider the universal i-th section σ_i:ℋ_g,n→𝒞_g,n, which induces a section σ_i=f∘σ_i of 𝒫_g,n→ℋ_g,n. Then, 𝒵_g,n^i,j is exactly the locus where σ_i=σ_j, i.e. σ_i and σ_j differ by the hyperelliptic involution.
	
	
		Recall that ℋ_g,n is integral and smooth for all g≥2 and all n. From the construction it follows that 𝒵_g,n^i,j is integral and smooth for all i≠j, and that 𝒵_g,n^i,j=𝒵_g,n^j,i. For this reason, we usually assume i<j. Moreover, the intersection of 𝒵_g,n^i,j and 𝒵_g,n^l,m is empty if and only if {i,j} and {l,m} intersect properly.
		See <cit.> for the details in a more general setting.
	
	The 𝒵_g,n^i,j define Cartier divisors on ℋ_g,n, whose classes are subject to the following relations.
	
		Let 1≠i<j. Then,
		
    [𝒵_g,n^i,j]=[𝒵_g,n^1,i]+[𝒵_g,n^1,j]+[𝒵_g,n^2,3]-[𝒵_g,n^1,2]-[𝒵_g,n^1,3].

		Moreover, there are no other relations between those divisors.
	
	
		See <cit.>.
	
	Now, we study the open substack ℋ_g,n^far⊂ℋ_g,n. For n≤2g+3, this is isomorphic to the quotient of an open subscheme of a representation of a linear algebraic group by the same group. We report here the precise statements. See <cit.> for details.
	
		It holds that
		
    ℋ_g≃[𝔸^2g+3∖Δ_g/GL_2/μ_g+1]

		where Δ_g is the locus of singular forms and a matrix A acts on a form f of degree 2g+2 as
		
    A· f(x,y)=f(A^-1(x,y)).

	
	
		We use the following notation. Let 3≤ n≤2g+3 and take a point
		
    (a_0,…,a_2g+3-n,s_1,s_2,s_3,t_1,…,t_n-3,p_1,…,p_n-3)∈𝔸^2g+3×(𝔸^n-3∖Δ)

		where Δ is the locus where p_1,…,p_n-3 are not mutually distinct and different from 0 and 1. Than, we think at the point as the data of a polynomial
		
    f(x,y)=a_0x^2g+2+a_1x^2g+1y+…+a_2g+2y^2g+2,

		where the there is no abuse of notation regarding the a_i's, such that
		
    f(0,1)=s_1^2,   f(1,0)=s_2^2,   f(1,1)=s_3^2,   f(1,p_i)=t_i^2 for all i.

		The polynomial f is clearly uniquely determined, see also <cit.>. We use a similar convention for n≤3.
	
	
		It holds that
		
    ℋ_g,1≃[𝔸^2g+3∖Δ_g,1/B_2/μ_g+1]

		where Δ_g,1 is the locus corresponding to singular forms, and a lower triangular matrix A=[ a 0; b c ] acts on (f,s_1) as
		
    A· (f,s_1)=(f(A^-1),c^-(g+1)s_1).

		
		Similarly,
		
    ℋ_g,2^far≃[𝔸^2g+3∖Δ_g,2/(𝔾_m×𝔾_m)/μ_g+1]

		where where Δ_g,2 is the locus corresponding to singular forms, and A=(a,c) acts on (f,s_1,s_2) as
		
    A· (f,s_1,s_2)=(f(A^-1),c^-(g+1)s_1,a^-(g+1)s_2).

	
	
		Let 3≤ n≤ 2g+3. Then
		
    ℋ_g,n^far≃(𝒫(2^2g+3-n,1^n)×(𝔸^n-3∖Δ))∖Δ_g,n

		where 𝒫(2^2g+3-n,1^n) is the weighted projective stack where the first 2g+3-n variables have degree 2, while the last n have degree 1, and Δ_g,n is the locus corresponding to singular forms. In particular, for n=2g+3,
		
    ℋ_g,2g+3^far≃(ℙ^2g+2×(𝔸^2g∖Δ))∖Δ_g,2g+3

		and it is a scheme.
	
	We will use these presentations to compute their Chow ring.
	We conclude the subsection defining one last important divisor of ℋ_g,n.
	
		Let n≥1. For all 1≤ i≤ n, define 𝒲_g,n^i to be the divisor of ℋ_g,n given by the points whose images under σ_i are in the ramification locus of the uniform cyclic covers parametrized by those points. Rigorously, it is defined as follows. Recall that 𝒞_g^n is the n-fold fibred product of 𝒞_g over ℋ_g. The ramification locus of the map 𝒞_g→𝒫_g is a closed substack
		
    𝒲_g,1^1⊂𝒞_g=ℋ_g,1

		which is étale of degree 2g+2 over ℋ_g. Then, for n≥2, define
		
    𝒲_g,n^1:=(𝒲_g,1^1×_ℋ_g𝒞_g^n-1)∩ℋ_g,n⊂ℋ_g,n⊂𝒞_g^n.

		For every other i the definition of 𝒲_g,n^i is analogous; alternatively, one can consider the automorphism of ℋ_g,n which exchanges the i-th section with the first.
	
	

 §.§ The Picard group and classes of divisors

	For our computation it will be crucial to know the Picard group of ℋ_g,n, which has been computed in <cit.> in a more general setting. We report the results here.
	
		Suppose that the ground field k is of characteristic not dividing 2g+2.
		
			
  * If g is odd, then  Pic(ℋ_g,n)≃ℤ^n⊕ℤ/(8g+4)ℤ.
			
  * If g is even, then  Pic(ℋ_g,n)≃ℤ^n⊕ℤ/(4g+2)ℤ.
		
		Moreover, the torsion part always comes from ℋ_g.
	
	For n≥3, it is easy to describe the generators of the Picard group.
	
		Let n≥3 and assume that the characteristic of k does not divide 2g+2. Then
		
    {[𝒵_g,n^1,j]}_2≤ j≤ n∪{[𝒵_g,n^2,3]}∪{[𝒲_g,n^1]}

		is a minimal set of generators of Pic(ℋ_g,n). Moreover,
		
    (8g+4)[𝒲_g,n^1]=(4g+2)(g+1)([𝒵_g,n^1,2]+[𝒵_g,n^1,3]-[𝒵_g,n^2,3])

		is the only relation, regardless of the parity of g.
	
	To address the case of n≤2 it is important to compute the classes of 𝒲_g,n^1 and ψ_1 in terms of a fixed base of Pic(ℋ_g,1). Recall that the psi class ψ_i is defined as the pullback of the relative differentials of the universal family under the i-th section.
	We use the conventions of Theorem 3.2 of <cit.>.
	
		Using the conventions of <cit.>, if g is odd then [𝒲_g,1^1] in Pic(ℋ_g,1) corresponds to
		
    (-(g+1)/2,1)∈ℤ⊕ℤ/(8g+4)ℤ

		while if g is even then it corresponds to
		
    (-(g+1),g/2+1)∈ℤ⊕ℤ/(4g+2)ℤ.

		The same holds for [𝒲_g,2^1] in Pic(ℋ_g,2^far).
		
		Similarly, if g is odd then [𝒲_g,2^2] in Pic(ℋ_g,2^far) corresponds to
		
    ((g+1)/2,1)∈ℤ⊕ℤ/(8g+4)ℤ

		while if g is even then it corresponds to
		
    (g+1,-g/2)∈ℤ⊕ℤ/(4g+2)ℤ.

	
	
		The first psi class ψ_1 in Pic(ℋ_g,1) corresponds to
		
    (-(g-1)/2,1)∈ℤ⊕ℤ/(8g+4)ℤ

		if g is odd, while if g is even it corresponds to
		
    (-(g-1),g/2)∈ℤ⊕ℤ/(4g+2)ℤ.

	
	
		Consider the linear group B_2×𝔾_m and its action on 𝔸^2g+2×𝔸^1×𝔸^3 described as follows. Let B∈B_2 and λ∈𝔾_m, and consider (a_0,…,a_2g+1,s_1,x_0,x_1,y) an element of 𝔸^2g+2×𝔸^1×𝔸^3, which we also denote by (f,s_1,x_0,x_1,y) with the usual convention. Then the action is defined by
		
    (B,λ)·(f,s_1,x_0,x_1,y)=(λ^-2f∘ B^-1,λ^-1c^-(g+1)s_1,B(x_0,x_1),λ^-1y)

		where
		
    B=[ a 0; b c ]

		and B(x_0,x_1)=(ax_0,bx_0+cx_1).
		Notice that we have homomorphisms
		
    B_2/μ_g+1[r]    B_2×𝔾_m[r]    B_2/μ_g+1

		sending [A]=[ a 0; b c ] to (c^-1A,c^g+1), which is easily seen to be well defined, and sending (B,λ) to the class of λ^1/(g+1)B. Notice that the composite is just the identity. Moreover, we can consider the morphism
		
    𝔸^2g+2×𝔸^1×𝔸^3[r]    𝔸^2g+2×𝔸^1

		simply given by the projection, which admits the obvious section
		
    𝔸^2g+2×𝔸^1[r]    𝔸^2g+2×𝔸^1×𝔸^3

		sending (f,s_1) to (f,s_1,0,1,s_1). Notice that both maps are equivariant with respect to the action of the two linear groups and the homomorphisms between them. Moreover, this last section has image in the closed subscheme X' of 𝔸^2g+2×𝔸^1×𝔸^3 defined by the equation f(x_0,x_1)=y^2. Notice that this locus is (B_2×𝔾_m)-invariant. Excising the locus of singular polynomials, we get an open invariant subscheme X of X'. It is not hard to show that the induced morphism
		
    φ:[X/B_2×𝔾_m][r]    ℋ_g,1

		is the universal family over ℋ_g,1. We can use this construction to compute the class of ψ_1 in Pic(ℋ_g,1).
		
		Notice that, using the exact sequence of differentials, the module of relative differentials of the morphism φ seen at the level of schemes, i.e. without passing to the quotient, is given by
		
    dx_0⊕ dx_1⊕ dy/⟨ f_x_0dx_0+f_x_1dx_1-2ydy⟩.

		Restricting to the section, we get
		
    dx_0⊕ dx_1⊕ dy/⟨ a_2g+1dx_0+(2g+2)s_1^2dx_1-2s_1dy⟩

		which is a rank 2 vector bundle. Take the determinant, which is generated by dx_0∧ dy over s_1≠0, and by dx_1∧ dy over a_2g+1≠0. Notice that, when both s_1 and a_2g+1 are non-zero,
		
    1/s_1^2dx_0∧ dy=2g+2/a_2g+2dx_1∧ dy.

		This defines a non-vanishing regular section (recall that the characteristic of the base field does not divide 2g+2), hence a trivialization of the pullback of the dual sheaf. Now, (dx_0∧ dy)/s_1^2 is an invariant section generating the line bundle. Notice that the action of B_2/μ_g+1 on it is via a^-1c^-g. We conclude by applying the linear base changes as in <cit.>.
	
	Together, the two results above yield the following Proposition.
	
		The classes ψ_1 and [𝒲_g,1^1] form a minimal set of generators of Pic(ℋ_g,1), and
		
    (4g+2)(g+1)ψ_1=(4g+2)(g-1)[𝒲_g,1^1]

		is the only relation. Adjoining [𝒵_g,2^1,2] one gets a minimal set of generators of Pic(ℋ_g,2), with the same relation.
	
	

§ THE CHOW RING OF ℋ_G,1

	In this Section we extend and correct the ideas of Pernice in <cit.> to compute the Chow ring of ℋ_g,1. The only difference in the result is the relation in degree 2, which essentially amounts to `half' of the one of Pernice in the even genus case, and to `one fourth' in the odd genus case.
	
	We begin by recalling some facts about Chow envelopes. For the proofs and more details see <cit.> and <cit.>; for the general theory on equivariant Chow rings, see <cit.>.
	
		Let f:𝒳→𝒴 be a proper, representable morphism of quotient stacks. We say that f is Chow-surjective if the morphism of groups
		
    f_*:CH_*(X)[r]    CH_*(Y)

		is surjective.
		
		We say that a morphism of algebraic stacks f:𝒳→𝒴 is a Chow envelope if f(K):𝒳(K)→𝒴(K) is essentially surjective for every extension of fields K/k.
	
	
		Let G be a group scheme over k and suppose we have an action on two algebraic spaces X and Y and a map f:X→ Y which is G-equivariant. We denote by f^G the induced morphism of quotient stacks and we assume it is proper and representable. If f^G is a Chow envelope, then f^G is Chow-surjective.
	
	
		If T is a special group acting on an algebraic space X, then the T-torsor X→[X/T] is a Chow envelope.
	
	
		Let G, T be two groups schemes over k and suppose we have an action of G× T on two algebraic spaces X and Y and a map f:X→ Y which is G× T-equivariant. Assume that T is a special group. Suppose that f^G is a Chow envelope and f^G× T is a proper representable morphism. Then f^G× T:[X/(G× T)]→[Y/(G× T)] is Chow-surjective.
	
	We set N=2g+2. Thanks to B_2/μ_g+1 being an extension of the maximal torus and a unipotent subgroup, using Proposition <ref> we reduce to the computation of the (𝔾_m×𝔾_m)/μ_g+1-equivariant Chow ring of (𝔸^2g+2×𝔸^1)∖Δ_g,1.
	
	Notice that there are isomorphisms
	
    (𝔾_m×𝔾_m)/μ_g+1[r,"φ"]    𝔾_m×𝔾_m[r,"φ_even"][d,"φ_odd"]    𝔾_m×𝔾_m
       𝔾_m×𝔾_m

	where
	
    φ(a,c)=(ac^-1,c^g+1), φ_even(λ,μ)=(λ^g/2+1μ,λ^g/2μ), φ_odd(λ,μ)=(λ^(g+1)/2μ,λ).

	The two composites are the isomorphisms induced by those in <cit.> and <cit.>, depending on the parity of the genus. Using these last identifications, for both B_2/μ_g+1 and GL_2/μ_g+1, the map
	
    𝔸^2g+3∖Δ_g,1∋(f,s_1)[r]     f∈𝔸^2g+3∖Δ_g

	inducing
	
    ℋ_g,1[r]    ℋ_g

	obtained by forgetting the section is equivariant in a natural way. We tacitly consider these identifications, depending on the parity of the genus.
	
	Since we want to use the theory of Chow envelopes, we pass to the projective setting. To do so, we consider the quotient (𝔸^2g+2×𝔸^1)∖0 by the action of 𝔾_m with weights (2,…,2,1), and let Δ_g,1 be the image of Δ_g,1 under the quotient. Notice that using the isomorphisms in (<ref>), we have to compute the quotient of the (𝔾_m×𝔾_m)-equivariant Chow ring of 𝒫(2^N,1) by the image of the (𝔾_m×𝔾_m)-equivariant Chow ring of Δ_g,1 under the inclusion. Then, one recovers the Chow ring of ℋ_g,1 simply by substituting the first Chern class ξ of the tautological bundle of 𝒫(2^N,1) with an expression in the obvious characters t_1 and t_2 of 𝔾_m×𝔾_m (the expression depends on the parity of g).
	
	
		Using the isomorphism φ in (<ref>), the action of 𝔾_m×𝔾_m on the space (𝔸^2g+2×𝔸^1)∖Δ_g,1 is with weights (-(2g+2),…,-1,0) and (-2,…,-2,-1), respectively. Therefore, one could simply study the quotient of the 𝔾_m-equivariant Chow ring of 𝒫(2^N,1) by the image of the 𝔾_m-equivariant Chow ring of Δ_g,1 under the inclusion, where 𝔾_m is the first component the product group scheme. Then, the first Chern class ξ of the tautological bundle of 𝒫(2^N,1) would correspond to the character -t_2 of the second factor in 𝔾_m×𝔾_m. However, in order to simplify the comparison between the Chow rings of ℋ_g,1 and ℋ_g, it is easier to consider (𝔾_m×𝔾_m)-equivariant Chow rings as explained above.
			
	The dependence on the parity of the genus is very annoying, but we can ignore it in some sense, thanks to the following Remark.
	
		We will follow and correct the strategy used by Pernice in <cit.>, which is based on comparing the Chow ring of ℋ_g,1 with the one of ℋ_g. Recall that, when g is even, GL_2/μ_g+1 is isomorphic to GL_2, while in the odd case it is isomorphic to 𝔾_m×PGL_2. The first is special, hence when computing CH^*(ℋ_g) one can reduce to the maximal torus T, while this cannot be done in the odd case, as PGL_2 is not special. Hence the strategy for the computation of the Chow ring of ℋ_g depends on the parity of g, see <cit.> and <cit.>. However, using the isomorphism φ in (<ref>), we get that the presentation of the T-equivariant Chow ring of 𝔸^2g+3∖Δ_g (see Proposition <ref>) does not depend on the parity of g. Indeed, one can notice that `half' of the generating degree 2 relation in <cit.> is a relation in the T-equivariant Chow ring. Thus, the map obtained by reducing to the maximal torus is not injective when g is odd, as expected. This observation allow us to ignore the parity of the genus. For CH^*(ℋ_g,1), we will show that in degree 2 the generating relation is `half' of the one in the even case, hence a `fourth' of the one in the odd case.
	
	We will use a slightly different notation from the one in <cit.>. Recall that Pernice has constructed maps
	
    [row sep=small]
    		a_r:ℙ^r×𝒫(2^N-2r,1)[r]    𝒫(2^N,1)
    
    		b_r:ℙ^r-1×𝒫(2^N-2r)[r]    𝒫(2^N,1)

	that factor through Δ_g,1, and he proved that a_r and b_r together give a Chow envelope of Δ_g,1; see <cit.>. The first map sends (f,(g,s_1))to (f^2g,f(0,1)s_1), where f is a polynomial of degree r+1 and g a polynomial of degree N-2r+1 such that g(0,1)=s_1^2,  while the second sends (f,g) to (x^2f^2g,0), with a similar notation.
	
	Moreover, he has shown that the image of the pushforward under each of those maps is generated by α_r,i:=(a_r)_*ξ_r^i and β_r,i:=(b_r)_*ξ_r-1^i, respectively, with i varying. In the first case, ξ_r is the pullback to ℙ^r×𝒫(2^N-2r,1) of the first Chern class of 𝒪_ℙ^r(1), while in the second case ξ_r-1 is the pullback to ℙ^r-1×𝒫(2^N-2r+1) of the first Chern class of 𝒪_ℙ^r-1(1).
	

 §.§ Problems with the previous proof

	The main problem of the computation in <cit.> is the following. The idea in that article is to consider cartesian diagrams
	
    X[r][d,"ψ"]    𝒫(2^N,1)[d,"ϕ"]
    ℙ^r×ℙ^N-2r[r,"π_r"]    ℙ^N

	and
	
    Y[r][d,"ψ'"]    𝒫(2^N,1)[d,"ϕ"]
    ℙ^r-1×ℙ^N-2r[r,"π_r'"]    ℙ^N

	where the first bottom horizontal map is part of a (𝔾_m×𝔾_m)-Chow envelope of Δ_g, see <cit.>, sending (f,g) to f^2g, while the second bottom horizontal map sends (f,g) to x_0^2f^2g.
	Then one considers the induced map from ℙ^r×𝒫(2^N-2r,1) to X, which factors a_r, and the one from ℙ^r-1×𝒫(2^N-2r) to Y, which factors b_r. These maps are again proper and representable. One then shows that the pushforward of 1 under those maps is again 1, and uses compatibility of proper pushforward and flat pullback of <cit.> to conclude that everything comes from ϕ^*, if 1 is in the image of the pullback under ψ'. While it is true that ϕ is flat, see Proposition <ref>, the pullback of 1 under ψ' is not necessarily 1. Indeed, for r=1, we will see that Y is irreducible but not smooth, not even generically reduced. Precisely, the fundamental class [Y] in CH_*(Y) corresponds to 2.
	Moreover, notice that the behaviour of ϕ is very similar to the root stack √((𝒪(1),a_2g+2)/ℙ^N) over ℙ^N. Indeed, over the locus where a_2g+2≠0, the map ϕ is an isomorphism, while over the zero locus it is ramified of order 2, and the automorphism groups are the same to those of the root stack. We show that they are indeed equal; this proves also the above facts. The result is slightly more general.
	
		Let N and r be positive integers. The morphism
		
    ϕ:𝒫(r^N,1)[r]    ℙ^N

		identifies 𝒫(r^N,1) with the r-th root stack √((𝒪(1),a_2g+2)/ℙ^N) over ℙ^N. In particular, ϕ is flat.
	
	
		The above discussion shows that the map ϕ:𝒫(r^N,1)→ℙ^N factors as
		
    𝒫(r^N,1)[r,"ψ"]    √((𝒪(1),a_2g+2)/ℙ^N)[r]    ℙ^N

		and ψ is an isomorphism over the locus where a_2g+2≠0. Moreover, ψ induces an isomorphism between the automorphism groups of geometric points, hence it is representable. Being also proper and birational between smooth stacks, ψ is an isomorphism.
		The flatness of ϕ follows from <cit.>.
	
	
		There is a commutative diagram with cartesian squares
		
    𝒫(2^N-1)[r,hookrightarrow,"i"]    𝒫(2^N-1)_ϵ[r,"b_1'"][d]    𝒫(2^N)_ϵ[r,"b_1”"][d]    𝒫(2^N,1)[d]
       𝒫(2^N-1)[r][d]    𝒫(2^N)[r][d]    𝒫(2^N+1)[d]
       ℙ^N-2[r]    ℙ^N-1[r]    ℙ^N.

		where b_1”∘ b_1'∘ i=b_1, and 𝒫(2^N)_ϵ (respectively, 𝒫(2^N-1)_ϵ) is a first order thickening of 𝒫(2^N) (respectively, of 𝒫(2^N-1)).
		
		In particular,
		
    [𝒫(2^N-1)_ϵ]=2∈CH_*(𝒫(2^N-1)_ϵ).

	
	
		The only thing to be proved is that 𝒫(2^N)_ϵ is a first order thickening of 𝒫(2^N). Notice that 𝒫(2^N)_ϵ→ℙ^N-1 is the locus of the root stack 𝒫(2^N,1) where the defining section a_2g+2 vanishes. Recall that when a n-root stack over an algebraic stack 𝒳 is defined by (𝒪_𝒳,f), where f is a global section of 𝒳, the root stack is isomorphic to the quotient stack of
		
    _𝒳(𝒪_𝒳[T]/(T^n-f))

		by the action of μ_n given by ζ· T=ζ T. In particular, when f=0 and n=2, we get
		
    [(_𝒳(𝒪_𝒳[T]/(T^2)))/μ_2]

		which is different from the root gerbe √(𝒪_𝒳/𝒳); see <cit.> or <cit.> for more on root stacks. In particular, 𝒫(2^N)_ϵ is Zariski-locally on ℙ^N-1 of the form described above.
	
	
		Consider the commutative diagram with cartesian square
		
    ℙ^r×𝒫(2^N-2r,1)[rd,"θ"'][rr,bend left=20,"a_r"][r,"a_r'"]     X[r][d,"ψ"]    𝒫(2^N,1)[d,"ϕ"]
       ℙ^r×ℙ^N-2r[r,"π_r"]    ℙ^N.

		Then ϕ, θ, a_r' and ψ are isomorphisms over the locus where a_2g+2≠0 (hence s_1≠0), with all spaces isomorphic to 𝔸^r×𝔸^N-2r.
	
	

 §.§ Computation of α_r,i

	First we focus on α_r,0.
	
		The classes α_r,0 are contained in the image of ϕ^*.
	
	
		It follows from Corollary <ref> that (a_r')_*1=ψ^*1=1, and we conclude by compatibility of representable proper pushforward and flat pullback.
	
	Now we consider α_r,i for 0<i≤ r. We have a commutative diagram
	
    ℙ^r-1×𝒫(2^N-2r,1)[rd,"θ"'][r,"i"]    ℙ^r×𝒫(2^N-2r,1)[r,"a_r"]    𝒫(2^N,1)
       ℙ^r-1×𝒫(2^N-2r+1)[ur,"b_r"']

	where i is the inclusion corresponding to the hyperplane of points in ℙ^r whose first coordinate is 0, and θ sends (f,(g,s_1)) to (f,g), with the obvious notation. Notice that the T-equivariant class of that hyperplane is the sum of ξ_r and the first Chern class of a character of T, thanks to Lemma 2.4 of <cit.>. This shows that α_r,1 is generated by α_r,0 and the image of (b_r-1)_*. One can argue inductively to obtain the same result for all α_r,i, proving the following Lemma.
	
		For 0<i≤ r, α_r,i is generated by α_r,0 and the image of (b_r-1)_*.
	
	This reduces us to the computation of the β_r,i's.
	

 §.§ Computation of β_r,i

	For all 1<r≤ N/2, consider the commutative diagram
	
    ℙ^r-1×𝒫(2^N-2r+1)[d,"b_r'"'][r,"b_r"]    𝒫(2^N,1)
    𝒫(2^N-1)[ru,"b_1"']

	where b_r' sends (f,g) in f^2g. This shows that it is enough to study (b_1)_*.
	
	Now, notice that
	
    b_1^*:ℤ[h]/(2^Nh^N+1)≃CH^*(𝒫(2^N,1))[r]    CH^*(𝒫(2^N-1))≃ℤ[h]/(2^N-1h^N-1)

	satisfies b_1^*h=h, hence it is surjective, and the same is true at the level of (𝔾_m×𝔾_m)-equivariant Chow rings. By the projection formula it follows that the image of (b_1)_* is the ideal generated by (b_1)_*1=β_1,0. Therefore we are reduced to study β_1,0.
	
		It holds that
		
    β_1,0=(s_1=0)·(a_2g+1=0)∈CH_𝔾_m×𝔾_m^2(𝒫(2^N,1)).

	
	
		It holds essentially by definition.
	
	
		In particular, this shows that in the setting of Subsection <ref> indeed (ψ')^*1≠1. Moreover, the defining relation of CH^*_T(𝒫(2^N,1)) is contained in the ideal generated by β_1,0. Indeed, the relation is given by the product of the T-equivariant classes of the N+1 coordinate hyperplanes, and β_1,0 is the product of two of those.
	
	

 §.§ Final computation

	Putting everything together, we have proved that the relations are generated by those coming from ℋ_g and
	
    β_1,0=(s_1=0)·(a_2g+1=0)∈CH_𝔾_m×𝔾_m^2(𝒫(2^N,1)).

	Computing this last class in terms of the characters of 𝔾_m×𝔾_m gives the following Theorem, where we use the notation of Pernice.
	
		Let k>2g. If g is even, then
		
    CH^*(ℋ_g,1)≃ℤ[t_0,t_1]/((4g+2)(t_0+t_1),g(g-1)/2(t_0^2+t_1^2)-g(g+3)t_0t_1+(2g+1)(t_0+t_1)t_1).

		If g is odd, then
		
    CH^*(ℋ_g,1)≃ℤ[τ,ρ]/((8g+4)τ,2τ^2+g(g+1)/2ρ^2-(2g+1)τρ).

	
	
		Notice that, for g even, multiplying the degree 2 relation and using the degree 1 relation we get back the degree 2 relation described in <cit.>. For the g odd case, we have to multiply by 4.
	
	

 §.§ Base changes

	We know that [𝒲_g,1^1] and ψ_1 generate the Chow ring, so we write those classes in terms of the generators t_0 and t_1, so that we can rewrite the above isomorphisms in terms of the classes we are interested in.
	
		With the notation of Theorem <ref>, we have the following. If g is even, then
		
    [𝒲_g,1^1]=-g/2t_0+(g/2+1)t_1,    ψ_1=-(g/2-1)t_0+g/2t_1.

		If g is odd, then
		
    [𝒲_g,1^1]=τ-g+1/2ρ,    ψ_1=τ-g-1/2ρ.

	
	
		It is a straightforward computation using our previous results <ref> and <ref> on the classes [𝒲_g,n^1] and ψ_1, respectively, together with <cit.>.
	
	Making this change of variables, we get the following result.
	
		Let g≥2 and chark>2g. Then,
		
    CH^*(ℋ_g,1)≃ℤ[ψ_1,[𝒲_g,n^1]]/((4g+2)((g+1)ψ_1-(g-1)[𝒲_g,1^1]),[𝒲_g,1^1]^2+ψ_1·[𝒲_g,1^1])
			
		regardless of the parity of g.
	
	There is another interesting basis that we can consider. Recall that we had isomorphisms
	
    (𝔾_m×𝔾_m)/μ_g+1[r,"φ"]    𝔾_m×𝔾_m[r,"φ_even"][d,"φ_odd"]    𝔾_m×𝔾_m
       𝔾_m×𝔾_m

	where
	
    φ(a,c)=(ac^-1,c^g+1), φ_even(λ,μ)=(λ^g/2+1μ,λ^g/2μ), φ_odd(λ,μ)=(λ^(g+1)/2μ,λ).

	Let l_1 and l_2 be the obvious characters of the copy of 𝔾_m×𝔾_m in the middle. Then, the above diagram induces the following base changes
	
    t_0=(g/2+1)l_1+l_2,     t_1=g/2l_1+l_2

	and
	
    τ=g+1/2l_1+l_2,    ρ=l_1

	and the equalities
	
    [𝒲_g,1^1]=l_2,    ψ_1=l_1+l_2.

	This allows us to rewrite everything in another symmetric way, which highlights again the fact that the presentation does not depend on the genus, even though when seen abstractly the two rings appear slightly different, in particular the Picard group.
	
		Using the base changes above, we have
		
    CH^*(ℋ_g,1)≃ℤ[l_1,l_2]/((4g+2)((g+1)l_1+2l_2),2l_2^2+l_1l_2).

	
	

§ THE CHOW RING OF ℋ_G,2^FAR

	In this section, we compute the Chow ring of ℋ_g,2^far. To do so, we follow the strategy used for the computation of the Chow ring of ℋ_g,1.
	
	First, recall that
	
    ℋ_g,2^far≃[𝔸^2g+3∖Δ_g,2/(𝔾_m×𝔾_m)/μ_g+1]

	where the action has weights
	
    [ -(2g+1)       …      -1       0  -(g+1);      -1       … -(2g+1)  -(g+1)       0 ].

	Again, we pass to the projective setting by considering the quotient of 𝔸^2g+3 by the action of 𝔾_m with weights (2,…,2,1,1), hence obtaining 𝒫(2^N-1,1,1) and its closed subscheme Δ_g,2, which is the image of Δ_g,2.
	
	Now, we construct morphisms that together will form a Chow envelope of Δ_g,2. We use the usual convention where, given an homogeneous form f of degree 2g+2 in two variables x_0,x_1, and such that f(0,1)=s_1^2 and f(1,0)=s_2^2, we denote by (f,s_1,s_2)∈𝔸^2g+3 the element (a_1,…,a_2g+1,s_1,s_2), where a_0,…,a_2g+2 are the coefficients of f. We use a similar convention for couples (f,s_1) satisfying f(0,1)=s_1^2 and couples (f,s_2) satisfying f(1,0)=s_2^2.
	
	First, we define
	
    a_r:𝔸^r+1×𝔸^2g+3-2r[r]    𝔸^2g+3

	as (f,(g,s_1,s_2))↦(f^2g,f(0,1)s_1,f(1,0)s_2), which is clearly well defined.
	In a similar way, we define
	
    b_r:𝔸^r×𝔸^2g+3-2r[r]    𝔸^2g+3

	as (f,(g,s_2))↦(x_0^2f^2g,0,f(1,0)s_2),
	
    c_r:𝔸^r×𝔸^2g+3-2r[r]    𝔸^2g+3

	as (f,(g,s_1)↦(x_1^2f^2g,f(0,1)s_1,0), and finally
	
    d_r:𝔸^r-1×𝔸^2g+3-2r[r]    𝔸^2g+3

	as (f,g)↦(x_0^2x_1^2f^2g,0,0).
	
		It is necessary to consider all these maps in order to have a Chow envelope, see <cit.>.
	
	Now, we define an action of G_g,2:=(𝔾_m×𝔾_m)/μ_g+1 on the source of the maps just defined so that they become G_g,2-equivariant.
	Then, we pass to the projective setting. For this, we identify G_g,2 with 𝔾_m×𝔾_m via the isomorphism φ of (<ref>), so that we do not have to worry about the parity of g.
	
	Given elements (a,c)∈𝔾_m×𝔾_m, the action on the source 𝔸^r+1×𝔸^2g+3-2r of a_r is defined as
	
    (a,c)·(f(x,y),(g(x,y),s_1,s_2))=(f(ax,y),(c^2g(ax,y),cs_1,a^(g+1)cs_2)).

	The definition of the other actions is analogous. For the sake of completeness, we describe them for b_r, c_r and d_r in this order:
	
    (a,c)·(f(x,y),(g(x,y),s_2))   =(f(ax,y),(c^2g(ax,y),a^(g+1)cs_2))
    
    		(a,c)·(f(x,y),(g(x,y),s_1))   =(f(ax,y),(c^2g(ax,y),cs_1))
    
    		(a,c)·(f(x,y),g(x,y))   =(f(ax,y),c^2g(ax,y)).

	It is a straightforward computation to check that the maps just defined are equivariant with respect to these actions. Notice that the first component of 𝔾_m×𝔾_m acts trivially on the first factor of each source.
	
	Now, we pass to the projective setting. To do so, we consider also the standard action of 𝔾_m on the first factor of each source, with weight one, and the morphism of group schemes
	
    𝔾_m×G_g,2[r]    G_g,2

	given by (λ,(a,c))↦(a,λ· c).
	
	It is again a straightforward calculation to show that the maps defined induce (𝔾_m×𝔾_m)-equivariant maps
	
    [column sep=normal, row sep=small]
    		a_r:ℙ^r×ℙ(2^2g+1-2r,1,1)[r]    ℙ(2^2g+1,1,1)
    
    		b_r:ℙ^r-1×ℙ(2^2g+2-2r,1)[r]    ℙ(2^2g+1,1,1)
    
    		c_r:ℙ^r-1×ℙ(2^2g+2-2r,1)[r]    ℙ(2^2g+1,1,1)
    
    		d_r:ℙ^r-2×ℙ(2^2g+3-2r)[r]    ℙ(2^2g+1,1,1).

	
		The maps a_r, b_r, c_r and d_r are proper and representable morphisms of quotient stacks.
	
	
		This is an analogue of <cit.>, and it can be proved in the same way.
	
	It is important to notice that for every r the morphisms a_r, b_r, c_r and d_r factor through Δ_g,2. To show it, one can construct for each map a commutative diagram as in <cit.>.
	
	If f is a morphism of algebraic stacks, we denote with X(f) the source of f.
	
		Suppose k>2g. Let ω_r be the co-product morphism of a_r, b_r, c_r and d_r, and define ω=_r=1^g+1ω_r.
		Then ω is surjective at the level of G_g,2-equivariant Chow ring, i.e. the map ω_* from
		
    ⊕_r=1^g+1[CH^*_G_g,2(X(a_r))⊕CH^*_G_g,2(X(b_r))⊕CH^*_G_g,2(X(c_r))⊕CH^*_G_g,2(X(d_r))]

		to CH^*(Δ_g,2)	is surjective.
	
	
		It follows by construction, and it is analogous to <cit.>. One has to subdivide the analysis in cases depending on whether an element (h,s_1,s_2) with h=f^2g is such that f(0,1) or f(1,0) are 0 or not. We then have four cases, one for each map. This shows that ω is a Chow envelope of Δ_g,2, hence the statement follows from Lemma <ref>.
		
		The proof of the following Lemma is standard.
	
	
		The following holds:
		
    CH_G_g,2^*(ℙ^r×ℙ(2^N-1-2r,1,1))   ≃ℤ[t_0,t_1,ξ_r,1,ξ_r,2]/(p(t_0,t_1,ξ_r,1),q(t_0,t_1,ξ_r,2))
    CH_G_g,2^*(ℙ^r-1×ℙ(2^N-2r,1))   ≃ℤ[t_0,t_1,ξ_r-1,1,ξ_r-1,2]/(p'(t_0,t_1,ξ_r-1,1),q'(t_0,t_1,ξ_r-1,2))
    CH_G_g,2^*(ℙ^r-2×ℙ(2^N-2r+1))   ≃ℤ[t_0,t_1,ξ_r-2,1,ξ_r-2,2]/(p”(t_0,t_1,ξ_r-2,1),q”(ξ_r-2,2)).

		Here, ξ_r,1 and ξ_r,2 are the pullback under the first and second projection, respectively, of the first Chern class of the tautological bundle 𝒪(1) of the respective projective stack, while p (respectively p', p”) is a monic polynomial in the variable ξ_r,1 (respectively ξ_r-1,1, ξ_r-2,1) of degree r+1 (respectively r, r-1), and similarly for q (respectively q', q”), which is of degree N+1-2r.
	
	Recall also that
	
    CH_G_g,2^*(ℙ(2^N-1,1,1))≃ℤ[t_0,t_1,h]/(h+p_N(t_0,t_1))(h+p_0(t_0,t_1))∏_i=1^N(2h+p_i(t_0,t_1))

	for some polynomials p_i(t_0,t_1) and h=c_1(𝒪(1)).
	
		Using the notation above,
		
    (a_r)^*(h)   =ξ_r,1+ξ_r,2,     (b_r)^*(h)   =ξ_r-1,1+ξ_r-1,2,
     (c_r)^*(h)   =ξ_r-1,1+ξ_r-1,2,     (d_r)^*(h)   =ξ_r-2,1+ξ_r-2,2.

	
	
		The proof is the same as the one of <cit.>.
	
	Therefore, we are left with computing the classes
	
    α_r,i   :=(a_r)_*ξ_r,1^i,    β_r,i   :=(b_r)_*ξ_r-1,1^i,
    γ_r,i   :=(c_r)_*ξ_r-1,1^i,    δ_r,i   :=(d_r)_*ξ_r-2,1^i

	in CH_G_g,2^*(ℙ(2^N-1,1,1)).
	
		The image of CH_G_g,2^*(Δ_g,2) in CH_G_g,2^*(ℙ(2^N-1,1,1)) is generated by
		
    α_r,0 for 1≤ r≤ N/2,  β_1,0,  γ_1,0,  δ_2,0.

	
	
		Notice that we have a commutative diagram of proper morphisms
		
    ℙ^r-1×𝒫(2^N-2r-1,1,1)[rd,"θ"][r,hookrightarrow,"i"]    ℙ^r×𝒫(2^N-2r-1,1,1)[r,"a_r"]    𝒫(2^N-1,1,1)
       ℙ^r-1×𝒫(2^N-2r,1)[ru,"b_r"']

		where i is the inclusion of the hyperplane where the first coordinate of ℙ^r is 0. Reasoning in the same way as in Lemma <ref>, we get that α_r,i is generated by α_r,0 and the image of (b_r-1)_*, for every 1≤ i≤ r.
		
		Now, consider the commutative diagram
		
    ℙ^r-1×𝒫(2^N-2r,1)[d,"b_r'"'][r,"b_r"]    𝒫(2^N-1,1,1)
    𝒫(2^N-2,1)[ru,"b_1"']

		where b_r' sends (f,(g,s_2)) to (f^2g,f(1,0)s_2). This shows that b_r factors through b_1. Moreover, the pullback homomorphism
		
    b_1^*:ℤ[h]/(2^N-1h^N+1)≃CH^*(𝒫(2^N-1,1,1))[r]    CH^*(𝒫(2^N-2,1))≃ℤ[h]/(2^N-2h^N-1)

		is surjective, and so is the pullback at the level of (𝔾_m×𝔾_m)-equivariant Chow rings. By the projection formula it follows that the image of (b_1)_* is generated by (b_1)_*1=β_1,0.
		
		A completely analogous argument works for γ_r,0 and δ_r,0. The statement follows.
	
	Let ϕ:𝒫(2^N-1,1,1)→𝒫(2^N,1) be the morphism induced by s_2↦ s_2^2. Notice that it is representable and flat. The proof of the following Lemma is straightforward.
	
		Let 𝔸^N⊂𝒫(2^N,1) be the open subscheme where s_1 does not vanish. Then, there is a cartesian diagram
		
    𝔸^N[r][d,"u"]    𝒫(2^N-1,1,1)[d,"ϕ"]
    𝔸^N[r]    𝒫(2^N,1)

		where u(a_1,…,a_N-1,s_2)=(a_1,…,a_N-1,s_2^2).
	
	
		The classes α_r,0 are in the image of ϕ^* for all r.
	
	
		Consider the commutative diagram with cartesian square
		
    ℙ^r×𝒫(2^N-2r-1,1,1)[rd][rr,bend left=20,"a_r"][r,"a_r'"]     X[r][d,"ψ"]    𝒫(2^N-1,1,1)[d,"ϕ"]
       ℙ^r×𝒫(2^N-2r,1)[r]    𝒫(2^N,1).

		Now, the set-theoretic image of a_r is the preimage under ϕ of the image of the bottom morphism. We prove that (a_r')_*1=ψ^*1=1. It is enough to show it after base changing the above diagram to an open substack of 𝒫(2^N,1) intersecting the image of a_r. Choose the complement of (s_1=0)∪(a_2g+1=0). Over that substack (actually, scheme), we have that a_r' is an isomorphism, thanks to the previous Lemma. Indeed, over s_1≠0 we have a diagram with cartesian square
		
    𝔸^r×𝔸^N-2r[r,"a_r'"]     X^o[r][d]    𝔸^N[d,"u"]
       𝔸^r×𝔸^N-2r[r]    𝔸^N

		inducing an isomorphism
		
    X^o≃ k[x_1,…,x_r,y_1,…,y_N-2r,t_N]/(t_N^2-x_r^2y_N-2r).

		Where x_r≠0, i.e. over a_2g+1≠0, a_r' is an isomorphism, and the statement follows.
	
	Let ϕ':𝒫(2^N-1,1,1)→𝒫(2^N,1) the arrow induced by s_1↦ s_1^2.
	
		The class β_1,0 is the intersection of the divisors (a_2g+1=0) and (s_1=0). Similarly, γ_1,0 is the intersection of the divisors (s_2=0) and (a_1=0). Moreover, δ_2,0=β_1,0·γ_1,0.
		In particular, all these classes are contained in the images of ϕ^* and ϕ'^*.
	
	
		It follows immediately from the definition of the maps.
	
	
		As in Remark <ref>, the above Lemma shows also that the defining relation of CH^*_T(𝒫(2^N-1,1,1)) is contained in the ideal generated by β_1,0.
	
	Notice that, using the notation of Corollary <ref>, it holds
	
    [𝒲_g,2^2]=(g+1)l_1+l_2,    ψ_2=gl_1+l_2

	so that
	
    [𝒲_g,2^2]·([𝒲_g,2^2]+ψ_2)   =((g+1)l_1+l_2)((2g+1)l_1+2l_2)
       =(g+1)(2g+1)l_1^2+2l_2^2+(4g+3)l_1l_2

	which is the new relation.
	
	Putting everything together we get the following Proposition.
	
		Let k>2g. Then,
		
    CH^*(ℋ_g,2^far)≃ℤ[l_1,l_2]/((4g+2)((g+1)l_1+2l_2),2l_2^2+l_1l_2,(g+1)(2g+1)l_1^2+(4g+2)l_1l_2).

	
	It is useful to write it also in terms of [𝒲_g,2^1] and ψ_1.
	
		Let k>2g. Then, CH^*(ℋ_g,2^far) is generated by ψ_1 and [𝒲_g,2^1], and the ideal of relations is generated by
		2pt
		
    (4g+2)((g+1)ψ_1-(g-1)[𝒲_g,2^1]),
     [𝒲_g,2^1]^2+ψ_1·[𝒲_g,2^1],  (2g+1)(g+1)ψ_1^2+(2g+1)(3g-1)[𝒲_g,2^1]^2.

		
	
		Using the computations done for Corollary <ref>, we get that
		
    [𝒲_g,2^2]=(g+1)ψ_1-g[𝒲_g,2^1],    ψ_2=gψ_1-(g-1)[𝒲_g,2^1]

		in Pic(ℋ_g,2^far). The statement follows.
	
	

§ GENERATORS AND SOME RELATIONS OF THE CHOW RING OF ℋ_G,N

	We have very explicit generators of the Picard group of ℋ_g,n, with evident geometric meaning. Together with our concrete description of ℋ_g,n, this should decisively help in the computation of the integral Chow ring ℋ_g,n, at least when this is generated in degree 1. We know that this does not always happen, since the (rational) Chow ring of the stack of n-pointed hyperelliptic curves is not always tautological. An example is given in <cit.>, where the authors construct a non-tautological class in ℳ_2,20. However, we shall prove that for 1≤ n≤2g+3 the Chow ring is generated in degree 1. Therefore, Proposition <ref> gives us also the generators of the Chow ring, and we are left with computing the intersection of those divisors. Thanks to their geometric nature, this is doable.
	

 §.§ Generators

	We show that the generators of the Picard group described in Proposition <ref> actually generate the Chow ring for 1≤ n≤2g+3. We already know this for n=1, see Section <ref>.
	We want to use induction on n. To do so, we use the fact that 𝒵_g,n^i,j is isomorphic to an open substack of ℋ_g,n-1.
	
		Let 1≤ i<j≤ n and define ℋ_g,n-1^i,j to be the open substack of ℋ_g,n-1 where the map
		
    ℋ_g,n-1[r,dashed]    ℋ_g,n

		induced by
		
    (C→ S,σ_1,…,σ_n-1)[r,mapsto]     (C→ S,σ_1,…,σ_j-1,α∘σ_i,σ_j,…,σ_n-1)

		is defined, where α is the hyperelliptic involution. This is exactly the complement of 𝒲_g,n-1^i∪⋃_l≠i𝒵_g,n-1^l,i, in ℋ_g,n-1.
	
	
		By construction, the above map defines a closed immersion
		
    u_i,j:ℋ_g,n-1^i,j[r,hookrightarrow]    ℋ_g,n

		identifying ℋ_g,n-1^i,j with 𝒵_g,n^i,j. See also <cit.>.
	
	
		Let 1≤ n≤2g+3. Then the Chow ring of ℋ_g,n^far is generated in degree 1. In particular, for n=1,2 the Chow ring is generated by [𝒲_g,1^1] and ψ_1, while for n≥3 it is generated by [𝒲_g,n^1].
	
	
		We know that this is true for n=1,2, see Theorem <ref> and Proposition <ref>. For 3≤ n≤ 2g+3, thanks to Proposition <ref>, we know that ℋ_g,n^far is an open substack of a weighted projective space, and the localization sequence concludes. The last part follows from the first and Subsection <ref>.
	
	
		For all 1≤ n≤2g+3, CH^*(ℋ_g,n) is generated in degree 1.
	
				
		We already know that the statement holds for n=1 and for ℋ_g,n^far. We prove the Proposition by induction on n.
		Suppose that the statement is true for n-1, and consider the following exact sequence
		
    ⊕_i,jCH^*(ℋ_g,n-1^i,j)[rr,"∑(u_i,j)_*"]       CH^*(ℋ_g,n)[r,"ι^*"]   CH^*(ℋ_g,n^far)[r]     0

		for all i<j. Since CH^*(ℋ_g,n-1) is generated in degree 1, the same holds for the Chow ring of ℋ_g,n-1^i,j, hence the restriction (u_i,j)^* is surjective. It follows from the projection formula that the image of ∑(u_i,j)_* is the ideal generated by the 𝒵_g,n^i,j.
		
		Now, let β be an element of CH^*(ℋ_g,n).
		Since [𝒲_g,n^1] and ψ_1 generate CH^*(ℋ_g,n^far), we can modify β with an element generated in degree 1 so that its restriction to ℋ_g,n^far is 0. Therefore, we may assume from now on that ι^*(β)=0. Thanks to the exactness of the sequence (<ref>), α is in the image of ∑_i,j(u_i,j)_*, which we have shown to be the ideal generated by the [𝒵_g,n^i,j]. Actually, more is true. Indeed, by the inductive hypothesis we know that every element of the Chow ring of ℋ_g,n-1^i,j is a product of classes of degree 1, and so it can be written as the pullback via (u_i,j)^* of a product of classes of degree 1 in the Chow ring of ℋ_g,n. It follows from the projection formula that every element in the image of ∑_i,j(u_i,j)_* is a product of classes of degree 1; in particular, the same holds for β. This concludes.
	
	
		Assume that k>2g. Then the classes [𝒲_g,n^1], ψ_1, [𝒵_g,2^1,2] form a minimal set of generators of CH^*(ℋ_g,2), and
		
    (4g+2)(g+1)ψ_1=(4g+2)(g-1)[𝒲_g,n^1]

		is the only relation in degree 1.
		
		Let n≥3. Then
		
    {[𝒵_g,n^1,j]}_2≤ j≤ n∪{[𝒵_g,n^2,3]}∪{[𝒲_g,n^1]}

		is a minimal set of generators of CH^*(ℋ_g,n), and
		
    (8g+4)[𝒲_g,n^1]=(4g+2)(g+1)([𝒵_g,n^1,2]+[𝒵_g,n^1,3]-[𝒵_g,n^2,3])

		is the only relation in degree 1.
	
		Notice that Proposition <ref> works for all n such that CH^*(ℋ_g,n^far) and CH^*(ℋ_g,n-1) are generated in degree 1. Thus, the previous observation tells us that the first examples of non-tautological classes are to be looked for in CH^*(ℋ_g,n^far).
	
	

 §.§ Intersection of Chow classes

	We are left with computing the intersections between the divisors in Corollary <ref>.
	
		Let n≥2. In CH^*(ℋ_g,n), the following holds.
		
    [𝒵_g,n^1,i]·[𝒲_g,n^1]   =0  for all i≠1
    
    			[𝒵_g,n^1,i]·[𝒵_g,n^1,j]   =0  for all 1<i<j
    
    			[𝒵_g,n^1,i]·[𝒵_g,n^2,3]   =0  for  i=2,3.

	
	
		It follows immediately from the fact that the intersection of those divisors is empty.
	
	
		Let n≥ i>3. Then,
		
    [𝒵_g,n^1,i]^2=-[𝒵_g,n^1,i]·[𝒵_g,n^2,3]=[𝒵_g,n^2,3]^2.

	
	
		Notice that, by Proposition <ref>,
		
    [𝒵_g,n^1,i]=-[𝒵_g,n^2,3]+[𝒵_g,n^1,2]+[𝒵_g,n^3,i].

		Multiplying by [𝒵_g,n^1,i] and [𝒵_g,n^2,3] we get the two equalities, respectively.
	
	
		Let n≥5 and i>3, then
		
    [𝒵_g,n^1,2]^2=[𝒵_g,n^1,3]^2=[𝒵_g,n^1,i]^2=-[𝒵_g,n^1,i]·[𝒵_g,n^2,3]=[𝒵_g,n^2,3]^2.

	
	
		It follows from the previous Lemma and the fact that the relation
		
    [𝒵_g,n^1,4]^2=[𝒵_g,n^1,5]^2

		has to hold even after permuting the indices.
	
	In the case n≥5, thanks to the above Lemmas, we are left with computing the relations [𝒵_g,n^2,3]^2, [𝒵_g,n^2,3]·[𝒲_g,n^1] and [𝒲_g,n^1]^2. Of course, this takes care only of the degree 2 component of CH^*(ℋ_g,n); however, we will see that, together with knowing CH^*(ℋ_g,n^far), this is enough.
	
		Let n≥2. Then, for all i≠j,
		
    [𝒵_g,n^i,j]|_𝒵_g,n^i,j=-ψ_i|_𝒵_g,n^i,j=-ψ_j|_𝒵_g,n^i,j

		in Pic(𝒵_g,n^i,j).
		In particular,
		
    [𝒵_g,n^i,j]^2=-[𝒵_g,n^i,j]·ψ_i=-[𝒵_g,n^i,j]·ψ_j

		in CH^*(ℋ_g,n).
	
	
		It can be proved in different ways. For instance, if g is even then it follows from the last two relations of Lemma <ref>, while for n≥3 it is a consequence of Proposition <ref>. In general, we can argue as follows. Clearly, it is enough to show it for n=2. Let
		
    𝒞_g,2[r,"f"]    𝒫_g,2[r,"π"]    ℋ_g,2

		be the universal family, with universal sections σ_1, σ_2, and universal `intermediate' sections σ_1:=f∘σ_1 and σ_2:=f∘σ_2. Let L_i^∨ be the ideal sheaf associated to σ_i(ℋ_g,2)=:𝒟_g,2^i⊂𝒫_g,2.  Thanks to <cit.>, we have σ_2^*𝒪_𝒟_g,2^1=𝒪_𝒵_g,2^1,2. Applying σ_2^* to the exact sequence
		
    0[r]     L_1^∨[r]    𝒪_𝒫_g,2[r]    𝒪_𝒟_g,2^1[r]     0

		we get a surjective map σ_2^*(L_1^∨)→𝒪_ℋ_g,2(-𝒵_g,2^1,2) between invertible sheaves, hence an isomorphism. It follows that 
    𝒪_ℋ_g,2(𝒵_g,2^1,2)≃σ_2^*L_1≃σ_1^*L_2.

		
		Let 𝒞:=𝒞_g,2, 𝒫:=𝒫_g,2 and ℋ:=ℋ_g,2. Recall that we have an exact sequence of sheaves
		
    0[r]     f^*ω_𝒫/ℋ[r]    ω_𝒞/ℋ[r]    Ω_𝒞/𝒫[r]     0

		where Ω_𝒞/𝒫 is supported on the Weierstrass divisor. Let u_1,2:𝒵_g,2^1,2↪ℋ_g,2 be the inclusion. Applying u_1,2^*∘σ_1^* to that exact sequence, we get
		
    u_1,2^*σ_1^*(ω_𝒫/ℋ)[r]     u_1,2^*σ_1^*(ω_𝒞/ℋ)[r]     u_1,2^*σ_1^*(Ω_𝒞/𝒫)[r]     0.

		Notice that the middle piece corresponds to the restriction of ψ_1 to 𝒵_g,2^1,2. Since the image of σ_1∘ u_1,2 does not intersect the Weierstrass divisor, u_1,2^*σ_1^*Ω_𝒞/𝒫=0, and we get an isomorphism
		
    u_1,2^*σ_1^*ω_𝒫/ℋ[r,"≃"]     u_1,2^*σ_1^*ω_𝒞/ℋ.

		Now, σ_1^*ω_𝒫/ℋ≃σ_1^*(L_1^∨), and clearly u_1,2^*σ_1^*≃ u_1,2^*σ_2^*. It follows that
		
    u_1,2^*𝒪_ℋ_g,2(-𝒵_g,2^1,2)≃ u_1,2^*σ_2^*L_1^∨≃ u_1,2^*σ_1^*L_1^∨≃ u_1,2^*σ_1^*ω_𝒫/ℋ≃ u_1,2^*σ_1^*(ω_𝒞/ℋ)

		which proves the Lemma.
	
	The following relation is very important, because it lives in degree 1.
	
		Let n≥3. Then,
		
    [𝒲_g,n^1]-[𝒲_g,n^2]=(g+1)([𝒵_g,n^1,3]-[𝒵_g,n^2,3]).

	
	
		The fact that it holds rationally follows from Proposition <ref> and the rational relation
		
    [𝒲_g,n^1]+[𝒲_g,n^2]=(g+1)[𝒵_g,n^1,2]

		proved in <cit.>, or in <cit.>. Therefore, it is enough to show that the image of [𝒲_g,n^1]-[𝒲_g,n^2] in Pic(ℋ_g,n^far) is 0. This follows immediately from Lemma <ref> and the definition of the map Pic(ℋ_g,2^far)→Pic(ℋ_g,3^far).
	
	
		Let n≥3. Then,
		
    [𝒲_g,n^1]·[𝒵_g,n^2,3]=-(g+1)[𝒵_g,n^2,3]^2.

	
	
		It follows from the previous Proposition intersecting the relation (<ref>) with [𝒵_g,n^2,3].
	
	
		Let n≥3. Then, for all i≠j,
		
    (4g+2)(g+1)[𝒵_g,n^i,j]^2=0
    
    			(4g+2)[𝒵_g,n^2,3]·[𝒲_g,n^1]=0
    
    			(8g+4)[𝒲_g,n^1]^2=0

	
	
		For the first part, it is enough to consider the case of 𝒵_g,n^1,2. Then it is enough to intersect
		
    (8g+4)[𝒲_g,n^1]=(4g+2)(g+1)([𝒵_g,n^1,2]+[𝒵_g,n^1,3]-[𝒵_g,n^2,3])

		with [𝒵_g,n^1,2]. The second part follows from the first and the previous Corollary. The third relation follows from the second, multiplying the relation above by [𝒲_g,n^1].
	
	
		Let n≥5. Then, CH^2(ℋ_g,n) is generated by the classes [𝒲_g,n^1]^2 and [𝒵_g,n^2,3]^2.
	
	
		It follows from Lemma <ref>, Lemma <ref> and Corollary <ref>.
	
	
		Let n≥5. For every q≥3 the Chow group CH^q(ℋ_g,n) is generated by [𝒲_g,n^1]^q. More precisely, the restriction homomorphism
		
    CH^q≥3(ℋ_g,n)[r]    CH^q≥3(ℋ_g,n^far)

		is an isomorphism of CH^*(ℋ_g,n)-modules.
	
	
		We know that the only (possibly) non trivial classes in degree 2 are generated by [𝒲_g,n^1]^2 and [𝒵_g,n^2,3]^2, see Corollary <ref>. Moreover,
		
    [𝒵_g,n^2,3]^3=[𝒵_g,n^1,2]^2·[𝒵_g,n^2,3]=0

		and similarly for [𝒲_g,n^1]·[𝒵_g,n^2,3]^2. Thanks to Corollary <ref>, we have
		
    [𝒲_g,n^1]^2·[𝒵_g,n^2,3]=-(g+1)[𝒲_g,n^1]·[𝒵_g,n^2,3]^2=0

		and the first part of the statement is proved. The second part follows immediately from the first and the exact sequence
		
    ⊕CH^*(ℋ_g,n^i,j)[rr,"∑(u_a,b)_*"]       CH^*(ℋ_g,n)[r,"ι^*"]    CH^*(ℋ_g,n^far)[r]     0.

		Indeed, the image of (u_i,j)_* is the ideal generated by [𝒵_g,n^i,j], hence the map is 0 in degree q≥3.
	
	We are left with computing the relations between [𝒵_g,n^2,3]^2 and [𝒲_g,n^1]^2. We have already obtained partial information in Corollary <ref>.
	
	The following are other interesting relations, particularly important for low n.
	
		Let n≥2. Then,
		
    g[𝒲_g,n^1]+[𝒲_g,n^2]=(g+1)ψ_1+(g+1)[𝒵_g,n^1,2].

	
	
		It follows from the computation of the classes ψ_1, [𝒲_g,n^1] and [𝒲_g,n^2] in Pic(ℋ_g,n^far), and from the rational relation [𝒲_g,n^1]+[𝒲_g,n^2]=(g+1)[𝒵_g,n^1,2].
	
	
		The first psi class ψ_2 in Pic(ℋ_g,2^far) corresponds to
		
    ((g-1)/2,1)∈ℤ⊕ℤ/(8g+4)ℤ

		if g is odd, while if g is even it corresponds to
		
    (g-1,-g/2+1)∈ℤ⊕ℤ/(4g+2)ℤ.

		Moreover, in Pic(ℋ_g,2) it holds
		
    (g-1)([𝒲_g,n^1]-[𝒲_g,n^2])   =(g+1)(ψ_1-ψ_2)
    
    			[𝒲_g,n^1]+[𝒲_g,n^2]   =ψ_1+ψ_2+2[𝒵_g,2^1,2]
    
    			(g-1)[𝒲_g,2^1]+ψ_2   =gψ_1+(g-1)[𝒵_g,2^1,2].

	
	
		The first part follows from the computation in Proposition <ref>, while the second follows from the first and the fact that the relations hold rationally.
	
	
		For all n≥2, it holds
		
    (2g+1)(g+1)ψ_1^2+(2g+1)(3g-1)[𝒲_g,n^1]^2-(2g+1)(g+1)[𝒵_g,n^1,2]^2.

		For all n≥3, i≠j and i'≠j', it holds
		
    (2g+1)(g+1)[𝒵_g,n^i,j]^2=(2g+1)(g+1)[𝒵_g,n^i',j']^2=0.

	
	
		Applying the two above Lemmas to the relation [𝒲_g,2^2]^2+ψ_2[𝒲_g,2^2]=0 we get the first part. The second follows from the first and Proposition <ref>.
	
	The following identity is very important.
	
		Let n≥3. Then,
		
    [𝒲_g,n^1]=ψ_1+[𝒵_g,n^1,2]+[𝒵_g,n^1,3]-[𝒵_g,n^2,3].

	
	
		From the definition of the map Pic(ℋ_g,2^far)→Pic(ℋ_g,3^far), we know that the image of ψ_1 in Pic(ℋ_g,n^far) is the same as [𝒲_g,n^1], for n≥3. It follows that their difference is expressible as a sum of the [𝒵_g,n^1,j]'s and [𝒵_g,n^2,3] in a unique way. Moreover, putting together Proposition <ref> and Lemma <ref>, we get that
		
    (g+1)[𝒲_g,n^1]=(g+1)ψ_1+(g+1)([𝒵_g,n^1,2]+[𝒵_g,n^1,3]-[𝒵_g,n^2,3]).

		Since there are no relations between the [𝒵_g,n^1,j]'s and [𝒵_g,n^2,3], the statement follows.
	
	Recall that in Corollary <ref> we have proved that [𝒲_g,1^1]^2=-[𝒲_g,1]·ψ_1. It is worth mentioning that it was possible to prove it using some results in <cit.>.
	
		The Chow ring of 𝒲_g,1^1 is
		
    CH^*(𝒲_g,1^1)=ℤ[ψ_1]/(4g(2g+1)ψ_1).

	
	Consider the restriction of [𝒲_g,1^1] to 𝒲_g,1^1; we want to express it in terms of ψ_1. This is done in the proof of the above Theorem, see the proofs of <cit.> and <cit.>.
	
		The restriction of [𝒲_g,1^1] to 𝒲_g,1^1 is
		
    [𝒲_g,1^1]|_𝒲_g,1^1=-ψ_1|_𝒲_g,1^1.

		In particular, in CH^*(ℋ_g,1) we have
		
    [𝒲_g,1^1]^2=-[𝒲_g,1^1]·ψ_1.

	
	
		Let n≥3. Then,
		
    2[𝒲_g,n^1]^2=-[𝒲_g,n^1]·[𝒵_g,n^2,3]=(g+1)[𝒵_g,n^2,3]^2.

	
	
		From Proposition <ref> and Corollary <ref>, we know that
		
    [𝒲_g,n^1]^2=[𝒲_g,n^1]·ψ_1-[𝒲_g,n^1]·[𝒵_g,n^2,3]=[𝒲_g,n^1]·ψ_1+(g+1)[𝒵_g,n^2,3]^2

		and one concludes using the previous Lemma.
	
	Since we will see that [𝒲_g,n^1]^2≠0 in CH^*(ℋ_g,n^far), we are left with computing the order of [𝒵_g,n^2,3]^2. We only know that it divides (4g+2)(g+1) when n≥3. We will manage to compute it completely only for n=2, and to find lower bounds greater than 1 when n=3,4.
	

§ COMPUTATION OF CH^*(ℋ_G,2)

	We have computed the Chow ring of ℋ_g,2^far in Section <ref>, Corollary <ref>.
	Recall that we have an exact sequence
	
    CH^*(ℋ_g,1^1,2)[r,"(u_1,2)_*"]    CH^*(ℋ_g,2)[r]    CH^*(ℋ_g,2^far)[r]     0

	where ℋ_g,1^1,2=ℋ_g,1∖𝒲_g,1^1, see Definition <ref>. Then we can complete the above exact sequence to a commutative diagram
	
    CH^*(ℋ_g,1^1,2)[r,"(u_1,2)_*"][dr,"(u_1,2)_*"][dr,hookrightarrow, bend right=30,"q_2^*"]    CH^*(ℋ_g,2)[r][d]    CH^*(ℋ_g,2^far)[r]     0
       CH^*(ℋ_g,2∖𝒲_g,n^1)[ul,twoheadrightarrow, bend left=60, "(u_1,2)^*"]

	where (u_1,2)^*q_2^* is the identity of CH^*(ℋ_g,1^1,2).
	
	This allows us to compute the Chow ring of ℋ_g,2∖𝒲_g,2^1, which in turn can be used to compute CH^*(ℋ_g,2).
	
		Suppose k>2g. Then,
		
    CH^*(𝒵^1,2_g,2)≃CH^*(ℋ_g,1^1,2)≃ℤ[ψ_1]/((4g+2)(g+1)ψ_1).

	
	
		Suppose k>2g. Then, CH^*(ℋ_g,2∖𝒲_g,2^1) is isomorphic to
		
    ℤ[ψ_1,[𝒵_g,2^1,2]]/((4g+2)(g+1)ψ_1,[𝒵_g,2^1,2]^2+ψ_1·[𝒵_g,2^1,2],(2g+1)(g+1)(ψ_1^2-[𝒵_g,2^1,2]^2)).

	
	
		We already know that CH^*(ℋ_g,2∖𝒲_g,2^1) is generated by ψ_1 and [𝒵_g,2^1,2], and that the only relation in degree 1 is (4g+2)(g+1)ψ_1=0.
		
		Now, we focus on degree 2 relations. We already know that the two degree 2 relations hold, see Lemma <ref> and Lemma <ref>.
		Suppose that there is an other general relation
		
    dψ_1^2+e[𝒵_g,2^1,2]^2=0.

		Restricting dψ_1^2+e[𝒵_g,2^1,2]^2 to CH^*(ℋ_g,2^far) one gets that d is divisible by (2g+1)(g+1), hence using the last relation we reduce to compute the order of [𝒵_g,2^1,2]^2. Now, we know that
		
    u_1,2^*[𝒵_g,2^1,2]=-u_1,2^*ψ_1=-ψ_1.

		Therefore, u_1,2^*[𝒵_g,2^1,2]^2 has order (4g+2)(g+1), thanks to the previous Lemma, and the relation is generated by those in the statement. A similar argument works in degrees greater than 2.
	
	
		Let g>2g. Then, CH^*(ℋ_g,2) is generated by the classes of ψ_1, 𝒲_g,2^1 and 𝒵_g,2^1,2, and the subgroup of relations is generated by
		2pt
		
    (4g+2)((g+1)ψ_1-(g-1)[𝒲_g,2^1]),
     [𝒵_g,2^1,2]^2+ψ_1·[𝒵_g,2^1,2],  [𝒲_g,2^1]^2+ψ_1·[𝒲_g,2^1],  
    			[𝒵_g,2^1,2]·[𝒲_g,2^1],
    
    			(2g+1)(g+1)ψ_1^2+(2g+1)(3g-1)[𝒲_g,2^1]^2-(2g+1)(g+1)[𝒵_g,2^1,2]^2.

		
	
		We already know that those are the generators, and that the relations in degree 1 are multiples of the first relation in the statement. Moreover, we already know that those are indeed relations.
		
		Now, suppose we have a relation in degree q≥2. Using the middle row, we get a linear relation between ψ_1^q, [𝒲_g,2^1]^q and [𝒵_g,2^1,2]^q. Modulo 𝒵_g,2^1,2 we get a relation in CH^*(ℋ_g,2^far). Since the ideal of relations in ℋ_g,2 surjects to the one in ℋ_g,2^far, we reduce to the computation of the order of (𝒵_g,2^1,2)^q. This is divisible by (4g+2)(g+1), thanks to Proposition <ref>, and it is exactly (4g+2)(g+1) as one can see by multiplying the relation in degree 1 with [𝒵_g,2^1,2] and using Lemma <ref>.
	
	

§ THE ALMOST COMPUTATION OF CH^*(ℋ_G,N) FOR 3≤ N≤2G+3

	

 §.§ The Chow ring of ℋ_g,n^far for 3≤ n≤2g+3

	In this Subsection we compute the Chow ring of ℋ_g,n^far for 3≤ n≤2g+2, and obtain the generator of the ring in the case n=2g+3. In this last case, the only thing we do not manage to compute is the multiplicative order of the generator.
	
	Recall that
	
    ℋ_g,n^far≃(ℙ(2^2g+3-n,1^n)×(𝔸^n-3∖Δ))∖Δ_g,n.

	Moreover,
	
    CH^*(ℙ(2^2g+3-n,1^n)×(𝔸^n-3∖Δ))≃ℤ[h]/(2^2g+3-nh^2g+3).

	
		Let k>2g. If 3≤ n≤ 2g+2, then
		
    CH^*(ℋ_g,n^far)≃ℤ[[𝒲_g,n^1]]/((8g+4)[𝒲_g,n^1],2[𝒲_g,n^1]^2).

	
	Moreover, the map induced by forgetting the n-th section, i.e. the last one, is an isomorphism for 4≤ n≤2g+2.
	
		First, we know that the above relations hold; see Proposition <ref> for the relation in degree 2. On the other hand, in <cit.> it has been shown that, for n≤2g+2, the Chow ring of the intersection of all 𝒲_g,n^i has Chow ring generated by the restriction of ψ_1, with ideal of relations generated by 2ψ_1. Since for 3≤ n≤2g+3 the restrictions to ℋ_g,n^far of the classes ψ_1 and [𝒲_g,n^i] agree for every i, this concludes.
	
	
		The case n=2g+3 is harder, and we do not compute CH^*(ℋ_g,2g+3^far). The difference with the case n≤2g+2 is that ℋ_g,n is now a scheme, hence the integral Chow ring has to be 0 in degrees above the dimension. However, some things follow easily from what we have done. Recall that
		
    ℋ_g,2g+3≃(ℙ^2g+2×(𝔸^2g∖Δ))∖Δ_g,2g+3

		and so the Chow ring is generated by [𝒲_g,2g+3^1], the usual relations hold, and [𝒲_g,2g+3^1]^2g+2=0. It is not clear what is the exact multiplicative order of [𝒲_g,2g+3^1]. Moreover, it is easy to convince ourself that it is not possible to use similar Chow envelopes as in the cases n=0,1,2.
	
	

 §.§ The Chow ring of ℋ_g,3

	We apply the same method as for n=2. Recall that we have a commutative diagram
	
    CH^*(ℋ_g,2^1,3)[r,"(u_1,3)_*"][dr,"(u_1,3)_*"][dr,hookrightarrow, bend right=20,"q_3^*"]    CH^*(ℋ_g,3)[r][d]    CH^*(ℋ_g,3^far)[r]     0
       CH^*(ℋ_g,3∖𝒲_g,n^1∪𝒵_g,3^1,2)[ul,twoheadrightarrow, bend left=40, "(u_1,3)^*"]

	where (u_1,3)^*q_3^* is the identity of CH^*(ℋ_g,2^1,3), and the first row is a complex.
	
	This allows us to compute the Chow ring of ℋ_g,3∖𝒲_g,3^1, which in turn can be used to compute CH^*(ℋ_g,3).
	
		Let k>2g. Then,
		
    CH^*(𝒵_g,3^1,3)≃CH^*(ℋ_g,2^1,3)≃ℤ[ψ_1]/((4g+2)(g+1)ψ_1,(2g+1)(g+1)ψ_1^2).

	
	
		It follows immediately from Theorem <ref>.
	
	
		Let k>2g. Then, CH^*(ℋ_g,3∖(𝒲_g,3^1∪𝒵_g,3^1,2)) is isomorphic to
		
    ℤ[ψ_1,[𝒵_g,3^1,3]]/((4g+2)(g+1)ψ_1,[𝒵_g,3^1,3]^2+ψ_1·[𝒵_g,3^1,3],(2g+1)(g+1)ψ_1^2,(2g+1)(g+1)[𝒵_g,3^1,3]^2).

	
	
		We already know that CH^*(ℋ_g,3∖(𝒲_g,3^1∪𝒵_g,3^1,2)) is generated by ψ_1 and [𝒵_g,3^1,3], and that the only relation in degree 1 is (4g+2)(g+1)ψ_1=0.
		
		We now focus on degree 2 relations; we already know that those in the statement hold (see Lemma <ref> for the last one). Suppose there is another general relation
		
    dψ_1^2+e[𝒵_g,3^1,3]^2=0.

		Now, recall that [𝒲_g,3^1]=ψ_1+[𝒵_g,3^1,2]+[𝒵_g,3^1,3]-[𝒵_g,3^2,3] in CH^*(ℋ_g,3), so using [𝒵_g,3^2,3] instead of ψ_1 we get the relation d[𝒵_g,3^2,3]^2+(d+e)[𝒵_g,3^1,3]^2=0. Restricting to 𝒵_g,3^1,3, we get that (2g+1)(g+1) divides d+e. Restricting to 𝒵_g,3^2,3 one gets that (2g+1)(g+1) divides d, and thus e too. The above follows from the fact that 𝒵_g,n^i,j are all isomorphic, thus they have the same Chow ring, which is described in the above Lemma. Therefore, using (2g+1)(g+1)[𝒵_g,3^1,3]^2=0 we reduce to computing the order of ψ_1^2, which is (2g+1)(g+1) thanks to the injectivity of q_3^*.
		
		A similar argument works in degrees greater than 2.
	
	
		Let k>2g. Then, CH^*(ℋ_g,3) is generated by the classes of 𝒲_g,3^1, 𝒵_g,3^1,2, 𝒵_g,3^1,3 and 𝒵_g,3^2,3, and the ideal of relations is generated by
		2pt
		
    (8g+4)[𝒲_g,3^1]-(4g+2)(g+1)([𝒵_g,3^1,2]+[𝒵_g,3^1,3]-[𝒵_g,3^2,3]),
     [𝒵_g,3^1,2]·[𝒵_g,3^1,3],  [𝒵_g,3^1,2]·[𝒵_g,3^2,3],  [𝒵_g,3^1,3]·[𝒵_g,3^2,3],  
    			[𝒵_g,3^1,j]·[𝒲_g,3^1] for j=2,3,
    
    			[𝒲_g,3^1]·[𝒵_g,3^2,3]+(g+1)[𝒵_g,3^2,3]^2,  2[𝒲_g,3^1]^2-(g+1)[𝒵_g,3^2,3]^2,
    
    			(2g+1)(g+1)([𝒵_g,3^i,j]^2+[𝒵_g,3^i',j']^2) for i≠j,i'≠j',   b_g[𝒵_g,3^i,j]^2 for i≠j.

		where b_g is either (2g+1)(g+1) or (4g+2)(g+1).
	
	
		We already know that those are the generators, and that the relations in degree 1 are multiples of the first relation in the statement. Moreover, we already know that those in the statement are relations, and that they generate those in degree ≥2 involving [𝒲_g,3^1].			
		We are left with computing the orders of [𝒵_g,3^i,j]^q for q≥2. Restricting to 𝒵_g,3^i,j we get that their order is divisible by (2g+1)(g+1), and we are done (notice that (2g+1)(g+1)[𝒵_g,3]^q=0 for all q≥3).
	
	

 §.§ The Chow ring of ℋ_g,4

	We start in the same way as before.
	
		Suppose k>2g. Then,
		
    CH^*(ℋ_g,3^1,4)≃ℤ[ψ_1]/((4g+2)(g+1)ψ_1,(g+1)ψ_1^2)≃ℤ[[𝒵_g,3^2,3]]/((4g+2)(g+1)[𝒵_g,3^2,3],(g+1)[𝒵_g,3^2,3]^2).

	
	
		It follows immediately from Theorem <ref>.
	
	
		Suppose k>2g. Then,
		
    CH^*(ℋ_g,4∖(𝒲_g,4^1∪𝒵_g,4^1,2∪𝒵_g,4^1,3))
       ≃ℤ[ψ_1,[𝒵_g,4^1,4]]/((4g+2)(g+1)ψ_1,[𝒵_g,4^1,4]^2+ψ_1·[𝒵_g,4^1,4],(g+1)ψ_1^2,[𝒵_g,4^1,4]^2-ψ_1^2)
       ≃ℤ[[𝒵_g,4^2,3],[𝒵_g,4^1,4]]/((4g+2)(g+1)[𝒵_g,4^2,3],[𝒵_g,4^1,4]^2+[𝒵_g,4^2,3]·[𝒵_g,4^1,4],[𝒵_g,4^2,3]^2+[𝒵_g,4^2,3]·[𝒵_g,4^1,4],(g+1)[𝒵_g,4^2,3]^2).

	
	
		It is proved in the same way as Proposition <ref>.
	
	
		Let k>2g. Then, CH^*(ℋ_g,4) is generated by the classes of 𝒲_g,4^1, 𝒵_g,4^1,2, 𝒵_g,4^1,3, 𝒵_g,4^1,4 and 𝒵_g,4^2,3, and the ideal of relations is generated by
		2pt
		
    (8g+4)[𝒲_g,4^1]-(4g+2)(g+1)([𝒵_g,4^1,2]+[𝒵_g,4^1,3]-[𝒵_g,4^2,3]),
     [𝒵_g,4^1,i]·[𝒵_g,4^1,j] for 1<i<j,  [𝒵_g,4^1,j]·[𝒵_g,4^2,3] for j=2,3,  
    			[𝒵_g,4^1,j]·[𝒲_g,3^1] for j>1,
    
    			[𝒵_g,4^2,3]·[𝒲_g,3^1]+(g+1)[𝒵_g,4^2,3]^2,  [𝒵_g,4^2,3]^2=-[𝒵_g,4^2,3]·[𝒵_g,4^1,4]=[𝒵_g,4^1,4]^2,
    
    			2[𝒲_g,4^1]^2-(g+1)[𝒵_g,4^2,3]^2,  
    			(g+1)[𝒵_g,4^i,j]^3 for all i≠j,
    
    			(g+1)([𝒵_g,4^i,j]^2-[𝒵_g,4^i',j']^2) for all i≠j,i'≠j',   b_g[𝒵_g,4^i,j]^2 for all i≠j

		for some integer b_g=b'_g(g+1) such that b'_g divides 4g+2.
	
	
		We already know that those are the generators and that the relations hold. The only relation which is less clear is the bottom left. However, notice that
		
    2[𝒲_g,4^1]^2=(g+1)[𝒵_g,4^2,3]^2

		and applying the automorphism exchanging the fourth and third section we get the desired relation. Finally, the same arguments as in Theorem <ref> apply to show that the relations in the statement generated the ideal of relations.
	
	

 §.§ The Chow ring of ℋ_g,n for 5≤ n≤2g+3

	
		Consider the map q_n:ℋ_g,n→ℋ_g,n-1 forgetting the last section. Define ℋ^o_g,n to be preimage of ℋ_g,n-1^1,n under q_n, which is an open substack of ℋ_g,n.
	
	
		We have already used this stack in the previous sections. For instance, ℋ_g,2^o=ℋ_g,2∖𝒲_g,2^1, while for n=3 we have ℋ_g,3^o=ℋ_g,3∖(𝒲_g,3^1∪𝒵_g,3^1,2).
	
	
		Let k>2g, and let 5≤ n≤2g+3. Then,
		
    CH^*(ℋ_g,n-1^1,n)≃ℤ[ψ_1]/((4g+2)(g+1)ψ_1,ψ_1^2)

		and
		
    CH^*(ℋ_g,n^o)   ≃ℤ[ψ_1,[𝒵_g,n^1,n]]/((4g+2)(g+1)ψ_1,ψ_1·[𝒵_g,n^1,n],ψ_1^2,[𝒵_g,n^1,n]^2)
       ≃ℤ[[𝒵_g,n^2,3],[𝒵_g,n^1,n]]/((4g+2)(g+1)[𝒵_g,n^2,3],[𝒵_g,n^2,3]·[𝒵_g,n^1,n],[𝒵_g,n^2,3]^2,[𝒵_g,n^1,n]^2).

	
	
		Everything follows from what we know and the fact that we are killing [𝒵_g,n^1,4] too, so that all the squares are 0.
	
	
		Let k>2g and 5≤ n≤2g+2. Then, CH^*(ℋ_g,n) is generated by the classes of 𝒲_g,n^1, 𝒵_g,n^1,i for 1<i≤ n and 𝒵_g,n^2,3, and the ideal of relations is generated by
		2pt
		
    (8g+4)[𝒲_g,n^1]-(4g+2)(g+1)([𝒵_g,n^1,2]+[𝒵_g,n^1,3]-[𝒵_g,n^2,3]),
     [𝒵_g,n^1,i]·[𝒵_g,n^1,j] for 1<i<j,  [𝒵_g,n^1,j]·[𝒵_g,n^2,3] for j=2,3,  
    			[𝒵_g,n^1,j]·[𝒲_g,n^1] for j≠1,
    
    			[𝒵_g,n^2,3]·[𝒲_g,n^1]+(g+1)[𝒵_g,n^2,3]^2,  2[𝒲_g,n^1]^2-(g+1)[𝒵_g,n^2,3]^2,
    
    			[𝒵_g,n^2,3]^2=-[𝒵_g,n^2,3]·[𝒵_g,n^1,i]=[𝒵_g,n^1,j]^2 for all i>3 and j≠1,
    
    			b_g[𝒵_g,n^i,j]^2 for all i≠j

		where b_g is a positive integer dividing (4g+2)(g+1).
	
	
		It can be proved in the same way as before.
	
	
		Notice that the major changes that happen varying n from 2 to 5 are due to the variations of the order of ψ_1^2 in CH(ℋ_g,n-1^1,n). This essentially measures how much movable the divisors [𝒵_g,n^i,j] are.
	
	In the same way one can show the following result.
	
		Let k>2g and n=2g+3. Then, CH^*(ℋ_g,2g+3) is generated by the classes of  𝒲_g,2g+3^1, 𝒵_g,2g+3^1,i for 1<i≤2g+3 and 𝒵_g,2g+3^2,3, and the ideal of relations is generated by
		2pt
		
    (8g+4)[W_g,2g+3^1]-(4g+2)(g+1)([𝒵_g,2g+3^1,2]+[𝒵_g,2g+3^1,3]-[𝒵_g,2g+3^2,3]),
     [𝒵_g,2g+3^1,i]·[𝒵_g,2g+3^1,j] for 1<i<j,  [𝒵_g,2g+3^1,j]·[𝒵_g,2g+3^2,3] for j=2,3,
    
    			[𝒵_g,2g+3^1,j]·[𝒲_g,2g+3^1] for j≠1,  [𝒵_g,2g+3^2,3]·[𝒲_g,2g+3^1]+(g+1)[𝒵_g,2g+3^2,3]^2,
    
    			2[𝒲_g,2g+3^1]^2-(g+1)[𝒵_g,2g+3^2,3]^2,
    
    			[𝒵_g,2g+3^2,3]^2=-[𝒵_g,2g+3^2,3]·[𝒵_g,2g+3^1,i]=[𝒵_g,2g+3^1,j]^2 for all i>3 and j≠1,
    
    			b_g[𝒵_g,n^i,j]^2 for all i≠j,  [𝒲_g,2g+3^1]^c_g

		where b_g is a positive integer dividing (4g+2)(g+1), and 1<c_g≤2g+2.
	
	
		For 3≤ n≤2g+2, we are left with computing b_g, which we do not do in this article. There are some difficulties in using higher Chow rings (with ℤ_l-coefficients) in this case, since it can be shown that the first higher Chow ring of ℋ_g,n^far is non-zero in the cases we are interested in. Notice that b_g is also the order of [𝒵_g,n^2,3]·[𝒵_g,n^1,4] for n≥4, so that at least we are dealing with a proper intersection.
	
	
	amsalpha
	
	
