Maximal Brill-Noether loci via degenerations and double covers
Andrei Bud Richard Haburcak

	empty
	

	
		Using limit linear series on chains of curves, we show that closures of certain Brill–Noether loci contain a product of pointed Brill–Noether loci of small codimension. As a result, we obtain new non-containments of Brill–Noether loci, in particular that dimensionally expected non-containments hold for expected maximal Brill–Noether loci. Using these degenerations, we also give a new proof that Brill–Noether loci with expected codimension -ρ≤g/2 have a component of the expected dimension. Additionally, we obtain new non-containments of Brill–Noether loci by considering the locus of the source curves of unramified double covers.

	
	
	
	Outage Probability Analysis of  Wireless Paths with Faulty Reconfigurable Intelligent Surfaces
    Mounir Bensalem and Admela Jukan
Technische Universität Braunschweig, Germany;
{mounir.bensalem,  a.jukan}@tu-bs.de







    Version of April 23, 2024
==============================================================================================================================



§ INTRODUCTION

	
	The main
	theorem of classical Brill–Noether theory <cit.>
	shows that if C is a general smooth projective curve of genus g, then C admits a
	nondegenerate (not lying in a hyperplane) map C →^r of degree d if and only if the
	Brill–Noether number
	
    ρ(g,r,d)  g - (r+1)(g-d+r) ≥ 0.

	A nondegenerate degree d map C →^r corresponds to a line bundle L∈(C) of degree d and a subspace V ⊆ H^0(C,L) of dimension r + 1. The pair (L, V) is called a linear system of degree d and dimension r on C, or a g^r_d on C for short.
	
	In the last few years, there has been a renewed focus on refined Brill–Noether theory, which aims to understand linear systems on a curve in a component of a Brill–Noether locus 
    grd = { C ∈_g  |  C  admits a rd}
 when ρ(g,r,d)<0. In particular, there have been major advances in a refined Brill–Noether theory for curves of fixed gonality <cit.>. Relatively little is known about the geometry of Brill–Noether loci in general. It is known that grd is a proper subvariety of _g, which can potentially have multiple components and satisfies grd≤max{0,-ρ(g,r,d)}, see <cit.>, where -ρ(g,r,d) is the expected codimension. See <Ref> for more details.
	
	By adding basepoints and subtracting non-basepoints, one obtains many trivial containments of Brill–Noether loci. The expected maximal Brill–Noether loci are precisely the loci which do not admit such trivial containments, for a detailed characterization see <Ref>. Inspired by work on lifting line bundles on K3 surfaces, Auel and the second author posed a conjecture in <cit.> concerning potential containments of the “largest" Brill–Noether loci.
	
	[Maximal Brill–Noether Loci Conjecture]
		 
		For any g ≥ 3, except for g=7,8,9, the expected maximal
		Brill–Noether loci are maximal with respect to containment.
	
	There has been a flurry of recent progress on this conjecture in work of Auel–Haburcak–Larson, Bud, and Teixidor i Bigas <cit.>. In particular, <Ref> holds in genus g≤ 23 and by work of Choi, Kim, and Kim <cit.>, in genus g such that 
    g+1  or  g+2∈ {lcm(1,2,…,n)  for some  n∈ℕ_≥ 3}.

	
	In this paper, we give new non-containments of Brill–Noether loci. One expects that a Brill–Noether locus of large expected dimension is not contained in a Brill–Noether locus of small expected dimension. We prove that this is indeed the case. 
	
	 
		Let grd and gse be expected maximal Brill–Noether loci. If ρ(g,s,e)<ρ(g,r,d), then grd⊈gse.
	
	We show that given an expected maximal Brill–Noether locus grd, we can find a curve in the closure of grd in _g that is not contained in the closure of any other expected maximal Brill–Noether locus gse with ρ(g,s,e)<ρ(g,r,d). To do this, we use limit linear series to show that the closure of grd contains a product of Brill–Noether loci with prescribed ramification having expected codimension 1 or 2. Then Brill–Noether additivity and a few base cases yield <Ref>.
	
	Furthermore, we give a new proof of the existence of a component of a Brill–Noether locus of the expected dimension.
	
	
		If d≤ 2g-2 and -ρ(g,r,d)≤g/2,
		then grd has a component of the expected dimension.
		
		
		We note that this does not improve the currently best known results on the existence of components of the expected dimension, which are given in <cit.>. However, our method has the advantage of avoiding many of the combinatorial intricacies appearing in the previous proofs. 
		
		We also study non-containments of Brill–Noether loci coming from restrictions on linear series on a curve C admitting an étale double cover C→ C of a curve of genus g. In particular, the image, Im(χ_g), of the map χ_g: _g →_2g-1 sending the double cover to the source curve interacts interestingly with the Brill–Noether stratification of _2g-1. For double covers, Bertram shows in <cit.> that Im(χ_g) is contained in certain Brill–Noether loci. Conversely, Schwarz shows in <cit.> that for a general double cover C→ C, letting g̃ be the genus of C, if ρ(g̃,r,d)<-r, then C admits no rd. Using these restrictions, as well as ideas of Aprodu and Farkas <cit.>, we show infinitely many non-containments of expected maximal Brill–Noether loci.
		
		
			Let g = 1 + r(r+1) + 2ε for some 0≤ε < r/2 and let s,d be positive integers satisfying either 
			
				
  * ρ(g,s,d) = -s-1, or 
				
  * ρ(g,s,d) = -s, d is odd and s ≢3 4,
			
			then there is a non-containment 
			
    ℳ^r_g,g-1⊈ℳ^s_g,d.

		
		
		Already taking ε=0 gives infinitely many non-containments of expected maximal Brill–Noether loci, see <Ref>.
		
		
		
		

 §.§ Outline
 In <Ref>, we recall facts about Brill–Noether loci, limit linear series, and Prym curves. In particular, we give more precise definitions of expected maximal Brill–Noether loci in <Ref>, including some useful facts for our proofs. In <Ref>, we prove non-containments of pointed Brill–Noether loci of small codimension which act as the base cases for our proof of <Ref>. In <Ref>, we prove our main technical result, <Ref> and give a proof of <Ref> as <Ref>. In <Ref>, we use an inductive argument and the argument of <Ref> to prove <Ref>. Finally, in <Ref>, we prove additional non-containments of Brill–Noether loci coming from Prym curves.
		
		

 §.§ Acknowledgements
 We are grateful to Gavril Farkas, Hannah Larson, and Martin
		Möller for helpful discussions and comments on this paper, and to the organizers of “Vector bundles and combinatorial algebraic geometry" at Goethe Universität Frankfurt am Main, where this project was started. The first author acknowledges support by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the Collaborative Research Centre TRR 326 Geometry and Arithmetic of Uniformized Structures, project number 444845124. The second author would like to thank the Hausdorff Research Institute for Mathematics funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2047/1 – 390685813, and the Max–Planck Institut für Mathematik Bonn for their hospitality and financial support.
		
		
		
		
		

§ BACKGROUND

		
		

 §.§ Brill–Noether loci

		Brill–Noether theory studies how curves map to projective space. A map C→^s factors as a non-degenerate map C→^r and the linear embedding ^r⊆^s. We restrict our attention to non-degenerate maps C→^r, which are determined by a rd, that is, an element of 
    G^r_d(C) {(L,V)  |  L∈^d(C),  V⊆
    		H^0(C,L),  V=r+1}.
 There is a natural globalization of G^r_d(C) to a moduli space ^r_g,d over the moduli space _g of smooth curves of genus g, where the natural map ^r_g,d→_g has fiber G^r_d(C) above C. The Brill–Noether loci
		
    grd{C∈_g  |  C  admits a
    			rd}
 
		are the images of the corresponding maps ^r_g,d→_g.
		
		Many classical theorems in Brill–Noether theory can be restated in terms of components of ^r_g,d. For example, the classical Brill–Noether theorem states that ^r_g,d has a unique component surjecting onto _g when ρ(g,r,d)≥ 0, and this component has relative dimension ρ(g,r,d) <cit.>. The expected relative dimension of ^r_g,d is ρ(g,r,d), in particular when ρ(g,r,d)<0, grd has expected codimension -ρ(g,r,d) in _g. 
		
		When Brill–Noether loci are equidimensional, perhaps even irreducible, one can use simple dimension arguments to prove non-containments of Brill–Noether loci, large loci cannot be contained in small loci. However, only Brill–Noether loci with ρ=-1 and g2d with ρ=-2 are known to be irreducible <cit.>. More is known about the existence of components of expected dimension, however not much is known about equidimensionality of components. It is known that the codimension of any component of grd is at most -ρ(g,r,d), and when -3 ≤ρ(g,r,d)≤ -1 (additionally assuming g≥ 12 when ρ(g,r,d)=-3), the Brill–Noether loci are equidimensional of the expected dimension <cit.>. Complicating the picture, components of larger than expected dimension can exist, examples include Castelnuovo curves, see for example <cit.>.
		
		When ρ is not too negative, avoiding the Castelnuovo curve examples, it is expected that there is a component of expected dimension. Recently, Pflueger and Teixidor i Bigas independently showed that when ρ≥ -g+3, grd has a component of expected dimension <cit.>. We give a new proof of the existence of a component of expected dimension for Brill–Noether loci of expected codimension ≤g/2.
		
		

 §.§ Expected maximal Brill–Noether loci

		Many statement of a refined Brill–Noether theory can be restated as studying the stratification of _g by Brill–Noether loci. There are trivial containments
		grd⊆grd+1 obtained by adding a basepoint
		to a rd on C; and grd⊆gr-1d-1
		when ρ(g,r-1,d-1)<0 by subtracting a non-basepoint <cit.>. The expected maximal Brill–Noether loci are defined as the Brill–Noether loci not admitting these trivial containments. Concretely, for fixed r≥ 1 a Brill–Noether locus grd is expected maximal if d is maximal such that ρ(g,r,d)<0 and ρ(g,r-1,d-1)≥ 0. Accounting for Serre duality, which shows grd=gg-d+r-12g-2-d, every Brill–Noether locus is contained in at least one expected maximal Brill–Noether locus. As observed in <cit.>, the expected maximal Brill–Noether loci are exactly the grd such that
		
    1≤ r ≤√(g)-1   if  g≥√(g)^2+√(g)
    √(g)-1   if  g<√(g)^2+√(g),
 
		and for each such r
		
    d = (g, r)   r+⌈gr/r+1⌉ -1.

		
		In <cit.>, Auel and the second author posed <Ref>, which
		says that the expected maximal Brill–Noether loci should be maximal
		with respect to containment, except when g = 7, 8, 9. Concretely, for any two grd and gse expected maximal, there should exist a curve C admitting a rd but no se. We note that the exceptional cases in genus
		7, 8, and 9, come from unexpected containments of Brill–Noether
		loci obtained from projections from points of multiplicity ≥ 2 in
		genus 7 and 9 <cit.>
		or from a trisecant line in genus 8, as shown by Mukai <cit.>.)  Following this, they proved <Ref> in genus g≤ 19, 22, and 23 using various K3 surface techniques and Brill–Noether theory for curves of fixed gonality. Moreover, work of Choi, Kim, and Kim <cit.> showing that Brill–Noether loci with ρ=-1,-2 are distinct verifies <Ref> in infinitely many genera, cf. <cit.>. More recently, Auel–Haburcak–Larson employed the gonality stratification and the refined Brill–Noether theory for curves of fixed gonality to verify the g=20 case <cit.>, and the first author has verified the g=21 case by employing a degeneration argument and studying strata of differentials <cit.>. Various non-containments of expected maximal Brill–Noether loci are also known, for details see <cit.>. 
		
		We end with a few useful facts about expected maximal Brill–Noether loci.
		
		
			Let g r+1 be the smallest non-negative representative. For an expected maximal Brill–Noether locus grd, we have -ρ(g,r,d)=r+1-(g r+1).
		
		
		Moreover, for r satisfying <Ref>, the expected maximal Brill–Noether loci are exactly the Brill–Noether loci with the largest expected dimension.
		
		
			For r satisfying <Ref> if -r-1≤ρ(g,r,d)≤ -1, then grd is expected maximal.
		
		
			A straightforward computation shows that if -r-1≤ρ(g,r,d)≤ -1, then d≥(g,r) and ρ(g,r,d+1)= ρ(g,r,d)+r+1≥ 0. For r satisfying <Ref> and ρ(g,r,d)<0, we have r+1≤ g-d+r, hence ρ(g,r-1,d-1)=ρ(g,r,d)+g-d+r≥ 0. Thus grd is expected maximal.
		
		
		
		

 §.§ Limit linear series and pointed Brill–Noether loci

		We recall the basics of limit linear series and pointed Brill–Noether loci. Let C be a smooth curve. We follow the standard terminology from <cit.> and <cit.>.
		
		Let g,r,d be positive integers satisfying d<g+r. Given a curve C of genus g, a linear series ℓ=(L,V)∈ G^r_d(C), and fixing a point p∈ C, we order the finite set {ord_p(σ)}_σ∈  V of vanishing orders of sections, giving a vanishing sequence 
    a^ℓ (p): 0≤ a_0^ℓ(p) < a_1^ℓ(p)⋯ < a_r^ℓ(p) ≤ d
 of non-negative integers. The ramification sequence of ℓ at p 
    0≤ b_0^ℓ(p)≤⋯≤ b_r^ℓ(p) ≤ d-r
 is given by b_i^ℓ(p)  a_i^ℓ(p)-i, and the weight of ℓ at p is 
    w^ℓ(p)=∑_i=1^r b_i^ℓ(p).
 When the linear series ℓ is understood, we omit it from the notation.
		
		We call a sequence of integers 0≤ b_0 ≤⋯ b_r ≤ d-r a ramification sequence of type (r,d) and weight w(b)=∑ b_i, and given two ramification sequences of type (r,d), we say (b_i)≤ (c_i) when b_i≤ c_i for all 0≤ i ≤ r. Similarly, we call a sequence of integers 0≤ a_0<a_1<⋯ < a_r≤ d a vanishing sequence of type (r,d). Given n smooth points p_1,…,p_n on a curve C and n ramification sequences b^1,…,b^n of type (r,d), we define 
    G^r_d(C,(p_1,b^1),…,(p_n,b^n)){ℓ∈ G^r_d(C)  |  b^ℓ(p_i)≥ b^i},
 which is a determinantal variety of expected dimension 
    ρ(g,r,d,b^1,…,b^n)ρ(g,r,d)-∑_i=1^n w(b^i),
 which is the adjusted Brill–Noether number. If the linear series ℓ and the vanishing sequences are understood, we sometimes abbreviate ρ(g,r,d,b^1,…,b^n)=ρ(ℓ,p_1,…,p_n) to emphasize the points rather than the ramification sequence.
		
		We will work mainly with vanishing sequences, hence given a ramification sequence (b_i) of type (r,d) we define the associated vanishing sequence as (a_i)(b_i+i).
		
		Similarly, one can define pointed versions of W^r_d(C), namely 
    W^r_d(C,(p_1,b^1),…,(p_n,b^n)){L∈^d(C)  |     h^0(C,L(-a_i^j p_j))≥ r+1-i
       for all  0≤ i ≤ r  and all 1≤ j ≤ n}.

		
		One may also globalize these constructions, as with ^r_d and ^r_g,d. Namely, given ramification sequences b^1,…,b^n of type (r,d), with a^1,…,a^n the associated vanishing sequences, we define the pointed Brill–Noether loci 
    grd(a^1,…,a^n){C∈_g,n |  G^r_d(C,(p_1,b^1),…,(p_n,b^n))≠∅}⊆_g,n.
 When the entries of the vanishing sequences are consecutive numbers, the corresponding point is simply a base-point of the linear series. In particular, by subtracting the base-point a_0 p, one sees that grd(a_0,a_0+1,…,a_0+r)=grd-a__0, viewed in ℳ_g,1.
		
		For a curve C of compact type (i.e. every node of C is disconnecting, or equivalently a curve whose dual graph is a tree or whose Jacobian is compact), a crude limit rd on C is a collection of ordinary linear series 
		
    ℓ={ℓ_Y=(L_Y,V_Y)∈ G^r_d(Y)  |  Y⊆ C  is an irreducible component}

		satisfying a compatibility condition on the intersections of components. Namely, if Y and Z are irreducible components of C with p=Y∩ Z, then 
    a^ℓ_Y_i(p) + a^ℓ_Z_r-i(p) ≥ d  for all 0≤ i ≤ r.
 When equality holds everywhere, we say that ℓ is a refined limit rd. The linear series ℓ_Y∈ G^r_d(Y) is called the Y-aspect of the limit linear series ℓ.
		
		In <cit.>, it is proven that the adjusted Brill–Noether number is additive. Namely 
    ρ(g,r,d)≥∑_Y⊆ Cρ(ℓ_Y,b^ℓ_Y(p_1),…,b^ℓ_Y(p_k)),
 where p_1,…,p_k are the intersections of Y with the other components of C, and equality holds exactly when ℓ is a refined limit linear series. Furthermore, due to the determinantal nature of G^r_d(C,(p_1,b^1),…,(p_n,b^n)), as shown in <cit.>, limit linear series that move in a space of the expected dimension smooth to nearby curves.
		
		
		

 §.§ Prym–Brill–Noether loci

		
		We recall some basic facts about the Prym moduli space _g of unramified double covers of curves of genus g, and Prym–Brill–Noether loci which are useful in <Ref>.
		
		Recall that the moduli space of Prym curves
		
    _g{[C,η]  |  C∈_g, η∈^0(C)∖{_C}, η^⊗2≅_C},
 
		introduced by Mumford in his seminal paper <cit.> and further popularized by Beauville in <cit.>, parameterizes smooth curves of genus g together with a 2-torsion point of the Jacobian of C. The data of such a pair [C,η]∈_g is equivalent to the datum of an unramified double cover f:C→ C where CSpec(_C⊕η). As the cover is unramified, we immediately see that the genus of C is given by g(C) = 2g(C)-1=2g-1. The étale double cover f:C→ C induces a norm map 
    Nm_f:^2g-2(C)→^2g-2(C), Nm_f ( _C(D)) _C(f(D)).
 The Prym moduli space _g parametrizing unramified double covers of curves of genus g, has many applications in the study of principally polarized Abelian varieties, _g, and Brill–Noether theory. In particular, Welters defined in <cit.> the Prym–Brill–Noether loci 
    V^r(fC→ C){L∈( C)  | Nm_f(L)≅ω_C,  h^0(C,L )≥ r+1 and h^0(C,L )≡ r+1  2}.
 It was subsequently shown in two papers <cit.> that when g≥r+12+1, the locus V^r(fC→ C) is non-empty of dimension at least g-1-r+12, and that equality is attained for generic [fC→ C]∈_g. Moreover, when g<r+12+1, then V^r(fC→ C) is empty for generic [fC→ C]. Recently, Schwarz investigated the Brill–Noether theory for general unramified cyclic covers of degree n, parameterized by _g,n, and showed that for general [fC→ C]∈_g,n, C admits no rd if ρ(g(C),r,d)<-r, where g(C)=n(g-1)+1 is the genus of C, see <cit.> for more details.
		
		In <Ref>, we consider the natural map 
    χ_g : _g →_2g-1, [fC→ C]↦ [C],
 which sends the étale double cover to the source curve, and investigate how the image, Im(χ_g), interacts with the Brill–Noether stratification of _2g-1.
		
		
		
		
		
		

§ NON-CONTAINMENTS OF POINTED BRILL–NOETHER LOCI OF SMALL CODIMENSION

		
		The goal of this section is to provide some preliminary results that will be used to prove <Ref> via degeneration techniques. We want to find curves in the closure of ℳ_g,d^r in _g that cannot be contained in the closure of another expected maximal Brill–Noether locus ℳ_g,e^s. As pointed Brill–Noether loci naturally appear in the boundary of Brill–Noether loci, in this section we will prove some non-containment results for them.
		
		One key statement is that pointed Brill–Noether loci of expected codimension 1 are not contained in pointed Brill–Noether loci of larger expected codimension.
		
		  Let g,r,d,s,e be positive integers and let a,b be vanishing sequences of type (r,d) and respectively (s,e), such that ρ(g,r,d,a) = -1 and ρ(g,s,e,b) ≤ -2. Then there is a non-containment 
			
    ℳ^r_g,d(a) ⊈ℳ^s_g,e(b)

		
		
			This result is an immediate consequence of <cit.>. The locus ℳ^r_g,d(a) is an irreducible divisor of ℳ_g,1 while the locus ℳ^s_g,e(b) has codimension 2 or higher.  
		
		
		This result can be extended to pointed Brill–Noether loci in ℳ_g,2.
		
		  Let g,r,d,s,e be positive integers and let a, b, c be vanishing sequences of type (r,d) and respectively (s,e), such that ρ(g,r,d,a) = -1 and ρ(g,s,e,b,c) ≤ -2. Then, letting πℳ_g,2→ℳ_g,1 be the map forgetting the second marking, there is a non-containment 
			
    π^-1ℳ^r_g,d(a) ⊈ℳ^s_g,e(b,c).
	
		
		
			Let [ℙ^1, p, p_1,p_2] ∈ℳ_0,3 and consider the clutching map 
			
    ℳ_g,1→ℳ_g,2

			sending a pointed curve [C,q] to [C∪_q∼ pℙ^1, p_1, p_2]. The pullback of π^-1ℳ^r_g,d(a) is simply ℳ^r_g,d(a), while the pullback of ℳ^s_g,e(b_1,b_2) consists of loci with Brill–Noether number strictly less than -1. <Ref> yields the conclusion.
		
		
		We want to show that containments are well-behaved with respect to the expected codimension, i.e., no Brill–Noether locus is contained in another Brill–Noether locus of higher expected codimension. We start with the case of codimension 2. 
		
		
			Let ℳ^r_g,d⊆ℳ_g be a Brill–Noether locus satisfying d < g+r,  ρ(g,r,d) = -2 and r + 1 ≤ g-d+r. If ℳ^s_g,e(b) is a pointed Brill–Noether locus with ρ(g,s,e,b) ≤ -3, then letting πℳ_g,1→ℳ_g be the forgetful map, there is a non-containment 
			
    π^-1ℳ^r_g,d⊈ℳ^s_g,e(b).

		
		
		
			If g ≥ 4r+2, we can consider a clutching map 
			
    ℳ_g_1,1×ℳ_g_2,1→ℳ_g

			with g_1 = (r+1)k_1-1 and g_2 = (r+1)k_2-1 for some k_1, k_2 ≥ 2. 
			
			The locus ℳ^r_g_1,d(rk_2-1, rk_2, …, rk_2+r-1) ×ℳ^r_g_2,d(rk_1-1, rk_1, …, rk_1+r-1) is a non-empty product of loci with Brill–Noether number -1, and appears in the pullback of grd via the clutching map as a result of <cit.>.
			
			We consider the diagram 
			
    ℳ_g_1,1×ℳ_g_2,2rιd   ℳ_g,1dπ
    ℳ_g_1,1×ℳ_g_2,1r   ℳ_g

			where the vertical maps are forgetful maps, while the horizontal maps are the obvious clutchings. 
			
			By Brill–Noether additivity (cf. <cit.>) and <Ref>, the pullback of ℳ^r_g_1,d(rk_2-1, rk_2, …, rk_2+r-1) ×ℳ^r_g_2,d(rk_1-1, rk_1, …, rk_1+r-1) to ℳ_g_1,1×ℳ_g_2,2 is not contained in ι^-1ℳ^s_g,e(b). This implies the required non-containment 
			
    π^-1ℳ^r_g,d⊈ℳ^s_g,e(b).

			
			We are left to treat the cases when g < 4r+2. In this situation, we have 
			
    4r+4 > g+2=(r+1)(g-d+r) ≥ (r+1)^2

			and hence 1≤ r ≤ 2. 
			
			If r = 1, then 4≤ g <6, and the condition ρ(g,r,d) = -2 implies g = 4 and d = 2, whereby grd is the hyperelliptic locus. 
			
			Let 𝒲_2 ⊆ℳ_2,1 be the Weierstrass divisor and consider the clutching 
			
    ℳ_2,1×ℳ_2,1→ℳ_4.

			The locus 𝒲_2×𝒲_2 appears in the pullback of ℳ^1_4,2 via the clutching. The rest of the proof follows analogously to the case g ≥ 4r +2. 
			
			When r = 2, we have 7 ≤ g < 10, and the condition ρ(g,2,d) = -2 implies g = 7 and d= 6. We consider the clutching 
			
    ℳ_2,1×ℳ_5,1→ℳ_7.

			We take the product of codimension 1 loci ℳ^2_2,6(2,4,6) ×ℳ^2_5,6(0,2,4). By <cit.>, this locus appears in the pullback of ℳ^2_7,6 via the clutching map. The proof of non-containment now follows as in the case g ≥ 4r+2.
		
		
		In fact, the same argument as in the proof of <Ref> can be used to extend the result to codimension 2 loci.
		
		  Let g,r,d,s,e be positive integers and let b, c be vanishing sequences of type (s,e) such that ρ(g,r,d) = -2 and ρ(g,s,e,b,c) ≤ -3. Then, letting πℳ_g,2→ℳ_g be the map forgetting the markings, there is a non-containment 
			
    π^-1ℳ^r_g,d⊈ℳ^s_g,e(b,c).
	
		
		
		This corollary, together with Brill–Noether additivity <cit.> will be the key results in proving <Ref>.
		
		
		
		
		
		

§ DIMENSIONALLY EXPECTED NON-CONTAINMENTS

		
		In this section, we prove that given two expected maximal Brill–Noether loci grd and gse satisfying ρ(g,r,d)>ρ(g,s,e) (i.e. the expected dimension of gse is smaller than the expected dimension of grd), we have grd⊈gse. Our approach is in two steps. We first construct a chain curve C_1 ∪ C_2 ⋯∪ C_k appearing in the boundary of grd by virtue of <cit.>. We then use Brill–Noether additivity to conclude that this curve does not admit a limit linear series of type se, thus proving the non-containment grd⊈gse.
		
		
			Let grd be a Brill–Noether locus  satisfying the numerical condition
			
				
  (∗) (2r+1)-ρ(g,r,d)+1/2--ρ(g,r,d)/2≤ g.
			
			Then the closure of this locus in ℳ_g contains a chain curve [C_1∪ C_2∪⋯ C_k] such that
			
				
  * Each irreducible component C_i is generic in a Brill–Noether locus g_ird_i with 
				
    -1≥ρ(g_i,r,d_i)≥ -2;
 
				
  * Each glueing point is generic on both irreducible components it connects.
			
		
		
		 Let k = -ρ(g,r,d)+1/2 and consider the clutching 
			
    φ:_g_1,1×(∏_i=2^k-1_g_i,2)×_g_k,1→_g,
 
			sending a tuple ( [C_1, p_1],[C_2, q_2^1, q_2^2],…, [C_k-1, q_k-1^1, q_k-1^2],[C_k,p_k] ) to the curve 
			
    C C_1 ∪_p_1∼ q_2^1 C_2 ∪_q_2^2∼ q_3^1C_3∪⋯∪_q_k-1^2∼ p_k C_k.
 
			We want to construct a chain curve [C_1∪ C_2⋯∪ C_k] admitting a smoothable limit g^r_d and respecting the conditions in the hypothesis. We remark that it is sufficient to find a limit g^r_d on this chain so that the vanishing orders are consecutive numbers for each node. Let (v_1, v_1+1, …, v_1+r) be the vanishing orders at p_1 for the C_1-aspect, (v^1_i, v^1_i+1,…, v^1_i+r) and (v^2_i, v^2_i+1,…, v^2_i+r) the vanishing orders at q^1_i and q^2_i for the C_i-aspect and (v_k, v_k+1, …, v_k+r) the vanishing orders at p_k for the C_k-aspect. 
			
			We treat first the case ρ(g,r,d) is even. 
			
			We show how to determine g_i and v_i^j from g,r,d. Note that ρ(g,r,d) = -2k≡ g r+1. Starting with (r-1,r-1,…,r-1)∈ (_>0)^⊕ k, we add r+1 to the first entry then the second, and so on, repeating cyclically until we obtain (g_1, g_2…, g_k) where g_i≡ -2r+1 and g=∑_i=1^k g_i. Let v_i=g_i+2/r+1+d-r-g_i. The vanishing orders are given inductively by 
			
    v_1   =g_1+2/r+1+d-r-g_1
    
    				v_2^1   =d-v_1-r
    
    				v_2^2   =v_2-v_2^1
    
    				v_i^1   =d-v_i-1^2-r
    
    				v_i^2   =v_i-v_i^1=g_i+2/r+1-g_i+v^2_i-1.

			
			By construction, the vanishing orders satisfy the compatibility condition to be a refined limit linear series for i≤ k-1, and at p_k we have 
			
    v_k+r+v_k-1^2    = ( ∑_i=1^kg_i+2/r+1-g_i ) +2d-r
       = g-ρ(g,r,d)/r+1-g+2d-r
       =d,
 thus the compatibility condition is satisfied at every clutching point. Moreover, by definition v_i=v_i^1+v_i^2 and one checks that 
			
    ρ(g,r,d,(v_1,…,v_1+r))   =-2,
    ρ(g,r,d,(v_i^1,…,v_i^1+r),(v_i^2,…,v_i^2+r))   =-2  for 2≤ i ≤ k-1,  and
    ρ(g,r,d,(v_k,…,v_k+r))   =-2.

			Finally, taking d_i=d-v_i, we note that the i^th aspect corresponds to a rd_i on C_i which satisfies ρ(g_i,r,d_i)=-2, thus g_ird_i is a Brill–Noether locus of codimension 2.
			
			The locus of curves in Im(φ) admitting a g^r_d with vanishing orders as above is of expected dimension and satisfies the conditions in the hypothesis. Finally, <cit.> implies that this locus appears in the closure of grd, as required.
			
			The condition (∗) was tacitly used to ensure that g_i > r-1 for all i and hence that g_ird_i is non-empty, see <cit.>.
			
			We now treat the case ρ(g,r,d) is odd. 
			
			We will keep the notations from the even case. In this situation, we have 
			
    ρ(g,r,d) =-2k+1 ≡ g r+1.

			
			Starting with (r-1,r-1,…,r-1, r)∈ (_>0)^⊕ k, we add r+1 to the first entry then the second, and so on, repeating cyclically until we obtain (g_1, g_2…, g_k) where g_i≡ -2r+1 for 1≤ i ≤ k-1, g_k ≡ -1 r+1 and g=∑_i=1^k g_i. Let v_i=g_i+2/r+1+d-r-g_i for 1≤ i ≤ k-1 and v_k = g_k+1/r+1+d-r-g_k. The vanishing orders are determined inductively by 
			
    v_1   =g_1+2/r+1+d-r-g_1
    
    				v_2^1   =d-v_1-r
    
    				v_2^2   =v_2-v_2^1
    
    				v_i^1   =d-v_i-1^2-r
    
    				v_i^2   =v_i-v_i^1=g_i+2/r+1-g_i+v^2_i-1.

			By construction, a g^r_d with these vanishing orders satisfies the compatibility condition to be a refined limit linear series for i≤ k-1, and at p_k we have 
			
    v_k+r+v_k-1^2    = ( ∑_i=1^k-1g_i+2/r+1-g_i ) + g_k+1/r+1-g_k +2d-r
       = g-ρ(g,r,d)/r+1-g+2d-r
       =d,
 thus the compatibility condition is satisfied at every clutching point. As before, v_i=v_i^1+v_i^2 by definition and one checks that ρ(g_i,r,d-v_i)=-2 for 1≤ i ≤ k-1 and ρ(g_k,r,d-v_k)=-1. Taking d_i=d-v_i we obtain the Brill–Noether loci g_ird_i having either codimension 1 or 2. By taking [C_i] ∈g_ird_i and glueing at generic points to form a chain [C_1∪ C_2∪⋯∪ C_k] we obtain our desired curve.
		
		
		The numerical condition (∗) ensures that all the Brill–Noether loci we consider are non-empty. The condition is very mild. We identify precisely when the numerical condition above holds. 
		
		
			Let grd be an expected maximal Brill–Noether locus. Then 
			
				
  (∗) (2r+1)-ρ(g,r,d)+1/2--ρ(g,r,d)/2≤ g
			 holds unless ρ(g,r,d)=-(r+1)=-√(g) is odd and g is not a square.
		
		
			We note that (∗) does not hold in general when ρ(g,r,d)=-r-1 is odd and r=√(g)-1, the expected maximal Brill–Noether locus 42641 provides such an example. In fact, for any genus of the form g=n^2-n with √(g)-1 even, the expected maximal Brill–Noether locus g√(g)d contradicts (∗).
		
		
			Assume that ρ(g,r,d) is even, then 
    (2r+1)-ρ(g,r,d)+1/2--ρ(g,r,d)/2 = -rρ(g,r,d),
 and since for expected maximal loci -ρ(g,r,d)≤ r+1, we have -rρ(g,r,d)≤ r(r+1). To see that this holds for expected maximal Brill–Noether loci, first note that r+1≤ g-d+r. We now compute 
			
    ρ(g,r,d)   ≥ -r-1
    
    				g+r+1    ≥ (r+1)(g-d+r) ≥ (r+1)^2
    
    				g   ≥ r(r+1),
 as was to be shown.
			
			Assume now that ρ(g,r,d) is odd. Then (∗) reads 
    -rρ(g,r,d)+r+1≤ g.
 As above, one sees that if -ρ(g,r,d)≤ r-1, then this holds. If -ρ(g,r,d)=r, then (∗) reads r^2+r+1≤ g, which clearly holds if r≤√(g)-1. Similarly, if -ρ(g,r,d)=r+1, then (∗) reads (r+1)^2≤ g, which holds if r≤√(g)-1. 
			
			Thus we may assume that r=√(g)-1 for g not a square, and r≤ -ρ(g,r,d)≤ r+1. 
			
			It remains to show that (∗) holds when -ρ(g,r,d)=r=√(g)-1=√(g) is odd, g is not a square, and g≥√(g)^2+√(g), see <Ref> in <Ref>. In this case, (∗) reads 
    r^2+r+1≤ g.
 We show that this holds. From <Ref>, we have 
    -ρ(g,r,d)=r≡ r+1-(g r+1),
 hence g≡ 1  r+1. Thus, as g≥√(g)^2+√(g)=r(r+1), we must have g≥ r^2+r+1, as claimed. 
		
		
		
			In particular, (∗) is satisfied for all but possibly one expected maximal Brill–Noether locus grd, the one with largest r and smallest ρ. Indeed, when (∗) is not satisfied, then we have ρ(g,r,d)<ρ(g,s,e) for all other expected maximal Brill–Noether loci gse.
		
		
		By imposing different requirements on the dimensions of the Brill–Noether loci g_ird_i we can obtain similar results. The proof of <Ref> can be adapted to conclude these new results.
		
			Let grd be a Brill–Noether locus  satisfying the numerical condition
			
    -ρ(g,r,d)·(2r+1)≤ g.
 
			Then the closure of this locus in ℳ_g contains a chain curve [C_1∪ C_2∪⋯ C_k] such that
			
				
  * Each curve C_i is generic in a Brill–Noether divisor g_ird_i of some ℳ_g_i;
				
  * Each glueing point is generic on both components it connects.
			
		 
		
		In fact, if we allow the “clutching components" g_ird_i of the expected maximal Brill–Noether loci to be of expected codimension 3, a similar proposition holds with no numerical requirement.
		
		
			Let grd be an expected maximal Brill–Noether locus. The closure of this locus in ℳ_g contains a chain curve [C_1∪ C_2∪⋯ C_k] such that
			
				
  * Each curve C_i is generic in a Brill–Noether locus g_ird_i with -1≥ρ(g_i,r,d_i)≥ -3; 
				
  * Each glueing point is generic on both components it connects.
			
		 
		
		With these results in hand we prove our main theorem, that the dimensionally expected non-containments of expected maximal Brill–Noether loci hold.
		
		
			Let grd and gse be expected maximal Brill–Noether loci. If ρ(g,s,e)<ρ(g,r,d), then grd⊈gse.
		
		
			As noted in <Ref>, the condition (∗) of <Ref> holds unless 
    -ρ(g,r,d)=r+1=√(g)-1+1
 is odd and g is not a square, whereby ρ(g,r,d)<ρ(g,s,e) for all expected maximal loci gse. By assumption, we have ρ(g,s,e)<ρ(g,r,d), thus we may assume (∗) holds.
			
			Consider a chain curve
			
    C C_1 ∪_p_1∼ q_2^1 C_2 ∪_q_2^2∼ q_3^1C_3∪⋯∪_q_k-1^2∼ p_k C_k

			in the boundary of grd as described in <Ref>. Each irreducible component C_i is generic in a Brill–Noether locus of codimension 1 or 2, depending on the parity of ρ(g,r,d) as in <Ref>. 
			
			Assume for contradiction that we have the containment grd⊆gse. This implies that C admits a limit se. Denoting the aspects of the limit se by l_i, <Ref>, <Ref> and <Ref> imply that ρ(l_1,p_1)≥ -2, ρ(l_i,q_i^1, q_i^2)≥ -2, and ρ(l_k,p_k)≥
				-1    if ρ(g,r,d) is odd

				-2    if ρ(g,r,d) is even. Brill–Noether additivity gives 
    ρ(g,s,e)≥ρ(l_1,p_1)+( ∑_i=2^k-1ρ(l_i,q_i^1, q_i^2))+ρ(l_k,p_k)≥ -2k+2 +
    				-1    if ρ(g,r,d) is odd
    
    				-2    if ρ(g,r,d) is even = ρ(g,r,d),
 contradicting ρ(g,s,e)<ρ(g,r,d).
		
		
		
		
		
		

§ EXISTENCE OF COMPONENTS OF EXPECTED DIMENSION
 
		
		The question of whether Brill–Noether loci, or more generally the schemes ^r_g,d, have components of the expected dimension has recently received attention in the work of many authors, in particular Pflueger and Teixidor i Bigas <cit.>. They show that when -ρ(g,r,d)≤ g-3, then there exists components of expected dimension (or expected relative dimension for ^r_g,d→_g) <cit.>, and in case d≠ g-1, then this also holds for -ρ(g,r,d)≤ g-2 <cit.>. We give a new proof of the existence of components of expected dimension in a smaller range.
		
		Reasoning as in <Ref> immediately gives components of the expected dimension.
		
		
			If d≤ 2g-2 and -ρ(g,r,d)≤g/2, then grd has a component of the expected dimension.   
		
		
		
			The low genus cases 2≤ g ≤ 7 are an immediate consequence of <cit.>, while the case r=1 is well-known in the literature, see <cit.> and <cit.>. We assume g≥ 8, r≥ 2 and prove the statement by reasoning inductively. We will consider two cases, depending on how large the value -ρ(g,r,d) is.
			
			Case I: We assume -ρ(g,r,d) ≥ r. 
			
			In this case, we consider a (hyperelliptic) curve [C_1] ∈r+2r2r and a curve [C_2]∈g-r-2rd-r and let p_1∈ C_1 and p_2∈ C_2.
			
			We know that the locus r+2r2r = r+212 is irreducible of codimension r. By induction, we also know that g-r-2rd-r has a component of expected dimension, as the numerical conditions in the hypothesis are satisfied:  
			 
				
  * The condition 
    ρ(g-r-2, r, d-r) = ρ(g,r,d) + r ≥ -g-r-2/2

				is an immediate consequence of r≥ 2 and the hypothesis ρ(g,r,d)≥ -g/2.
				
  * For the condition d-r ≤ 2(g-r-2)-2, i.e. d≤ 2g-r-6, we assume d≤ g-1 by Serre duality. If the condition is not satisfied, we obtain the inequality 
				
    2g-r-5 ≤ d ≤ g-1
 
				and hence g ≤ r+4 and d≤ r+3. Clifford's inequality 2r≤ d implies r≤ 3 and hence g≤ 7, contradicting our assumption.
			
			
			By taking [C_2] in a component of expected dimension of g-r-2rd-r and reasoning as in the proof of <Ref> we obtain that [C_1∪_p_1∼ p_2C_2] ∈ℳ^r_g,d. In particular, we found a locus having expected codimension in the boundary of ℳ_g. This locus must be contained in a component of grd of expected codimension -ρ(g,r,d).
			
			Case II: Assume that -ρ(g,r,d) ≤ r-1. 
			
			In this situation, we consider 
    [C_1]∈3r+3+ρ(g,r,d)r4r+ρ(g,r,d) and  [C_2] ∈g-3r-3-ρ(g,r,d)rd-3r-ρ(g,r,d).

			We note that the genus g-3r-3-ρ(g,r,d) is nonnegative. Indeed, from <Ref>, we see that grd is expected maximal,  hence 
			
    r ≤√(g)-1   if  g≥√(g)^2+√(g)
    √(g)-1   if  g<√(g)^2+√(g),

			and we note that the inequality 
			
    g≤ 3r+2 + ρ(g,r,d)

			cannot be satisfied for g≥ 8. We also note that since grd is expected maximal and g≥ 8, the degree d-3r-ρ(g,r,d) is non-negative.
			
			Reasoning inductively we see that 3r+3+ρ(g,r,d)r4r+ρ(g,r,d) has a component of codimension -ρ(g,r,d) in ℳ_3r+3+ρ(g,r,d). 
			
			Moreover, as 
			
    ρ(g-3r-3-ρ(g,r,d), r, d-3r-ρ(g,r,d)) = 0,

			we obtain g-3r-3-ρ(g,r,d)rd-3r-ρ(g,r,d) = ℳ_g-3r-3-ρ(g,r,d), hence the Brill–Noether locus has codimension 0, and has a component of expected dimension.
			
			Reasoning as in the proof of <Ref> we get that [C_1∪_p_1∼ p_2 C_2] ∈ℳ^r_g,d when [C_1] is contained in a component of expected dimension of g-3r-3-ρ(g,r,d)rd-3r-ρ(g,r,d). 
			
			In particular, we found a locus having expected codimension -ρ(g,r,d) in the boundary. This locus must be in the intersection of the boundary with a component of grd having codimension -ρ(g,r,d) in ℳ_g.
		
		
		
		
		
		

§ NON-CONTAINMENTS OBTAINED FROM PRYM

		
		In this section, we look at the Prym moduli space ℛ_g parametrizing unramified double covers [fC→ C] of genus g curves, and consider the map 
		
    χ_gℛ_g →ℳ_2g-1

		sending the double cover [fC→ C] to the source curve C. In analogy to <cit.>, where gonality loci were used to distinguish Brill–Noether loci, we consider how Im(χ_g) intersects the Brill–Noether stratification of _2g-1, thereby obtaining new non-containments of Brill–Noether loci.
		
		The following proposition is an immediate consequence of <cit.>.
		
			Let g = 1 + r(r+1)/2+ε for 0≤ε < r/2. Then 
    Im(χ_g) ⊆ℳ^r_g, 2g-2.

			where g = 2g-1 =  1 + r(r+1) + 2ε.
		
		
		 We have the following obvious containment between Prym–Brill–Noether and Brill–Noether spaces: 
			
    V^r(fC→ C) ⊆ W^r_2g-2(C)

			By <cit.>, V^r(fC→ C) ≠∅ for any [fC→ C] ∈ℛ_g, it follows that any C in the image of χ_g admits a g^r_2g-2, i.e. 
    Im(χ_g) ⊆ℳ^r_2g-1,2g-2.

		
		
		We remark that 2g-1r2g-2 is expected maximal. Indeed, as g̃=2g-1, we have 
    -r-1≤ρ(2g-1,r,2g-2) = 2g-1 - (r+1)(r+1) = 2ε -r  ≤ -1

		and hence as r≤√(2g-1), we see that r satisfies <Ref> (with genus g̃=2g-1), hence <Ref> shows that 2g-1r2g-2 is expected maximal.
		
		Conversely, <cit.> shows that Im(χ_g) is not contained in certain Brill–Noether loci.
		
		
			Let g = 2g-1 and r, d two numbers such that ρ(g,r,d) = -r-1. Then we have the non-containment 
			
    Im(χ_g) ⊈ℳ^r_g,d.

		  
		
		Using the method of <cit.> we can prove that Im(χ_g) is not contained in certain Brill–Noether loci.
		
		
			Let g = 2g-1 and r, d two numbers such that ρ(g,r,d) = -r and either 
			
				
  * r is even and d is odd, or 
				
  * r≡ 1 4 and d is odd.
			
			Then we have the non-containment 
    Im(χ_g) ⊈ℳ^r_g,d.

		
		
		
			We assume Im(χ_g) ⊆ℳ^r_g,d and we will reach a contradiction. For this, we will provide a curve in the closure Im(χ_g) that does not admit a limit g^r_d. 
			
			As in the proof of <cit.>, let π_EE→ E be an étale double cover of an elliptic curve, p ∈ E and {x,y}π_E^-1(p). Taking [C_1,p_1] and [C_2, p_2], two copies of a generic pointed curve [C,p] ∈ℳ_g-1,1, we obtain a double cover 
			
    [C_1∪_p_1∼ xE∪_y∼ p_2C_2→ C∪_p E] ∈ℛ_g

			and see that C[C_1∪E∪ C_2/p_1∼ x, p_2∼ y] ∈Im(χ_g), see the boundary description of ℛ_g in <cit.> and <cit.>.
			Assume that C admits a limit g^r_d and denote by l_1, l and l_2 its aspects over the curves C_1, E and C_2. Moreover, we denote by w_i the vanishing orders of l_i at the node p_i for i =1,2 and by w_1, w_2 the vanishing orders of l at the points x and y. 
			
			By Brill–Noether additivity, we have 
			
    ρ(2g-1,r,d) = -r ≥ρ(l_1, p_1) + ρ(l_2,p_2) + ρ(l, x, y) ≥ 0 + 0 + (-r) = -r

			We have used here that the Brill–Noether number is non-negative for every  linear series on a generic pointed curve [C,p] ∈ℳ_g-1,1, see <cit.>, and that ρ(l, x, y) ≥ -r for every g^r_d and every two points on an elliptic curve, see <cit.>. 
			
			This double inequality implies that ρ(l_1,p_1) = ρ(l_2,p_2) = 0 and ρ(l, x, y) = -r and the limit linear series is refined. Let (a_0, …, a_r) and (b_0,…, b_r) be the entries of w_1 and w_2, respectively.
			
			Because ρ(l, x, y) = -r, we must have a_i + b_r-i = d for every 0≤ i ≤ r. Moreover, because 2x≡ 2y all the a_i's have the same parity. Implicitly, all the b_i's have the same parity. 
			
			Because the limit linear series is refined, we must have w_2 = (a_0,…, a_r) and w_1 = (b_0,…, b_r). 
			
			Because ρ(g-1,r,d,w_1) = ρ(g-1,r,d, w_2) = 0 we get that 
			
    ∑_i=0^ra_i = ∑_i=0^r b_i = (r+1)d/2

			
			When r is even and d is odd, this is impossible.
			
			When r≡ 1 4 and d is odd, we obtain the contradiction 
			
    0 ≡∑_i=0^r a_i ≡(r+1)d/2≡ 1 2.
 Therefore the curve  C does not admit any limit g^r_d.
		
		
		As a consequence of <Ref> and <Ref>, we obtain new non-containments of Brill–Noether loci.
		 
			Let g = 1 + r(r+1) + 2ε for some 0≤ε < r/2 and let s,d be positive integers satisfying either 
			
				
  * ρ(g,s,d) = -s-1, or 
				
  * ρ(g,s,d) = -s, d is odd and s ≢3 4,
			
			then there is a non-containment 
			
    ℳ^r_g,g-1⊈ℳ^s_g,d.

		
		
			Let g'  1 + r(r+1)/2 + ε. By <Ref>, a generic element in the locus Im(χ_g') is contained in ℳ^r_g,g-1 but <Ref> or <Ref> show that  Im(χ_g')⊈ℳ^s_g,d. The conclusion follows. 
		
		
		This gives infinitely many non-containments of expected maximal Brill–Noether loci of the form grd⊈gse with s<r, which has been heretofore out of reach of other techniques in general. We give an example of an infinite family of non-containments by taking ε=0.
		
		
			Let r be an even integer not divisible by 4 and let g=r^2+r+1. Then we have a non-containment of expected maximal Brill–Noether loci 
    grg-1⊈gr-1g-3.

		
		
			One checks that ρ(g,r,g-1)=-r, and ρ(g,r-1,g-3)=-r+1. The result follows from <Ref>.
		
		
		By taking larger values of ε, one might potentially obtain further families of non-containments of expected maximal Brill–Noether loci.
		
		
		
			These results, however, cannot show the conjectured non-containments of expected maximal Brill–Noether loci of the form 
    r^2+rrr^2+r-1⊈r^2+rr-1r^2+r-3.
 In fact, at present, these non-containments remain out of reach in general for all known techniques.
		
		
		
		
		
		
		
		
		alpha
		
		
		