TROPICAL INDEPENDENCE II: THE MAXIMAL RANK CONJECTURE FOR QUADRICS

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1 TROPICAL INDEPENDENCE II: THE MAXIMAL RANK CONJECTURE FOR QUADRICS DAVID JENSEN AND SAM PAYNE Abstact. Building on ou ealie esults on topical independence and shapes of divisos in topical linea seies, we give a topical poof of the maximal ank conjectue fo quadics. We also pove a topical analogue of Max Noethe s theoem on quadics containing a canonically embedded cuve, and state a combinatoial conjectue about topical independence on chains of loops that implies the maximal ank conjectue fo algebaic cuves. 1. Intoduction Let X P be a smooth cuve of genus g, and ecall that a linea map between finite dimensional vecto spaces has maximal ank if it is eithe injective o sujective. The kenel of the estiction map ρ m : H 0 (P, O(m)) H 0 (X, O(m) X ) is the space of homogeneous polynomials of degee m that vanish on X. The conjectue that ρ m should have maximal ank fo sufficiently geneal embeddings of sufficiently geneal cuves, attibuted to Noethe in [AC83, p. 4], 1 was studied classically by Sevei [Sev15, 10], and populaized by Hais [Ha8, p. 79]. Maximal Rank Conjectue. Let V L(D X ) be a geneal linea seies of ank 3 and degee d on a geneal cuve X of genus g. Then the multiplication maps have maximal ank fo all m. µ m : Sym m V L(mD X ) Ou main esult gives a combinatoial condition on the skeleton of a cuve ove a nonachimedean field to ensue the existence of a linea seies such that µ has maximal ank. In paticula, wheneve the geneal cuve of genus g admits a nondegeneate embedding of degee d in P then the image of a geneal embedding is contained in the expected numbe of independent quadics. Let Γ be a chain of loops connected by bidges with admissible edge lengths, as defined in 4. See Figue 1 fo a schematic illustation, and note that ou conditions on the edge lengths ae moe estictive than those in [CDPR1, JP14]. Theoem 1.1. Let X be a smooth pojective cuve of genus g ove a nonachimedean field such that the minimal skeleton of the Bekovich analytic space X an is isometic to Γ. Suppose 3, ρ(g,, d) 0, and d < g +. Then thee is a vey ample complete linea seies L(D X ) of degee d and ank on X such that the multiplication map µ : Sym L(D X ) L(D X ) has maximal ank. 1 Noethe consideed the case of space cuves in [Noe8, 8]. See also [CES5, pp ] fo hints towad Noethe s undestanding of the geneal poblem. 1

2 DAVID JENSEN AND SAM PAYNE Such cuves do exist, ove fields of abitay chaacteistic, and the condition that X an has skeleton Γ ensues that X is Bill Noethe Peti geneal [JP14]. As explained in, to pove the maximal ank conjectue fo fixed g,, d, and m it is enough to poduce a single linea seies V L(D X ) on a single Bill Noethe Peti geneal cuve fo which µ m has maximal ank. In paticula, the maximal ank conjectue fo m =, and abitay g,, and d, follows fom Theoem 1.1. Sujectivity of µ m fo small values of m can often be used to pove sujectivity fo lage values of m. See, fo instance, [ACGH85, pp ]. In chaacteistic zeo, when Theoem 1.1 gives sujectivity of µ, we apply standad aguments fom linea seies on cuves to deduce sujectivity of µ m fo all m. Theoem 1.. Let X and D X be as in Theoem 1.1, and suppose µ is sujective. Then µ m is sujective fo all m. This confims the maximal ank conjectue fo all m in the ange whee µ is sujective. In paticula, a geneal nondegeneate embedding of a geneal cuve ove an algebaically closed field of chaacteistic zeo is pojectively nomal if and only if ( ) + d g + 1. Remak 1.3. The maximal ank conjectue is known, fo all m, when = 3 [BE87a], and in the non-special case d + g [BE87b]. Thee is a ich histoy of patial esults on the maximal ank conjectue fo m =, including some with significant applications. Voisin poved the case of adjoint bundles of gonality pencils and deduced the sujectivity of the Wahl map fo geneic cuves [Voi9, 4]. Teixido poved that µ is injective fo all linea seies on the geneal cuve when d < g + [TiB03]. Fakas poved the case whee ρ(g,, d) is zeo and dim Sym L(D X ) = dim L(D X ), and used this to deduce an inifinite sequence of counteexamples to the slope conjectue [Fa09, Theoem 1.5]. Anothe special case is Noethe s theoem on canonically embedded cuves, discussed below. Futhemoe, Lason has poved an analogue of the maximal ank conjectue fo hypeplane sections of cuves [La1]. This is only a small sampling of pio wok on the maximal ank conjectue. We note, in paticula, that the liteatue contains many shot aticles by Ballico based on classical degeneation methods, and the maximal ank conjectue fo quadics in chaacteistic zeo appeas as Theoem 1 in [Bal1]. Two key tools in the poof of Theoem 1.1 ae the lifting theoem fom [CJP15] and the notion of topical independence developed in [JP14]. The lifting theoem allows us to ealize any diviso D of ank on Γ as the topicalization of a diviso D X of ank on X. Ou undestanding of topical linea seies on Γ, togethe with the nonachimedean Poincaé-Lelong fomula, poduces ational functions {f 0,..., f } in the linea seies L(D X ) whose topicalizations {ψ 0,..., ψ } ae a specific wellundestood collection of piecewise linea functions on Γ. We then show that a lage subset of the piecewise linea functions {ψ i + ψ j } 0 i j is topically independent. Since ψ i + ψ j is the topicalization of f i f j, the size of this subset is a lowe bound fo the ank of µ, and this is the bound we use to pove Theoem 1.1. Thee is no obvious obstuction to poving the maximal ank conjectue in full geneality using this appoach, although the combinatoics become moe challenging as the paametes incease. We state a pecise combinatoial conjectue in 4, which, fo any given g,, d, and m, implies the maximal ank conjectue fo the same g,, d, and m. We pove this conjectue not only fo m =, but also fo md < g + 4. (See Theoem 5.3.) We also pesent advances in undestanding multiplication maps by topical methods on skeletons othe than a chain of loops. Recall that Noethe s theoem on

3 TROPICAL INDEPENDENCE II: THE MAXIMAL RANK CONJECTURE FOR QUADRICS 3 canonically embedded cuves says that µ : Sym L(K X ) L(K X ) is sujective wheneve X is not hypeelliptic. This may be viewed as a stong fom of the maximal ank conjectue fo quadics in the case whee = g 1 and d = g. On the puely topical side, we pove an analogue of Noethe s theoem fo tivalent, 3-edge-connected gaphs. Theoem 1.4. Let Γ be a tivalent, 3-edge-connected metic gaph. Then thee is a topically independent set of 3g 3 functions in R(K Γ ). Futhemoe, we pove the appopiate lifting statements to leveage this topical esult into a maximal ank statement fo canonical embeddings of cuves with tivalent and 3-connected skeletons. Theoem 1.5. Let X be a smooth pojective cuve of genus g ove a nonachimedean field such that the minimal skeleton Γ of X an is tivalent and 3-edgeconnected with fist Betti numbe g. Then thee ae 3g 3 ational functions in the image of µ : Sym L(K X ) L(K X ) whose topicalizations ae topically independent. In paticula, µ is sujective. The last statement, on sujectivity of µ, also follows fom Noethe s theoem, because tivalent, 3-edge connected gaphs ae neve hypeelliptic [BN09, Lemma 5.3]. Remak 1.6. The pesent aticle is a sequel to [JP14], futhe developing the method of topical independence. This is just one aspect of the topical appoach to linea seies, an aay of techniques fo handling degeneations of linea seies ove a one paamete family of cuves whee the special fibe is not of compact type, combining discete methods with computations on skeletons of Bekovich analytifications. Seminal woks in the development of this theoy include [BN07, Bak08, AB15]. Combined with techniques fom p-adic integation, this method also leads to unifom bounds on ational points fo cuves of fixed genus with small Modell Weil ank [KRZB15]. This topical appoach is in some ways analogous to the theoy of limit linea seies, developed by Eisenbud and Hais in the 1980s, which systematically studies the degeneation of linea seies to singula cuves of compact type [EH86]. This theoy led to simplified poofs of the Bill Noethe and Gieseke Peti theoems [EH83], along with many new esults about the geomety of cuves, linea seies, and moduli [EH87a, EH87b, EH87c, EH89]. Topical methods have also led to new poofs of the Bill Noethe and Gieseke Peti theoems [CDPR1, JP14]. Some pogess has been made towad building famewoks that include both classical limit linea seies and also genealizations of limit linea seies fo cuves not of compact type [AB15, Oss14a, Oss14b], which ae helpful fo explaining connections between the topical and limit linea seies poofs of the Bill Noethe theoem. These elations ae also addessed in [JP14, Remak 1.4] and [CLMTiB14]. The natue of the elations between the topical appoach and moe classical appoaches fo esults involving multiplication maps, such as the Gieseke Peti theoem and othe maximal ank esults, emain unclea, as do the elations between such basic and essential facts as the Riemann Roch theoems fo algebaic and topical cuves. Note that seveal families of cuves appeaing in poofs of the Bill Noethe and Gieseke Peti theoems ae not contained in the open subset of M g fo which the maximal ank condition holds. Fo example, the sections of K3 sufaces used by Lazasfeld in his poof of the Bill Noethe and Gieseke Peti theoems without degeneations [Laz86] do not satisfy the maximal ank conjectue fo m = [Voi9, Theoem 0.3 and Poposition 3.]. Futhemoe, the stabilizations of the flag cuves used by Eisenbud and Hais ae limits of such cuves [FP05, Poposition 7.].

4 4 DAVID JENSEN AND SAM PAYNE Acknowledgments. We thank D. Abamovich, E. Ballico, and D. Ranganathan fo helpful comments on an ealie vesion of this pape. The second autho was suppoted in pat by NSF CAREER DMS and is gateful fo ideal woking conditions at the Institute fo Advanced Study in Sping Peliminaies Recall that a geneal cuve X of genus g has a linea seies of ank and degee d if and only if the Bill Noethe numbe ρ(g,, d) = g ( + 1)(g d + ) is nonnegative, and the scheme Gd (X) paametizing its linea seies of degee d and ank is smooth of pue dimension ρ(g,, d). This scheme is ieducible when ρ(g,, d) is positive, and monodomy acts tansitively when ρ(g,, d) = 0. Theefoe, if U M g is the dense open set paametizing such Bill Noethe Peti geneal cuves, then Gd (U), the univesal linea seies of ank and degee d ove U, is smooth and ieducible of elative dimension ρ(g,, d). The geneal linea seies of degee d and ank on a geneal cuve of genus g appeaing in the statement of the maximal ank conjectue efes simply to a geneal point in the ieducible space Gd (U). When X is Bill Noethe Peti geneal and D X is a basepoint fee diviso of ank at least 1, the basepoint fee pencil tick shows that its multiples md X ae nonspecial fo m (see Remak.1). Theefoe, by standad uppe semincontinuity aguments fom algebaic geomety and the fact that Gd is defined ove Spec Z, to pove the maximal ank conjectue fo fixed g,, d, and m, ove an abitay algebaically closed field of given chaacteistic, it suffices to poduce a single Bill Noethe Peti geneal cuve X of genus g ove a field of the same chaacteistic with a linea seies V L(D X ) of degee d and ank such that µ m has maximal ank. As mentioned in the intoduction, the maximal ank conjectue is known when the linea seies is nonspecial. In the emaining cases, the geneal linea seies is complete, so we can and do assume that V = L(D X ). Remak.1. Suppose D X is a basepoint fee special diviso of ank 1 on a Bill Noethe Peti geneal cuve X. The fact that md X is nonspecial fo m is an application of the basepoint fee pencil tick, as follows. Choose a basepoint fee pencil V L(D X ). Then the tick identifies L(K X D X ) with the kenel of the multiplication map µ : V L(K X D X ) L(K X ). The Peti condition says that this multiplication map is injective, even afte eplacing V by L(D X ). Theefoe, thee ae no sections of K X that vanish on D X and hence no sections that vanish on md X fo m, which means that md X is nonspecial. Remak.. When 3 and ρ(g,, d) 0, the geneal linea seies of degee d on a geneal cuve of genus g defines an embedding in P, and hence the conjectue can be ephased in tems of a geneal point of the coesponding component of the appopiate Hilbet scheme. One can also conside analogues of the maximal ank conjectue fo cuves that ae geneal in a given ieducible component of a given Hilbet scheme, athe than geneal in moduli. Howeve, the maximal ank condition can fail when the Hilbet scheme in question does not dominate M g. Suppose, fo example, that X is a cuve of genus 8 and degee 8 in P 3. Then h 0 (O X ()) = 9, and hence X is contained in a quadic suface. It follows that the kenel of µ 3 has dimension at least 4, and theefoe µ 3 is not sujective. This does

5 TROPICAL INDEPENDENCE II: THE MAXIMAL RANK CONJECTURE FOR QUADRICS 5 not contadict the maximal ank conjectue, since the geneal cuve of genus 8 has no linea seies of ank 3 and degee 8. Poof of Theoem 1.. Suppose 4. We begin by showing that µ 3 is sujective. Note the polynomial identity ( ) ( ) ( ) + d g g d + (d g + 1) = ρ(g,, d). (This identity eappeas as Lemma 8., in the special case ρ(g,, d) = 0.) By assumption, the left hand side is nonnegative, as ae ρ(g,, d) and g d +. It follows that d g. By [ACGH85, Execise B-6, p. 138], it follows that the dimension of the linea seies spanned by sums of divisos in D X and D X is at least min{4d g, 3d g} = 3d g. Theefoe, if µ is sujective then µ 3 is also sujective. We now show, by induction on m, that µ m is sujective fo all m > 3. Let V L(D X ) be a basepoint fee pencil. By the basepoint fee pencil tick, we have an exact sequence 0 L((m 1)D X ) V L(mD X ) L((m + 1)D X ). Since (m 1)D X and md X ae both nonspecial, the image of the ight hand map has dimension (md g + 1) ((m 1)d g + 1) = (m + 1)d g + 1, hence it is sujective. It emains to conside the cases whee = 3. By assumption, the diviso D X is special, so d < g + 3. Futhemoe, µ is sujective, so d g , and ρ(g,, d) 0, so 3g 4d 1. This leaves exactly two possibilities fo (g, d), namely (4, 6) and (5, 7). In each of these cases, h 1 (O(D X )) = 1 and, since X is Bill Noethe geneal, Cliff(X) = g 1. Then d = g + 1 h 1 (O(D X )) Cliff(X) and hence O(D X ) gives a pojectively nomal embedding, by [GL86, Theoem 1]. Remak.3. In the above agument, the chaacteistic zeo assumption is used only to show that µ 3 is sujective. Even in positive chaacteistic, if µ k is sujective fo some k >, then µ m is sujective fo all m > k. Since we ae tying to poduce a single sufficiently geneal cuve of each genus ove a field of each chaacteistic, we may, fo simplicity, assume that we ae woking ove an algebaically closed field that is spheically complete with espect to a valuation that sujects onto the eal numbes. Any metic gaph Γ of fist Betti numbe g appeas as the skeleton of a smooth pojective genus g cuve X ove such a field (see, fo instance, [ACP1]). Recall that the skeleton is a subset of the set of valuations on the function field of X, and evaluation of these valuations, also called topicalization, takes each ational function f on X to a piecewise linea function with intege slopes on Γ, denoted top(f). Ou pimay tool fo using the skeleton of a cuve and topicalizations of ational functions to make statements about anks of multiplication maps is the notion of topical independence developed in [JP14]. The statement of the execise is missing a necessay hypothesis, that D has ank at least 3. The solution following the hint equies the unifom position lemma, which is known fo 3 in chaacteistic zeo [Ha80] and, ove abitay fields, when 4 [Rat87].

6 6 DAVID JENSEN AND SAM PAYNE Definition.4. A set of piecewise linea functions {ψ 0,..., ψ } on a metic gaph Γ is topically dependent if thee ae eal numbes b 0,..., b such that fo evey point v in Γ the minimum min{ψ 0 (v) + b 0,..., ψ (v) + b } occus at least twice. If thee ae no such eal numbes then {ψ 0,..., ψ } is topically independent. One key basic popety of this notion is that if {top(f i )} i is topically independent on Γ, then the coesponding set of ational functions {f i } i is linealy independent in the function field of X [JP14, 3.1]. Note also that if f and g ae ational functions, then top(f g) = top(f) + top(g). Remak.5. Adding a constant to each piecewise linea function does not affect the topical independence of a given collection. When {ψ 0,..., ψ } is topically dependent, we often eplace each ψ i with ψ i + b i and assume that the minimum of the set {ψ 0 (v),..., ψ (v)} occus at least twice at evey point v Γ. Lemma.6. Let D X be a diviso on X, and let {f 0,..., f } be ational functions in L(D X ). If thee exist k multisets I 1,..., I k {0,..., }, each of size m, such that { i I j top(f i )} j is topically independent, then the multiplication map has ank at least k. µ m : Sym m L(D X ) L(mD X ) Poof. The topicalization of i I j f i is the coesponding sum i I j top(f i ). If these sums { i I j top(f i )} j ae topically independent then the ational functions { i I j f i } j ae linealy independent. These k ational functions ae in the image of µ m, and the lemma follows. Remak.7. If f 0,..., f ae ational functions in a linea seies L(D X ), and b 0,..., b ae eal numbes, then the pointwise minimum θ = min{top(f 0 ) + b 0,..., top(f ) + b } is the topicalization of a ational function in L(D X ). The ational function may be chosen of the fom a 0 f a f whee a i is a sufficiently geneal element of the gound field such that val(a i ) = b i. We will also epeatedly use the following basic fact about the shapes of divisos associated to a pointwise minimum of functions in a topical linea seies. Shape Lemma fo Minima. [JP14, Lemma 3.4] Let D be a diviso on a metic gaph Γ, with ψ 0,..., ψ piecewise linea functions in R(D), and let θ = min{ψ 0,..., ψ }. Let Γ j Γ be the closed set whee θ is equal to ψ j. Then div(θ) + D contains a point v Γ j if and only if v is in eithe (1) the diviso div(ψ j ) + D, o () the bounday of Γ j. This shape lemma fo minima is combined with anothe lemma about shapes of canonical divisos to each the contadiction that poves the Gieseke Peti theoem in [JP14].

7 TROPICAL INDEPENDENCE II: THE MAXIMAL RANK CONJECTURE FOR QUADRICS 7 3. Max Noethe s theoem Hee we examine functions in the canonical and -canonical linea seies using tivalent and 3-edge-connected gaphs. This section is not logically necessay fo the poof of Theoem 1.1, and can be safely skipped by a eade who is inteested only in the poof of the maximal ank conjectue fo quadics. Nevetheless, the two ae not unelated and we include this section because, as explained in the intoduction, Noethe s theoem is a stong fom of one case of the maximal ank conjectue fo quadics. Also, the aguments pesented hee illustate the potential fo applying ou methods to the study of linea seies and multiplication maps using skeletons othe than a chain of loops, which may be impotant fo futue wok. Ou aguments in this section depend on a caeful analysis of the loci whee piecewise linea functions attain thei minima. Recall that, fo a diviso D on a metic gaph Γ, the topical linea seies R(D) is the set of piecewise linea functions with intege slope ψ on Γ such that div(ψ)+d is effective. The topical linea seies R(D) is a topical module, which means that it is closed unde scala addition and pointwise minimum [HMY1, Lemma 4]. Fo v Γ, we wite deg v (D) fo the coefficient of v in the diviso D, and fo a piecewise linea function ψ, we wite Γ ψ = {v Γ ψ(v) = min w Γ ψ(w)} fo the subgaph on which ψ attains its global minimum. Lemma 3.1. Let D be a diviso on Γ with ψ R(D). Then, fo any point v Γ ψ, outdeg Γψ (v) deg v (D). Poof. Since ψ obtains its minimum value at v, all of the outgoing slopes of ψ at v ae nonnegative, and those along edges that ae not in Γ ψ ae stictly positive. Since all of these slopes ae integes and div(ψ) + D is effective, it follows that outdeg Γψ (v) is at most deg v (D). Recall that the canonical diviso K Γ is given by deg v (K Γ ) = val(v), whee val(v) is the valence (o numbe of outgoing edges) of v in Γ. The following lemma esticts the loci on which functions in R(K Γ ) attain thei minimum. Lemma 3.. Let ψ be a piecewise linea function in R(K Γ ). Then the subgaph Γ ψ on which ψ attains its minimum is a union of edges in Γ and has no leaves. Poof. By Lemma 3.1, the outdegee outdeg v (Γ ψ ) is at most deg(v) at each point v Γ ψ. It follows that any edge which contains a point of Γ ψ in its inteio is entiely contained in ψ, and the numbe of edges in Γ ψ containing any vetex v is at least two, so Γ ψ has no leaves. As a fist application, we show that evey loop in Γ is the locus whee some function in R(K Γ ) attains its minimum, and that this function lifts to a canonical section on any totally degeneate cuve whose skeleton is Γ. Hee, a loop is an embedded cicle in Γ o, equivalently, a connected subgaph in which evey point has valence. Poposition 3.3. Let Γ be a metic gaph and let Γ Γ be a loop. Then thee is a function ψ R(K Γ ) such that the subgaph Γ ψ on which ψ attains its minimum is exactly Γ. Futhemoe, if X is a smooth pojective cuve ove a nonachimedean field such that the minimal skeleton of the Bekovich analytic space X an is isometic to Γ,

8 8 DAVID JENSEN AND SAM PAYNE and K X is a canonical diviso that topicalizes to K Γ, then ψ can be chosen to be top(f) fo some f L(K X ). Poof. Let g be the fist Betti numbe of Γ. Choose points v 1,..., v g 1 of valence in Γ Γ such that Γ {v 1,..., v g 1 } is connected. Since K Γ has ank g 1, thee is a diviso D K Γ such that D v 1 v g 1 is effective. Let ψ be a piecewise linea function such that K Γ + div(ψ) = D. By Lemma 3.1, the subgaph Γ ψ Γ whee ψ attains its minimum is a union of edges of Γ and has no leaves. Since od vi (ψ) is positive fo 1 i g 1, it follows that Γ ψ does not contain any of the points v 1,..., v g 1. Being a subgaph of Γ {v 1,..., v g 1 }, the fist Betti numbe of Γ ψ is at most 1. On the othe hand, evey point has valence at least two in Γ ψ. It follows that Γ ψ is a loop, and hence must be the unique loop Γ contained in Γ {v 1,..., v g 1 }. We now pove the last pat of the poposition. Let p 1,..., p g 1 be points in X specializing to v 1,..., v g 1, espectively. Since K X has ank g 1, thee is a ational function f L(K X ) such that div(f) + K Γ p 1 p g 1 is effective. Fom this we see that div(top(f)) + K Γ v 1 v g 1 is effective, and the poposition follows. Ou next lemma contols the locus whee a piecewise linea function in R(K Γ ) attains its minimum, when Γ is tivalent. Lemma 3.4. Suppose Γ is tivalent and let ψ be a piecewise linea function in R(K Γ ). Then Γ ψ is a union of edges in Γ. Poof. If v Γ ψ lies in the inteio of an edge of Γ then, by Lemma 3.1, we have outdeg Γψ (v) = 0, so Γ ψ contains the entie edge. On the othe hand, if v Γ ψ is a tivalent vetex of Γ then Lemma 3.1 says that outdeg Γψ (v). It follows that Γ ψ contains at least one of the thee edges adjacent to v. We conclude this section by applying this lemma and the peceding poposition togethe with Menge s theoem to pove Theoem 1.4, the analogue of Noethe s theoem fo tivalent 3-connected gaphs. Remak 3.5. A simila application of Menge s theoem is used to pove an analogue of Noethe s theoem fo gaph cuves in [BE91, 4]. Poof of Theoems 1.4 and 1.5. Assume Γ is tivalent and 3-edge-connected. Let e Γ be an edge with endpoints v and w. Since Γ is 3-edge-connected, Menge s theoem says that thee ae two distinct paths fom v to w that do not shae an edge and do not pass though e. Equivalently, thee ae two loops Γ 1 and Γ in Γ such that Γ 1 Γ = e. By Poposition 3.3 thee ae functions ψ 1 and ψ in R(K Γ ) such that Γ ψi = Γ i. We wite ψ e = ψ 1 + ψ, which is a piecewise linea function in R(K Γ ). Note that Γ ψe = e. Futhemoe, again by Poposition 3.3, if X is a cuve with skeleton Γ and K X is a canonical diviso topicalizing to K Γ, then we can choose f 1 and f in L(K X ) such that ψ i = top(f i ), and hence ψ e = top(f 1 f ) is the topicalization of a function in the image of µ : Sym (L(K X )) L(K X ). We claim that the set of 3g 3 functions {ψ e } e is topically independent. Suppose not. Then thee ae constants b e such that min e {ψ e + b e } occus twice at evey point of Γ. Let θ = min{ψ e + b e }, e which is a piecewise linea function in R(K Γ ). By Lemma 3.4, the function θ achieves its minimum along an edge, and hence thee must be two functions in the set {ψ e + b e } e that achieve thei minima along this edge. Howeve, by constuction, the functions ψ e + b e achieve thei minima along distinct edges, which is a contadiction. We conclude that {ψ e } e is topically independent, as claimed.

9 TROPICAL INDEPENDENCE II: THE MAXIMAL RANK CONJECTURE FOR QUADRICS 9 4. Special divisos on a chain of loops Fo the emainde of the pape, we focus ou attention on a chain of loops with bidges Γ, as pictued in Figue 1. Hee, we biefly ecall the classification of special divisos on Γ fom [CDPR1], along with the chaacteization of vetex avoiding classes and thei basic popeties. The gaph Γ has g + vetices, one on the lefthand side of each bidge, which we label w 0,..., w g, and one on the ighthand side of each bidge, which we label v 1,..., v g+1. Thee ae two edges connecting the vetices v k and w k, the top and bottom edges of the kth loop, whose lengths ae denoted l k and m k, espectively, as shown schematically in Figue 1. Fo 1 k g + 1 thee is a bidge connecting l k w 0 v 1 v n k w g 1 w 1 v g+1 v g w g m k Figue 1. The gaph Γ. w k and v k+1, which we efe to as the kth bidge β k, of length n k. Thoughout, we assume that Γ has admissible edge lengths in the following sense, which is stonge than the geneicity conditions in [CDPR1, JP14]. Definition 4.1. The gaph Γ has admissible edge lengths if 4gm k < l k min{n k 1, n k } fo all k, and thee ae no nontivial linea elations c 1 m c g m g = 0 with intege coefficients of absolute value at most g + 1. Remak 4.. The inequality 4gm k < l k is equied to ensue that the shapes of the functions ψ i and ψ ij ae as descibed in 6-7. Both inequalities ae used in the poof of Lemma 6., and the equied uppe bound on l k depends on the size of the multisets. Fo multisets of size m, we assume ml k < min{n k 1, n k }. In paticula, fo Theoem 1.1, the inequality 4l k < min{n k 1, n k } would suffice. The condition on intege linea elations is used in the poof of Poposition 7.6. The special diviso classes on a chain of loops, i.e. the classes of effective divisos D such that (D) > deg(d) g, ae explicitly classified in [CDPR1]. Evey effective diviso on Γ is equivalent to an effective w 0 -educed diviso D 0, which has d 0 chips at the vetex w 0, togethe with at most one chip on evey loop. We may theefoe associate to each equivalence class the data (d 0, x 1, x,... x g ), whee x i R/(l i + m i )Z is the distance fom v i to the chip on the ith loop in the counteclockwise diection, if such a chip exists, and x i = 0 othewise. The associated lingeing lattice path in Z, whose coodinates we numbe fom 0 to 1, is a sequence of points p 0,..., p g stating at p 0 = (d 0, d 0 1,..., d 0 + 1), with the ith step given by ( 1, 1,..., 1) if x i = 0, e p i p i 1 = j if x i = (p i 1 (j) + 1)m i mod l i + m i and both p i 1 and p i 1 + e j ae in C, 0 othewise,

10 10 DAVID JENSEN AND SAM PAYNE whee e 0,... e 1 ae the basis vectos in Z and C is the open Weyl chambe C = {y Z y 0 > > y 1 > 0}. By [CDPR1, Theoem 4.6], a diviso D on Γ has ank at least if and only if the associated lingeing lattice path lies entiely in the open Weyl chambe C. The steps in the diection 0 ae efeed to as lingeing steps, and the numbe of lingeing steps cannot exceed the Bill Noethe numbe ρ(g,, d). In the case whee ρ(g,, d) = 0, such lattice paths ae in bijection with ectangula tableaux of size ( + 1) (g d + ). This bijection is given as follows. We label the columns of the tableau fom 0 to and place i in the jth column when the ith step is in the diection e j, and we place i in the last column when the ith step is in the diection ( 1,..., 1). An open dense subset of the special diviso classes of degee d and ank on Γ ae vetex avoiding, in the sense of [CJP15, Definition.3], which means that the associated lingeing lattice path has exactly ρ(g,, d) lingeing steps, fo any i, x i m i mod (l i + m i ), and fo any i and j, x i (p i 1 (j))m i mod (l i + m i ). Vetex avoiding classes come with a useful collection of canonical epesentatives. If D is a diviso of ank on Γ whose class is vetex avoiding, then thee is a unique effective diviso D i D such that deg w0 (D i ) = i and deg vg+1 (D i ) = i. Equivalently, D i is the unique diviso equivalent to D such that D i iw 0 ( i)v g+1 is effective. Futhemoe, the diviso D i has no points on any of the bidges, fo i <, the diviso D i fails to have a point on the jth loop if and only if the jth step of the associated lingeing lattice path is in the diection e i, the diviso D fails to have a point on the jth loop if and only if the jth step of the associated lingeing lattice path is in the diection ( 1,..., 1). Notation 4.3. Thoughout, we let X be a smooth pojective cuve of genus g whose analytification has skeleton Γ. Fo the emainde of the pape, we let D be a w 0 -educed diviso on Γ of degee d and ank whose class is vetex avoiding, D X a lift of D to X, and ψ i a piecewise linea function on Γ such that D + div(ψ i ) = D i. By a lift of D to X, we mean that D X is a diviso of degee d and ank on X whose topicalization is D. Note that ψ i is uniquely detemined up to an additive constant, and fo i < the slope of ψ i along the bidge β j is p j (i). In this context, being w 0 -educed means that D = D, so the function ψ is constant. In paticula, the functions ψ 0,..., ψ have distinct slopes along bidges, so {ψ 0,..., ψ } is topically independent. Fo convenience, we set p j () = 0 fo all j. Poposition 4.4. Thee is a ational function f i L(D X ) such that top(f i ) = ψ i. Poof. The poof is identical to the poof of [JP14, Poposition 6.5], which is the special case whee ρ(g,, d) = 0. When ρ(g,, d) = 0, all diviso classes of degee d and ank ae vetex avoiding. Note that, since {ψ 0,..., ψ } is topically independent of size + 1, the set of ational functions {f 0,..., f } is a basis fo L(D X ). Fo a multiset I {0,..., } of size m, let D I = i I D i and let ψ I be a piecewise linea function such that md+div ψ I = D I. By constuction, the function ψ I is in R(mD) and agees with i I ψ i up to an additive constant. Conjectue 4.5. Suppose 3, ρ(g,, d) 0, and d < g +. Then thee is a diviso D of ank and degee d whose class is vetex avoiding on a chain of loops Γ

11 TROPICAL INDEPENDENCE II: THE MAXIMAL RANK CONJECTURE FOR QUADRICS 11 with geneic edge lengths, and a topically independent subset A {ψ I #I = m} of size {( ) } + m #A = min, md g + 1. m The conjectue is tivial fo = 0 and easy fo = 1, since the functions kψ 0 have distinct nonzeo slopes on evey bidge and hence {0, ψ 0,..., mψ 0 } is topically independent. Yet anothe easy case is m = 1, since {ψ 0,... ψ } is topically independent. In the emainde of the pape we pove the conjectue fo m = and fo md < g + 4. Poposition 4.6. Fo any fixed g,, d, and m, the maximal ank conjectue follows fom Conjectue 4.5. Poof. Choose a smooth pojective cuve X ove a nonachimedean field whose skeleton is Γ. Then X is Bill Noethe Peti geneal [JP14] and D lifts to a diviso D X of degee d and ank on X [CJP15]. We may assume 1, and it follows that md X is nonspecial fo m by Remak.1. By Lemma.6, the ank of µ m is at least as lage as any set A such that {ψ I I A} is topically independent. Theefoe, Conjectue 4.5 implies that µ m : Sym m L(D X ) L(mD X ) has maximal ank and, as discussed in, the maximal ank conjectue fo g,, d, and m follows. 5. Two points on each loop In this section, we show that any nontivial topical dependence among the piecewise linea functions ψ I, fo multisets I of size m, gives ise to a diviso equivalent to md with degee at least at w 0, degee at least at v g+1, and degee at least on each loop. As a consequence, we deduce Theoem 5.3, which confims Conjectue 4.5 and the maximal ank conjectue fo md < g + 4. We begin with the following obsevation. Lemma 5.1. Let I and J be distinct multisets of size m. Then, fo each loop γ in Γ, the estictions D I γ and D J γ ae distinct. Poof. Suppose γ is the jth loop. Let q i be the point on γ whose distance fom v j in the counteclockwise diection is x j p j 1 (i)m j. Then the degee of q i in D I is equal to the multiplicity of i in the multiset I, unless the jth step of the lingeing lattice path is in the diection e i, in which case the degee of q i in D I is zeo. It follows that the multiset I can be ecoveed fom the estiction D I γ. Let θ be the piecewise linea function θ = min I {ψ I }, which is in R(mD), and let be the coesponding effective diviso = md + div θ. By Lemma 5.1, no two functions ψ I can agee on an entie loop, so if the minimum occus eveywhee at least twice on a loop, then thee ae at least thee functions ψ I that achieve the minimum at some point of the loop. We will study θ and by systematically using obsevations like this one, examining behavio on each piece of Γ and contolling which functions ψ I can achieve the minimum at some point in each loop. Recall that, fo 0 k g, the kth bidge β k connects w k to v k+1. Let u k be the midpoint of β k 1. We then decompose Γ into g + locally closed subgaphs γ 0,..., γ g+1, as follows. The subgaph γ 0 is the half-open inteval [w 0, u 1 ). Fo

12 1 DAVID JENSEN AND SAM PAYNE 1 i g, the subgaph γ i, which includes the ith loop of Γ, is the union of the two half-open intevals [u i, u i+1 ), which contain the top and bottom edges of the ith loop, espectively. Finally, the subgaph γ g+1 is the closed inteval [u g+1, v g+1 ]. We futhe wite γ i fo the ith embedded loop in Γ, which is a closed subset of γ i, fo 1 i g. The decomposition is illustated by Figue. Γ = γ 0 γ g+1 γ 0 γ 1 γ γ g γ g+1 w 0 u 1 u ug v g+1 Figue. Decomposition of the gaph Γ into locally closed pieces {γ k }. Poposition 5.. Suppose the minimum of {ψ I (v)} I occus at least twice at evey point v in Γ. Then deg w0 ( ), deg vg+1 ( ), and deg( γ i ) ae all at least. Poof. Note that exactly one function ψ I has slope m on the fist bidge; this is the function coesponding to the multiset I = {0,..., 0}. Similaly, the only multiset that gives slope m 1 is {1, 0,..., 0}. Theefoe, if the minimum occus twice along the fist bidge, then the outgoing slope of θ at w 0 is at most m, and hence deg w0 ( ), as equied. Similaly, we have deg vg+1 ( ). It emains to show that deg( γ i ) fo 1 i g. Choose a point v γi. By assumption, thee ae at least two distinct multisets I and I such that both ψ I and ψ I obtain the minimum on some closed inteval containing v. By Lemma 5.1, the functions ψ I and ψ I do not agee on all of γi, so thee is anothe point v γi whee at least one of these two functions does not obtain the minimum. Without loss of geneality, assume that ψ I does not obtain the minimum at v. Then ψ I obtains the minimum on a pope closed subset of γi, and since γ i is a loop, this set has outdegee at least two. By the shape lemma fo minima (see ), it follows that (div(θ) + md) γ i has degee at least two. As an immediate application of this poposition, we pove Conjectue 4.5 fo md < g + 4. Theoem 5.3. If md < g + 4 then {ψ I #I = m} is topically independent. Poof. Suppose that {ψ I } I is topically dependent. Afte adding a constant to each ψ I, we may assume that the minimum θ(v) = min I ψ I (v) occus at least twice at evey point v in Γ. By Poposition 5., the estiction of = md + div(θ) to each of the g + locally closed subgaphs γ k Γ has degee at least two. Theefoe the degee of is at least g + 4, and the theoem follows. In paticula, the maximal ank conjectue holds fo md < g + 4. This patially genealizes the case whee m = and d < g +, poved by Teixido i Bigas [TiB03]. Note, howeve, that [TiB03] poves that the maximal ank condition holds fo all divisos of degee less than g +, wheeas Theoem 5.3 implies this statement only fo a geneal diviso.

13 TROPICAL INDEPENDENCE II: THE MAXIMAL RANK CONJECTURE FOR QUADRICS Pemissible functions In the peceding section, we intoduced a decomposition of Γ as the disjoint union of locally closed subgaphs γ 0,..., γ g+1 and poved that if θ(v) = min I ψ I (v) occus at least twice at evey point v in γ i then the degee of = md + div(θ) esticted to γ i is at least. These degees of estictions γi appea epeatedly thoughout the est of the pape, so we fix δ i = deg( γi ). By Poposition 5., we have δ i fo all i. We now discuss how the nonnegative intege vecto δ = (δ 0,..., δ g+1 ) esticts the multisets I such that ψ I can achieve the minimum on the kth loop of Γ. Fo a b, let Γ [a,b] be the locally closed, connected subgaph Γ [a,b] = γ a γ b. Note that the degees of divisos in a topical linea seies esticted to such subgaphs ae govened by the slopes of the associated piecewise linea functions, as follows. Suppose Γ Γ is a closed connected subgaph and ψ is a piecewise linea function with intege slopes on Γ. Then div(ψ Γ ) has degee zeo and the multiplicity of each bounday point v Γ is the sum of the incoming slopes at v, along the edges in Γ. Now div(ψ) Γ agees with div(ψ Γ ) except at the bounday points and a simple computation at the bounday points of the locally closed subgaph γ k, fo 1 k g shows that deg(div(ψ) γk ) = σ k (ψ) σ k+1 (ψ), whee σ k (ψ) is the incoming slope of ψ fom the left at u k. Similaly, deg(div(ψ) Γ[0,k] ) = σ k+1 (ψ). Ou indexing conventions fo lingeing lattice paths ae chosen fo consistency with [CDPR1], and with this notation we have σ k (ψ i ) = p k 1 (i). These slopes, and the conditions on the edge lengths on Γ, lead to estictions on the multisets I such that ψ I achieves the minimum at some point in the kth loop γ k. Definition 6.1. Let I {0,..., } be a multiset of size m. We say that ψ I is δ-pemissible on γk if and deg(d I Γ k 1 ) δ δ k 1 deg(d I Γ k ) δ δ k. We say that ψ I is δ-pemissible on Γ [a,b] if thee is some k [a, b] such that ψ I is δ-pemissible on γ k. Lemma 6.. If ψ I (v) = θ(v) fo some v γ k then ψ I is δ-pemissible on γ k. Poof. Recall that the edge lengths of Γ ae assumed to be admissible, in the sense of Definition 4.1. Suppose ψ I (v) = θ(v) fo some point v in γ k. We claim that the slope of ψ I along the bidge β k 1 to the left of the loop is at most the incoming slope of θ fom the left at u k 1. Indeed, if the slope of ψ I is stictly geate than that of θ then, since ψ I (u k 1 ) θ(u k 1 ) and the slope of θ can only decease when going fom u k 1 to v k, the diffeence ψ I (v k ) θ(v k ) will be at least the distance fom u k 1 to v k, which is n k 1 /.

14 14 DAVID JENSEN AND SAM PAYNE The slopes of ψ I and θ along the bottom edge ae between 0 and mg, and the slopes along the top edge ae between 0 and m. Since l k > 4gm k by assumption, it follows that ψ I θ changes by at most ml k between v k and any othe point in γ k. Assuming ml k < n k 1, this poves the claim. Note that the incoming slopes of ψ I and θ fom the left at u k ae deg(md Γ[0,k 1] ) deg(d I Γ[0,k 1] ), and deg(md Γ[0,k 1] ) δ 0 δ k 1, espectively. Theefoe, the claim implies that deg(d I Γ[0,k 1] ) δ δ k 1. A simila agument using slopes along the bidge β k to the ight of γk and assuming ml k < n k shows that deg(d I Γ k ) δ δ k, and the lemma follows. Ou geneal stategy fo poving Conjectue 4.5 is to choose the set A caefully, assume that the minimum occus eveywhee at least twice, and then bound δ δ i inductively, moving fom left to ight acoss the gaph. By induction, we assume a lowe bound on δ 0 + +δ i. Then, fo a caefully chosen j > i, we conside δ 0 + +δ j. If this is too small, then Lemma 6. seveely esticts which functions ψ I can achieve the minimum on loops in Γ [i+1,j], making it impossible fo the minimum to occu eveywhee at least twice unless the bottom edge lengths m i+1,..., m j satisfy a nontivial linea elation with small intege coefficients. We deduce a lowe bound on δ δ j and continue until we can show δ δ g+1 > d, a contadiction. We give a fist taste of this type of agument in Lemma 6.4 and Example 6.6. Example 6.7 illustates how simila techniques may be applied to undestand the kenel of µ m when it is not injective. A moe geneal (and moe technical) vesion of the key step in this agument, using the assumption that a small numbe of functions ψ I achieve the minimum eveywhee at least twice on Γ [i+1,j] to poduce a nontivial linea elation with small intege coefficients, appeas in the poof of Poposition 7.6. Notation 6.3. Fo the emainde, we fix m =, and I and I j will always denote multisets of size in {0,..., }, which we identify with pais (i, j) with 0 i j. We wite ψ ij fo the piecewise linea function ψ i + ψ j coesponding to the multiset I = {i, j}. Lemma 6.4. Suppose that δ k = and θ(v) = min{ψ I1 (v), ψ I (v), ψ I3 (v)} occus at least twice at evey point in γ k. Then, θ γ k = ψ I j γ k, fo some 1 j 3. Poof. By Lemma 5.1, no two of the functions may obtain the minimum on all of γk. Afte enumbeing, we may assume that ψ I 3 obtains the minimum on some but not all of the loop. Let v be a bounday point of the locus whee ψ I3 obtains the minimum. Since thee ae only thee functions that obtain the minimum, one must obtain the minimum in a neighbohood of v. Afte enumbeing we may assume that this is ψ I1. We claim that θ is equal to ψ I1 on the whole loop. If not, then by the shape lemma fo minima, D + div θ would contain the two points in the bounday of the locus whee ψ I1 obtains the minimum, in addition to v, contadicting the assumption that δ k, the degee of D + div θ on γ k, is. Remak 6.5. It follows fom Lemma 6.4 that, unde the given hypotheses, the topical dependence on the kth loop is essentially unique, in the sense that if b 1, b, and b 3 ae eal numbes such that θ(v) = min{ψ I1 (v) + b 1, ψ I (v) + b, ψ I3 (v) + b 3 } occus at least twice at evey point on the kth loop, then b 1 = b = b 3. Futhemoe, since each ψ Ij has constant slope along the bottom edge of γ k and no two agee on the entie top edge, thee must be one pai that agees on the full bottom edge and

15 TROPICAL INDEPENDENCE II: THE MAXIMAL RANK CONJECTURE FOR QUADRICS 15 pat of the top edge and anothe pai that agees on pat of the top edge, as shown in Figue 3. Note that the diviso D + div θ consists of two points on the top edge and one (but not both) of these points may lie at one of the end points, v k o w k. ψ I1 ψ I ψ I1 ψ I3 Figue 3. An illustation of the egions whee diffeent functions obtain the minimum in the situation of Lemma 6.4. Befoe we tun to the poof of the main theoem, we illustate the techniques involved with a pai of examples. Example 6.6. Suppose g = 10, and let D be the diviso of ank 4 and degee 1 coesponding to the tableau pictued in Figue 4. We note that this special case of the maximal ank conjectue fo m = is used to poduce a counteexample to the slope conjectue in [FP05] Figue 4. The tableau coesponding to the diviso D in Example 6.6. Assume that the minimum θ = min{ψ I } occus at least twice at evey point of Γ. By Poposition 5., the diviso = div(θ) + D has degee at least two on each of the 1 locally closed subgaphs γ k. Since deg(d) = 4, the degee of on each of these subgaphs must be exactly. In othe wods, δ = (,..., ). In the lingeing lattice path fo D, we have p 4 = (6, 5,, 1, 0), p 5 = (6, 5, 3, 1, 0), p 6 = (6, 5, 4, 1, 0). The δ-pemissible functions ψ ij on Γ [5,6] ae those such that eithe p 4 (i) + p 4 (j) 6 and p 5 (i) + p 5 (j) 6, p 5 (i) + p 5 (j) 6 and p 6 (i) + p 6 (j) 6. Thee ae only 3 such pais: (0, 4), (1, 3), and (, ). By Lemma 6.4, in ode fo the minimum to occu at least twice at evey point of Γ [5,6], on each of the two loops thee must be a single function ψ ij that obtains the minimum at evey point. A simple case analysis shows that fo both loops this function must be ψ 13 and that ψ 04 must achieve the minimum on both bottom edges. Let q 5 and q 6 be the points of D on γ 5 and γ 6, espectively, as shown in Figue 5. The egions of the gaph ae labeled by the pais of functions ψ ij, ψ i j that obtain the minimum on that egion. Fo each i, the change in value ψ i (q 6 ) ψ i (q 5 ) may o

16 16 DAVID JENSEN AND SAM PAYNE ψ, ψ 13 q 5 ψ, ψ13q 6 ψ 13, ψ 04 Figue 5. Regions of Γ [5,6] on which the functions obtain the minimum. be expessed as a function of the enties in the lattice path and the lengths of the edges in Γ. Specifically, as we tavel fom q 5 to q 6, the slopes of ψ and ψ 13 diffe by 1 on an inteval of length m 5 along the top edge of γ 5, and again on an inteval of length m 6 along the top edge of γ 6. This computation shows that (ψ (q 5 ) ψ 13 (q 5 )) (ψ (q 6 ) ψ 13 (q 6 )) = m 5 m 6. Since ψ 13 and ψ agee at q 5 and q 6, it follows that m 5 must equal m 6, contadicting the hypothesis that Γ has admissible edge lengths in the sense of Definition 4.1. We conclude that the minimum cannot occu eveywhee at least twice, so {ψ I } I is topically independent. Theefoe, fo any cuve X with skeleton Γ and any lift of D to a diviso D X of ank 4, the map is injective. µ : Sym L(D X ) L(D X ) We now conside an example illustating ou appoach via topical independence when µ is not injective. Recall that the canonical diviso on a non-hypeelliptic cuve of genus 4 gives an embedding in P 3 whose image is contained in a unique quadic. This is the special case of the maximal ank conjectue whee g,, d, and m ae 4, 3, 6, and, espectively. Example 6.7. Suppose g = 4 and m =. Note that the class of the canonical diviso D = K Γ is vetex avoiding of ank 3. Since Γ is the skeleton of a cuve whose canonical embedding lies on a quadic, the functions ψ I ae topically dependent, and we may assume min I ψ I (v) occus at least twice at evey point v Γ. Let θ(v) = min I ψ I (v), and let = K Γ + div θ. By Poposition 5., the degee δ k of on γ k is at least fo k = 0,..., 5. Since deg( ) = 1, it follows that δ = (,..., ). The lingeing lattice path associated to K Γ is given by p 0 = (3,, 1, 0), p 1 = (4,, 1, 0), p = (4, 3, 1, 0), p 3 = (4, 3,, 0), p 4 = (3,, 1, 0). Since δ 0 = δ 1 =, the δ-pemissible functions ψ ij on γ 1 ae those such that p 0 (i) + p 0 (j) 4 and p 1 (i) + p 1 (j) 4. Thee ae only thee such pais: (0, ), (1, 1), and (0, 3). In a simila way, we see that thee ae pecisely thee δ-pemissible functions on each loop γ k. By Lemma 6.4 and Remak 6.5, the topical dependence among the thee functions that achieve the minimum on each loop is essentially unique. Figue 6 illustates the combinatoial stuctue of this dependence.

17 TROPICAL INDEPENDENCE II: THE MAXIMAL RANK CONJECTURE FOR QUADRICS 17 ψ 11, ψ 1 ψ 1, ψ ψ 0, ψ 03 ψ 03, ψ 13 ψ 11, ψ 03 ψ, ψ 03 ψ 1, ψ 03 ψ 11, ψ 0 ψ, ψ 13 Figue 6. The unique topical dependence fo the canonical linea system when g = 4 and m =. Since this dependence among the functions that ealize the minimum at some point in Γ is essentially unique, omitting any one of the six functions that appea leaves a topically independent set of size 9. Theefoe, the map µ : Sym L(D X ) L(D X ) has ank at least 9. Since L(D X ) has dimension 9, it follows that µ is sujective. 7. Shapes of functions, excess degee, and linea elations among edge lengths In this section, and in 8, below, we assume that ρ(g,, d) = 0. All of the essential difficulties appea aleady in this special case. The case whee ρ(g,, d) > 0 is teated in 9 though a mino vaiation on these aguments. We now poceed with the moe delicate and pecise combinatoial aguments equied to pove Theoem 1.1. With g,, and d fixed, and assuming d g, we must poduce a diviso D of degee d and ank on Γ, togethe with a set of size A {(i, j) 0 i j } {( ) } + #A = min, d g + 1, such that the coesponding collection of ational functions is topically independent. {ψ ij (i, j) A} Notation 7.1. The quantity g d + appeas epeatedly thoughout, so we fix the notation s = g d +, which simplifies vaious fomulas. g = ( + 1)s. The condition that ρ(g,, d) = 0 means that We now specify the diviso D that we will use to pove Conjectue 4.5 fo m = when ρ(g,, d) = 0. The set A is descibed in 8. Notation 7.. Fo the emainde of this section and 8, let D be the diviso of degee d and ank on Γ coesponding to the standad tableau with + 1 columns and s ows in which the numbes 1,..., s appea in the leftmost column; s+1,..., s appea in the next column, and so on. We numbe the columns fom zeo to, so the lth column contains the numbes ls + 1,..., (l + 1)s. The specific case g = 10, = 4, d = 1 is illustated in Figue 4 fom Example 6.6.

18 18 DAVID JENSEN AND SAM PAYNE Remak 7.3. Ou choice of diviso is paticulaly convenient fo the inductive step in the poof of Theoem 1.1, in which we divide the gaph Γ into the + 1 egions Γ [ls+1,(l+1)s], fo 0 l, and move fom left to ight acoss the gaph, one egion at a time, studying the consequences of the existence of a topical dependence. Since the numbes ls + 1,..., (l + 1)s all appea in the lth column, the slopes of the functions ψ i, fo i l, ae the same along all bidges and bottom edges, espectively, in the subgaph Γ [ls+1,(l+1)s]. Only the slopes of ψ l ae changing in this egion. Hee we descibe the shape of the function ψ i, by which we mean the combinatoial configuation of egions on the loops and bidges on which ψ i has constant slope, as well as the slopes fom left to ight on each egion. These data detemine (and ae detemined by) the combinatoial configuations of the points in D i = D + div(ψ i ). Fix 0 l. Suppose ls + 1 k (l + 1)s, so γk is a loop in the subgaph Γ [ls+1,(l+1)s]. Recall fom 4 and 6 that D contains one point on the top edge of γk, at distance p k 1(l) = σ k (l) in the counteclockwise diection fom w k, whee σ k (l) is the slope of ψ l along the bidge β k. Case 1: The shape of ψ i, fo i < l. If i < l then D i = D + div ψ i contains one point on the top edge of γ k, at distance ( + s i 1 σ k(l)) m k fom v k, the left endpoint of γ k. This is illustated schematically in Figue 7. The point of D i s i + s i + s i 1 Figue 7. The shape of ψ i, fo i < l. on the top edge of γk is maked with a black cicle, and the point of D is maked with a white cicle. Each egion of constant slope is labeled with the slope of ψ i fom left to ight. The slope of ψ i fom left to ight along each bidge adjacent to γk is + s i, and the slope along the bottom edge is + s i 1. Case : The shape of ψ j, fo j > l. If j > l then D j = D + div ψ j contains one point on the top edge of γ k, at distance (σ k(l) + j) fom w k, as shown in Figue 8. The slope of ψ j along the bottom edge and both adjacent bidges is j. Case 3: The shape of ψ l. The diviso D l has no points on γk, as shown in Figue 9. Note that this is the only case in which the slope is not the same along the two bidges adjacent to γk. We use the shapes of the functions ψ i to contol the set of pais (i, j) such that ψ ij is δ-pemissible on cetain loops, as follows. Suppose {ψ ij } is topically dependent, so thee ae constants b ij such that min{ψ ij (v) + b ij } occus at least twice at evey