FINITE HILBERT STABILITY OF (BI)CANONICAL CURVES

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1 FINITE HILBERT STABILITY OF (BICANONICAL CURVES JAROD ALPER, MAKSYM FEDORCHUK, AND DAVID ISHII SMYTH* To Joe Harrs on hs sxteth brthday Abstract. We prove that a generc canoncally or bcanoncally embedded smooth curve has semstable m th Hlbert ponts for all m 2. We also prove that a generc bcanoncally embedded smooth curve has stable m th Hlbert ponts for all m 3. In the canoncal case, ths s accomplshed by provng fnte Hlbert semstablty of specal sngular curves wth G m-acton, namely the canoncally embedded balanced rbbon and the canoncally embedded balanced double A 2k+1 -curve. In the bcanoncal case, we prove fnte Hlbert stablty of specal hyperellptc curves, namely Wman curves. Fnally, we gve examples of canoncally embedded smooth curves whose m th Hlbert ponts are non-semstable for low values of m, but become semstable past a defnte threshold. Contents 1. Introducton 2 Acknowledgements 4 2. GIT background 4 3. Curves wth G m -acton: Rbbons, A 2k+1 -curves, and rosares Canoncal case, odd genus: The balanced rbbon wth G m -acton Canoncal case, even genus: The balanced double A 2k+1 -curve wth G m -acton Bcanoncal case, odd genus: The rosary wth G m -acton Monomal bases and semstablty Canoncally embedded rbbon Canoncally embedded A 2k+1 -curve Bcanoncally embedded rosary Non-semstablty results Canoncally embedded rosary Canoncally embedded bellptc curves Stablty of bcanoncal curves Wman curves Key combnatoral lemmas Monomal multbases and stablty 30 References 34 *The thrd author was partally supported by NSF grant DMS durng the preparaton of ths work. 1

2 2 ALPER, FEDORCHUK, AND SMYTH 1. Introducton Geometrc Invarant Theory (GIT was developed by Mumford n order to construct quotents n algebrac geometry, and n partcular to construct modul spaces. To use GIT to construct a modul space one must typcally prove that a certan class of embedded varetes has stable or semstable Hlbert ponts. The prototypcal example of a stablty result s Geseker and Mumford s asymptotc stablty theorem for plurcanoncally embedded curves [Mum77, Ge82, Ge83]: Theorem 1.1 (Asymptotc Stablty. Suppose C PH 0( C, K n C s a smooth curve embedded by the complete lnear system K n C, where n 1. Then the mth Hlbert pont of C s stable for all m 0. Geseker and Mumford s arguments are non-effectve, and there s no known bound on how large m must be n order to obtan the concluson of the theorem. In lght of ths theorem, t s natural to ask: for whch fnte values of m do plurcanoncally embedded smooth curves have stable or semstable Hlbert ponts? Ths has been a basc open problem n GIT snce the poneerng work of Geseker and Mumford, but has ganed renewed nterest from recent work of Hassett and Hyeon on the log mnmal model program for M g. Indeed, Hassett and Hyeon observed that a stablty result for fnte Hlbert ponts of canoncally and bcanoncally embedded smooth curves would enable one to use GIT to construct a sequence of new projectve bratonal models of M g that would consttute steps of the log mnmal model program for M g [HH08]. In ths paper, we prove the requste stablty result. Theorem 1.2 (Man Result. (1 If C s a generc canoncally or bcanoncally embedded smooth curve, then the m th Hlbert pont of C s semstable for every m 2. (2 If C s a generc bcanoncally embedded smooth curve, then the m th Hlbert pont of C s stable for every m 3. Part (1 of the man result s proved n Corollares 4.2 (odd genus canoncal, 4.11 (even genus canoncal, and Theorem 6.2 (bcanoncal case. Part (2 of the man result s proved n Theorem 6.2. Ths s, to our knowledge, the frst example of a result n whch the (semstablty of all Hlbert ponts of a gven varety s establshed by a unform method. In the case of canoncally and bcanoncally embedded curves, we recover a weak form of the asymptotc stablty theorem by a much smpler proof. Furthermore, as a sdelght to our man result, we gve an example of an embedded smooth curve whose m th Hlbert pont changes from semstable to non-semstable as m decreases (Theorem 5.2. We wll explan our method of proof n the next secton. Frst, however, let us conclude ths ntroducton by descrbng a fascnatng applcaton of the man result, antcpated n the work of Hassett and Hyeon [HH08], and by consderng prospects for future generalzatons. Fx g 2, n 1, m 2, and set r = (2n 1(g 1 1 f n 2, and r = g 1 f n = 1. To an n-canoncally embedded smooth genus g curve C we assocate ts m th Hlbert pont [C] m PW m ; these are defned n more detal n Secton 2 below. We denote by H m g,n the closure n PW m of the locus of m th Hlbert ponts of n-canoncally embedded smooth curves of genus g. Then the SL(r + 1-acton on H m g,n admts a natural lnearzaton O(1, whch defnes an open locus (Hg,n m ss H m g,n of semstable ponts. Assumng that (Hg,n m ss s

3 FINITE HILBERT STABILITY OF (BICANONICAL CURVES 3 non-empty, one obtans a GIT quotent (Hg,n m ss // SL(r + 1 := Proj H 0( Hg,n, m O(k SL(r+1 k 0 as a projectve varety assocated to the algebra of SL(r + 1-nvarant functons n the homogenous coordnate rng of H m g,n. When m 0, the crtcal assumpton ( H m g,n ss s satsfed by Theorem 1.1, and the correspondng quotents have been analyzed usng GIT [Ge82, Ge83, Sch91, HH09, HH08, HL10, HM10]. The results of ths analyss can be summarzed as follows: Here, M ps g by cusps, and M hs g ( H m g,n ss// SL(r + 1 M g f n 5, m 0, M ps g f n = 3, 4, m 0, M hs g f n = 2, m 0. s the modul space of pseudostable curves, n whch ellptc tals have been replaced s the modul space of h-semstable curves, n whch ellptc brdges have been replaced by tacnodes. Furthermore, the bratonal transformatons M g M ps g M hs g consttute the frst two steps of the log mnmal model program, namely the frst dvsoral contracton and the frst flp [HH09, HH08]. The key pont s that the next stage of the log mnmal model program cannot be constructed usng an asymptotc stablty result. Indeed, an examnaton of the formula for the dvsor class of the polarzaton on the GIT quotent ( H m g,n ss// SL(r+1 suggests that the next model occurrng n the log mnmal model program should be ( H 6 g,2 ss// SL(3g 3. Thus, n marked contrast to the cases n 3, where fnte Hlbert lnearzatons are not expected to yeld new bratonal models of M g, t s wdely antcpated that n the cases n = 1, 2, there wll exst several values of m at whch the correspondng GIT quotents undergo nontrval bratonal modfcatons caused by the fact that curves wth worse than nodal sngulartes become semstable for low values of m. For n = 1 we expect the number of threshold values of m at whch ( H m g,n ss changes to grow wth g, whle for n = 2 the only nterestng values are m 6, rrespectvely of g; for a detaled analyss of the expected threshold values of m see [FS10] and [AFS10]. Untl now, the man obstacle to verfyng these expectatons has been provng ( H m g,n ss for explct, fnte values of m and arbtrary genus g. Theorem 1.2 removes ths obstacle, and thus opens the door to analyzng a whole menagere of new GIT quotents ( H m g,n ss// SL(r + 1. Fnally, let us dscuss a slght sharpenng of our man result whch follows naturally from the methods employed n ths paper. We observe that the canoncally embedded curve of even genus for whch we establsh fnte Hlbert semstablty n Secton 4.2 s n fact trgonal,.e. t les n the closure of the locus of canoncally embedded smooth trgonal curves. Smlarly, n Secton 4.3, we prove the fnte Hlbert semstablty of the bcanoncally embedded curve of odd genus, whch s easly seen to be n the closure of the locus of bcanoncally embedded smooth bellptc curves. From these observatons, we obtan the followng result: Theorem 1.3 (Stablty of trgonal and bellptc curves. (1 Suppose C PH 0( C, K C s a generc canoncally embedded smooth trgonal curve of even genus. Then the m th Hlbert pont of C s semstable for every m 2.

4 4 ALPER, FEDORCHUK, AND SMYTH (2 Suppose C PH 0( C, K 2 C s a generc bcanoncally embedded smooth bellptc curve of odd genus. Then the m th Hlbert pont of C s semstable for every m 2. Ths result naturally rases the questons: Is t true that all canoncally embedded smooth trgonal curves have semstable m th Hlbert ponts for m 2? Smlarly, do other curves wth low Clfford ndex, such as canoncal bellptc curves, have ths property? Surprsngly, the answer to both questons s no. In Secton 5 of ths paper, we prove that the m th Hlbert pont of a canoncally embedded smooth bellptc curve s non-semstable below a certan defnte threshold value of m (dependng on g, whle the m th Hlbert pont of a generc canoncally embedded bellptc curve of odd genus s semstable for large values of m. As for trgonal curves, t s not dffcult to see that the 2 nd Hlbert pont of a canoncally embedded trgonal curve wth postve Maron nvarant s non-semstable; see [FJ11, Corollary 3.2]. On the other hand, n Secton 5 we gve heurstc reasons for belevng that a canoncally embedded smooth trgonal curve should have semstable m th Hlbert ponts for m 3. Notaton and conventons. We work over the feld of complex numbers C. In partcular, we denote G m := Spec C[t, t 1 ]. In Secton 6, we use the term multset to denote a collecton of elements wth possbly repeatng elements. Acknowledgements. We learned about the problem of GIT stablty of fnte Hlbert ponts many years ago from Brendan Hassett s talks on the log mnmal model program for M g. Over the past several years we learned about many aspects of GIT from conversatons wth Ian Morrson and Davd Hyeon, as well as through ther many papers on the topc. In addton, we ganed a great deal from conversatons wth Ase Johan de Jong, Anand Deopurkar, Davd Jensen, and Davd Swnarsk. 2. GIT background The proof of our man result s surprsngly smple. In the canoncal (resp., bcanoncal case, we exhbt a curve C such that the acton of Aut(C on V = H 0( C, ω C (resp., V = H 0( C, ωc 2 s multplcty-free,.e. no representaton occurs more than once n the decomposton of V nto rreducble Aut(C-representatons. As Ian Morrson observed some thrty years ago, under ths hypothess, powerful results of Kempf mply that the m th Hlbert pont of C s semstable f and only f t s semstable wth respect to one-parameter subgroups of SL(V whch act dagonally on a fxed bass of V. Verfyng stablty wth respect to the resultng fxed torus of SL(V s a dscrete combnatoral problem whch we solve explctly for every m 2. We thus prove the semstablty of all Hlbert ponts of C and deduce the semstablty of a generc smooth curve by openness of the semstable locus. In Secton 3 we wll gve a precse descrpton of the (rather exotc curves C appearng n our argument. In ths secton, we recall the relevant defntons from GIT and explan the general framework for provng semstablty of Hlbert ponts due to Mumford, as well as the aforementoned refnements of Kempf. Let us begn by recallng the defnton of the m th Hlbert pont of an embedded scheme. If X PV s a closed subscheme such that the restrcton map H 0( PV, O(m H 0( X, O X (m s surjectve (equvalently, h 1( X, I X (m = 0, set h (X,O 0 X (m W m := H 0 ( PV, O(m.

5 FINITE HILBERT STABILITY OF (BICANONICAL CURVES 5 The m th Hlbert pont of X PV s a pont [X] m PW m, defned as follows. Frst, consder the surjecton H 0( PV, O(m H 0( X, O X (m 0. Takng the h 0( X, O X (m -fold wedge product and dualzng, we obtan the m th Hlbert pont: (X,O h0 X (m [X] m := H 0 ( PV, O(m h (X,O 0 X (m H 0 ( X, O X (m 0 P(W m. Recall that f W s any lnear representaton of SL(V, a pont x P(W s semstable f the orgn of W s not contaned n the closure of the orbt of x W, where x s any lft of x. Thus, to show that a Hlbert pont [X] m P(W m s semstable, we must prove that 0 W m s not n the closure of SL(V [X] m, where [X] m s any lft of [X] m. An obvous necessary condton s that for any one-parameter subgroup ρ: Spec C[t, t 1 ] SL(V, we have lm t 0 ρ(t [X] m 0. A foundatonal theorem of Mumford asserts that ths necessary condton s suffcent. Proposton 2.1 (Hlbert-Mumford Numercal Crteron. Let X PV be as above. The Hlbert pont [X] m s semstable f and only f lm t 0 ρ(t [X] m 0 for every one-parameter subgroup ρ: Spec C[t, t 1 ] SL(V. Gven a one-parameter subgroup ρ: Spec C[t, t 1 ] SL(V, we may reformulate the condton lm t 0 ρ(t [X] m 0 as follows. Frst, we may choose a bass {x } r =0 of V whch dagonalzes the acton of ρ. Then ρ(t x = t ρ x for some ntegers ρ satsfyng r =0 ρ = 0. We call {x } r =0 a ρ-weghted bass. If we set N m := h 0( X, O X (m, a bass for W m = N m H 0( P r, O P r(m dagonalzng the ρ-acton conssts of N m -tuples e 1... e Nm of dstnct monomals of degree m n the varables x s. If e l = r, then ρ acts on =0 xa l e 1... e Nm wth weght N m r l=1 =0 a lρ. Now the condton that lm t 0 ρ(t [X] m 0 s equvalent to the exstence of one such coordnate whch s non-vanshng on [X] m and on whch ρ acts wth non-postve weght. The condton that a coordnate e 1... e Nm s non-zero on [X] m s precsely the condton that the restrctons of {e l } Nm l=1 to X form a bass of H 0( X, O X (m. Ths dscusson leads us to the followng defnton. Defnton 2.2. If {x } r =0 s a ρ-weghted bass of V, a monomal bass of H0( X, O X (m s a set B = {e l } Nm l=1 of degree m monomals n the varables {x } r =0 such that B maps onto a bass of H 0( X, O X (m va the restrcton map H 0( PV, O(m H 0( X, O X (m. Moreover, f e l = r =0 xa l, we defne the ρ-weght of B to be w ρ (B := N m r l=1 =0 a lρ. Wth ths termnology, we have the followng crteron. Proposton 2.3 (Numercal Crteron for Hlbert ponts. [X] m s semstable (resp., stable f and only f for every ρ-weghted bass of V, there exsts a monomal bass of H 0( X, O X (m of non-postve (resp., negatve ρ-weght. The Hlbert-Mumford crteron reduces the problem of provng semstablty of [X] m to a concrete algebro-combnatoral problem concernng the defnng equatons of X PV. However, ths problem s not dscretely computable snce t requres checkng all one-parameter subgroups of SL(V. A theorem of Kempf allows us, under certan hypotheses on Aut(X,

6 6 ALPER, FEDORCHUK, AND SMYTH to check only those one-parameter subgroups of SL(V whch act dagonally on a fxed bass. Ths reduces the problem to one whch s dscretely computable. In order to state the next proposton, let us establsh a bt more termnology. Gven an embeddng X PV by a complete lnear system, there s a natural acton of Aut(X on V = H 0( X, O X (1. Gven a lnearly reductve subgroup G Aut(X, we say that V s a multplcty-free G-representaton (or smply multplcty-free f G s understood f t contans no rreducble G-representaton more than once n ts decomposton nto rreducble G-representatons. We say that a bass of V, say {x } r =0, s compatble wth the rreducble decomposton of V f each rreducble G-representaton n V s spanned by a subset of the x s. We may now state the reformulaton of Kempf s results that we wll use. We keep the assumpton that X s embedded by a complete lnear system O X (1 and that the restrcton map H 0( PV, O(m H 0( X, O X (m s surjectve. Proposton 2.4 (Kempf-Morrson Crteron. Suppose G Aut(X s a lnearly reductve subgroup, and that V = H 0( X, O X (1 s a multplcty-free representaton of G. Let {x } r =0 be a bass of V whch s compatble wth the rreducble decomposton of V. Then [X] m s semstable (resp., stable f and only f for every one-parameter subgroup ρ: Spec C[t, t 1 ] SL(V actng dagonally on {x } r =0, we have lm t 0 ρ(t [X] m 0 (resp., lm t 0 ρ(t [X] m does not exst. Equvalently, for every weghted bass {x } r =0 of V, there exsts a monomal bass of H 0( X, O X (m of non-postve (resp., negatve weght. Proof. If [X] m s not semstable, then [Kem78, Theorem 3.4 and Corollary 3.5] mples that there s a one-parameter subgroup ρ : Spec C[t, t 1 ] SL(V wth lm t 0 ρ (t [X] m = 0 such that the parabolc subgroup P SL(V assocated to the ρ -weght fltraton 0 = U 0 U 1 U k 1 U k = V contans Aut(X. Let V = j V j be the decomposton nto rreducble G-representatons. Snce V s multplcty-free, each U can be wrtten as a drect sum of some of the V j s. The maxmal torus T SL(V assocated to the bass {x } r =0 fxes each V j and thus the fltraton. Therefore, T P. By [Kem78, Theorem 3.4 (c(4], there exsts a one-parameter subgroup ρ: Spec C[t, t 1 ] T such that lm t 0 ρ(t [X] m = 0. The statement for semstablty follows. The statement for stablty follows by the same argument by replacng the concept of semstablty (0-stablty n Kempf s termnology by a more general concept of S-stablty; see [Kem78]. We are grateful to Ian Morrson for pontng ths out. For the sake of concreteness, let us reterate the Kempf-Morrson crteron n the case of a canoncally (resp., bcanoncally embedded curve C P r. In order to prove that [C] m s semstable, we must frst check that V = H 0( C, K C (resp., V = H 0 ( C, K 2 C s a multplctyfree representaton of some lnearly reductve G Aut(C. Second, we fx a bass {x } r =0 of V compatble wth the rreducble decomposton of V. Now any one-parameter subgroup ρ actng dagonally on {x } r =0 s gven by an nteger weght vector (ρ 0,..., ρ r satsfyng r =0 ρ = 0. To show that [C] m s semstable wth respect to ρ, we must fnd a monomal bass B of H 0( C, O C (m such that w ρ (B 0. Note that for a fxed monomal bass B, the ρ-weght functon w ρ (B s lnear n (ρ 0,..., ρ r. Therefore, each monomal bass determnes a half-space of weght vectors for whch [C] m s ρ-semstable, namely the half-space w ρ (B 0. It follows that as soon as one produces suffcently many monomal bases such that the unon

7 FINITE HILBERT STABILITY OF (BICANONICAL CURVES 7 of these half-spaces contans all weght vectors (ρ 0,..., ρ r satsfyng r =0 ρ = 0, the proof of semstablty for [C] m s completed. We summarze ths dscusson n the followng lemma: Lemma 2.5. Let G Aut(C be a lnearly reductve subgroup such that V = H 0( C, O C (1 s a multplcty-free representaton of G, and let {x } r =0 be a bass of V whch s compatble wth the rreducble decomposton of V. Suppose there exsts a fnte set {B j } j J of monomal bases of H 0( C, O C (m and {c j } j J Q (0, such that c j w ρ (B j = 0 j J for every ρ: G m SL(V actng on {x } r =0 dagonally. Then [C] m s semstable. The dea of applyng these results of Kempf to the semstablty of fnte Hlbert ponts of curves s due to Morrson and Swnarsk [MS11]. In ther paper, they consder the so-called hyperellptc Wman curve C wth ts bcanoncal embeddng. They check that the automorphsm group, whch s cyclc of order 4g+2, acts on H 0( C, KC 2 wth 3g 3 dstnct characters. They fx a bass H 0( C, KC 2 = {x0,..., x r } compatble wth the decomposton of H 0( C, KC 2 nto characters, and then, for low values of g and m, use a computer to enumerate monomal bases of H 0( C, O C (m untl the assocated half-spaces cover the hyperplane r =0 ρ = 0. In ths paper, we apply the Kempf-Morrson crteron to canoncally embedded rbbons of odd genus (Secton 4.1, canoncally embedded balanced double A 2k+1 -curves of even genus (Secton 4.2, bcanoncally embedded rosares of odd genus (Secton 4.3, and bcanoncally embedded Wman curves (Secton 6.3. For each m 2, we wrte down by hand suffcently many monomal bases to establsh the requste (semstablty result. 3. Curves wth G m -acton: Rbbons, A 2k+1 -curves, and rosares As dscussed n the prevous secton, the key to our proof s to fnd a sngular Gorensten curve C such that H 0( C, ω C (resp., H 0 ( C, ωc 2 s a multplcty-free representaton of Aut(C n the canoncal case (resp., bcanoncal case. In ths secton, we descrbe the curves we wll use. In the odd genus canoncal case, we wll use a certan rbbon wth G m -acton, the socalled balanced rbbon. In the even genus canoncal case, we wll use the balanced double A 2k+1 -curve,.e. a curve comprsed of three P 1 s meetng n two hgher tacnodes wth trval crmpng. In the bcanoncal case, we wll use the so-called rosary,.e. a cycle of P 1 s attached by tacnodes, ntroduced by Hassett and Hyeon n ther classfcaton of asymptotcally stable bcanoncal curves [HH08]. A word of motvaton as to where on earth these curves come from may be useful. That some class of canoncally embedded rbbons should be GIT-semstable s ntutvely plausble, snce rbbons arse as flat lmts of famles of canoncally embedded smooth curves degeneratng abstractly to a hyperellptc curve. The fact that the balanced rbbon of odd genus s the only rbbon wth G m -acton that has the potental to be Hlbert semstable was proved n [AFS10, Theorem 7.2]. Hence, t was natural to attempt to prove that ths curve s, n fact, semstable. Our motvaton for consderng double A 2k+1 -curves comes from the log mnmal model program for M 2k, where we expect the 2k 4 dmensonal locus of double A 2k+1 -curves to replace the locus n the boundary dvsor k M 2k consstng of nodal curves C 1 C 2 such that each C s a hyperellptc curve of genus k. Indeed, ths predcton has already been verfed n g = 4 by the second author who showed that the dvsor 2 M 4 s contracted to the pont correspondng to the unque genus 4 double A 5 -curve n the fnal non-trval log

8 8 ALPER, FEDORCHUK, AND SMYTH canoncal model of M 4 [Fed12]. In the bcanoncal case, we made use of the classfcaton of asymptotcally semstable curves n [HH08]. We smply looked through the curves on ther lst for one wth a large enough symmetry group to satsfy the hypotheses of Proposton 2.4. The rosary was the frst curve we checked, and t worked! 3.1. Canoncal case, odd genus: The balanced rbbon wth G m -acton. In ths secton we wll construct, for every odd g 3, a specal non-reduced curve C of arthmetc genus g whose canoncal embeddng satsfes the hypotheses of Proposton 2.4. Gven a postve odd nteger g = 2k + 1, where k 1, set U := Spec C[u, ɛ]/(ɛ 2, V := Spec C[v, η]/(η 2, and dentfy U {0} and V {0} va the somorphsm u v 1 v k 2 η, ɛ v g 1 η. The resultng scheme C s evdently a complete, locally planar curve of arthmetc genus g; see [BE95, Secton 3] for more detals on such curves. Note that C admts G m -acton by the formulae t u = tu, t v = t 1 v, t ɛ = t k+1 ɛ, t η = t k 1 η. Snce C s locally planar, t s Gorensten and ts dualzng sheaf ω C s a lne bundle. Usng adjuncton, we may dentfy global sectons of ω C wth regular functons f(u, ɛ on U. To be precse, the global sectons of ω C consst of all dfferentals du dɛ f(u, ɛ ɛ 2 whch transform to dfferentals h(v, η dv dη wth h(v, η regular on V. One easly wrtes η 2 down a bass of g functons satsfyng ths condton to obtan the followng lemma, whch s a specal case of a more general [BE95, Theorem 5.1]. Lemma 3.1. A bass for H 0( C, ω C s gven by dfferentals f(u, ɛ du dɛ where f(u, ɛ runs ɛ 2 over the followng lst of g functons: x := u, 0 k, y k+ := u k+ + u 1 ɛ, 1 k. Lemma 3.2. ω C s very ample. Proof. Usng the bass of H 0( C, ω C from Lemma 3.1, we see that ωc separates ponts of C red P 1 and defnes a closed embeddng when restrcted to U and V. The clam follows. Proposton 3.3. H 0( C, ω C s a multplcty-free representaton of Gm {x 0,..., x k, y k+1,..., y 2k } s compatble wth ts rreducble decomposton. Aut(C and Proof. The bass {x 0,..., x k, y k+1,..., y 2k } dagonalzes the acton of G m on H 0( C, ω C wth the 2k + 1 dstnct weghts k,..., 1, 0, 1,..., k.

9 FINITE HILBERT STABILITY OF (BICANONICAL CURVES 9 In order to apply Proposton 2.4, we wll need an effectve way of determnng when a set of monomals n the g varables {x 0,..., x k, y k+1,..., y 2k } forms a monomal bass of H 0( C, ωc m. To do ths, observe that the global sectons of ω m C are easly dentfed wth (du dɛm regular functons on U va f(u, ɛ f(u, ɛ. Wth ths conventon, we record the followng observaton used throughout the paper. Lemma 3.4 (Rbbon Product Lemma. The expanson n u and ɛ of the degree m monomal x 1... x l y l+1... y m s u a + (a bu a k 1 ɛ, where ɛ 2m a = m, b = l + k(m l. The followng proposton determnes a bass for H 0( C, ω m C under the above dentfcaton. Proposton 3.5. For m 2, the product map Sym m H 0( C, ω C H 0 ( C, ωc m s surjectve. A bass for H 0( C, ωc m s gven by dfferentals f(u, ɛ (du dɛ m where f(u, ɛ runs over the ɛ 2m followng (2m 1(g 1 functons on U: {u } 2mk (k+1 =0, {u + ( ku k 1 ɛ} 2mk =k+1. Proof. We wll show that the mage of the product map Sym m H 0( C, ω C H 0 ( C, ωc m contans the gven functons. Snce h 0 (C, ωc m = (2m 1(g 1 by Remann-Roch, and because the gven functons are lnearly ndependent, ths wll prove the proposton. Lemma 3.4 gves u a = x m 1 0 x a for 0 a k, u 2mk k + (2mk 2ku 2mk 2k 1 ɛ = y m 1 2k x k, and u (2m 1k+a + ( (2m 2k+a u (2m 2k+a 1 ɛ = y m 1 2k y a for 1 a k. For the ntermedate u-degrees, note smply that snce the dmenson of the space {cu +du k 1 ɛ : c, d C} s two, we need to exhbt two lnearly ndependent functons of ths form as degree m monomals n {x 0,..., y 2k }. Usng Lemma 3.4, ths s an easy exercse whch we leave to the reader. Ths result gves a very smple way of checkng whether a set B of degree m monomals n {x 0,..., y 2k } projects to a bass for H 0( C, O C (m. If we smply vew the monomals n B as polynomals n C[u, ɛ]/(ɛ 2 va the dentfcaton precedng Lemma 3.4, then B s a monomal bass of H 0( C, O C (m f and only f (1 B contans one polynomal of each u-degree 0,..., k, (2 B contans two lnearly ndependent polynomals of each u-degree k + 1,..., (2m 1k 1, (3 B contans one polynomal of each u-degree 2mk k,..., 2mk. We can rephrase ths as follows. Lemma 3.6. A set of degree m monomals {x 1 x l y l+1 y m } (1,..., m S forms a monomal bass of H 0( C, O C (m f and only f the followng two condtons hold: (1 For 0 a k and (2m 1k a 2mk, there s exactly one ndex vector ( 1,..., m S wth m = a. (2 For k < a < (2m 1k, there are exactly two ndex vectors ( 1,..., m S satsfyng m = a. Furthermore, for these two ndex vectors, the assocated ntegers l m k(m l are dstnct. Proof. Immedate from the precedng observatons and the Rbbon Product Lemma 3.4.

10 10 ALPER, FEDORCHUK, AND SMYTH 3.2. Canoncal case, even genus: The balanced double A 2k+1 -curve wth G m -acton. In ths secton we wll construct specal sngular curves of even genus, whose canoncal embeddngs satsfy the hypotheses of Proposton 2.4. We defne a double A 2k+1 -curve to be any curve obtaned by glung three copes of P 1 along two A 2k+1 sngulartes (Fgure 1. The arthmetc genus of a double A 2k+1 -curve s g = 2k, and double A 2k+1 -curves have 2k 4 modul correspondng to the crmpng of the A 2k+1 -sngulartes,.e. deformatons that preserve the analytc types of the sngulartes as well as the normalzaton of the curve (see [vdw10] for a comprehensve treatment of crmpng modul. Indeed, the modul space of crmpng for an A 2k+1 -sngularty wth automorphsm-free branches has dmenson k, but the presence of automorphsms of the ponted P 1 s n our stuaton reduces the dmenson of crmpng modul by 4. Among double A 2k+1 -curves, there s a unque double A 2k+1 -curve wth G m -acton, correspondng to the trval choce of crmpng for both A 2k+1 -sngulartes. We call ths curve the balanced double A 2k+1 -curve. Now let us gve a more precse descrpton of the balanced double A 2k+1 -curve: Let C 0, C 1, C 2 denote three copes of P 1, and label the unformzers at 0 (resp., at by u 0, u 1, u 2 (resp., by v 0, v 1, v 2. Fx an nteger k 2, and let C be the arthmetc genus g = 2k curve obtaned by glung three P 1 s along two A 2k+1 sngulartes wth trval crmpng. More precsely, we mpose an A 2k+1 sngularty at ( C 0 (0 C 1 by glung C 0 \ 0 and C 1 \ nto an affne sngular curve (3.1 Spec C[(v 0, u 1, (v k+1 0, u k+1 1 ] Spec C[x, y]/(y 2 x 2k+2. Smlarly, we mpose an A 2k+1 sngularty at ( C 1 (0 C 2 by glung C 1 \ 0 and C 2 \ nto (3.2 Spec C[(v 1, u 2, (v k+1 1, u k+1 2 ] Spec C[x, y]/(y 2 x 2k+2. A 2k+1 C 1 A 2k+1 C 0 C 2 Fgure 1. Double A 2k+1 -curves The automorphsm group of C s gven by Aut(C = G m Z 2 where Z 2 acts va u v 2 and G m = Spec C[t, t 1 ] acts va t u 0 = tu 0, t u 1 = t 1 u 1, t u 2 = tu 2. Usng the descrpton of the dualzng sheaf on a sngular curve as n [Ser88, Ch.IV] or [BHPVdV04, Ch.II.6], we can wrte down a bass of H 0( C, ω C as follows: ( ( (3.3 x = u du 0 0, u du 1 1, 0, y = 0, u du 1 1, u du 2 2, 1 k. u 0 u 1 u 1 u 2 It s straghtforward to generalze ths descrpton to the spaces of plurcanoncal dfferentals.

11 FINITE HILBERT STABILITY OF (BICANONICAL CURVES 11 Lemma 3.7. For m 2, the product map Sym m H 0( C, ω C H 0 ( C, ωc m s surjectve and a bass of H 0( C, ωc m conssts of the followng (2m 1(2k 1 dfferentals: ( ω j = u j (du 0 m 0, u j (du 1 m ( 1, 0, η j = 0, u j (du 1 m 1, u j (du 2 m 2, m j mk. and u m 0 χ l = ( 0, u l 1 u m 1 (du 1 m u m 1 u m 1, 0, k(m l k(m 1 1. Proof. By Remann-Roch formula, h 0( C, ωc m = (2m 1(2k 1. Thus, t suffces to observe that the gven (2m 1(2k 1 dfferentals all le n the mage of the map Sym m H 0( C, ω C H 0( C, ωc m. Usng the bass of H 0 ( C, ω C gven by (3.3, one easly checks that the dfferentals {ω j } mk j=m are precsely those arsng as m-fold products of x s, the dfferentals {η j } mk j=m are those arsng as m-fold products of y s, and the dfferentals {χ l } k(m 1+1 l= k(m 1+1 are those arsng as mxed m-fold products of x s and y s. Next, we show that ω C s a very ample lnear system, so that C admts a canoncal embeddng, and the correspondng Hlbert ponts are well defned. Proposton 3.8. ω C s very ample. The complete lnear system ω C embeds C as a curve on a balanced ratonal normal scroll P 1 P 1 O(1,k 1 P g 1. Moreover, C 0 and C 2 map to (1, 0-curves on P 1 P 1, and C 1 maps to a (1, k + 1 curve. In partcular, C s a (3, k + 1 curve on P 1 P 1 and has a g3 1 cut out by the (0, 1 rulng. Proof. To see that the canoncal embeddng of C les on a balanced ratonal normal scroll n P 2k 1, recall that the scroll s the determnantal varety (see [Har92, Lecture 9] defned by: ( x1 x (3.4 rank 2 x k 1 y k y k 1 y 2 1. x 2 x 3 x k y k 1 y k 2 y 1 From our explct descrpton of the bass of H 0( C, ω C gven by (3.3, one easly sees that the dfferentals x s and y s on C satsfy the determnantal condton (3.4. Moreover, we see that ω C embeds C 0 and C 2 as degree k 1 ratonal normal curves n P 2k 1 lyng n the class (1, 0 on the scroll. Also, we see that ω C embeds C 1 va the very ample lnear system span{1, u 1,..., u k 1 1, u k+1 1,..., u 2k 1 } O P 1(2k as a curve n the class (1, k + 1. It follows that ω C separates ponts and tangent vectors on each component of C. We now prove that ω C separates ponts of dfferent components and tangent vectors at the A 2k+1 -sngulartes. Frst, observe that C 0 and C 2 span dsjont subspaces. Therefore, beng (1, 0 curves, they must be dstnct and non-ntersectng. Second, C 0 and C 1 are the mages of the two branches of an A 2k+1 -sngularty and so have contact of order at least k + 1. However, beng (1, 0 and (1, k + 1 curves on the scroll, they have order of contact at most k+1. It follows that the mages of C 0 and C 1 on the scroll meet precsely n an A 2k+1 -sngularty. We conclude that ω C s a closed embeddng at each A 2k+1 -sngularty. u m 2

12 12 ALPER, FEDORCHUK, AND SMYTH We can also drectly verfy that ω C separates tangent vectors at an A 2k+1 sngularty of C, say the one wth unformzers v 0 and u 1. The local generator of ω C at ths sngularty s ( x k = dv 0 v k+1 0, du 1, 0 On the open affne chart Spec C[(v 0, u 1, (v0 k+1, u k+1 1 ] defned n Equaton (3.1, we have x k 1 = (v 0, u 1 x k and y 1 = (0, u k+1 1 x k. Under the dentfcaton C[(v 0, u 1, (v0 k+1, u k+1 1 ] C[x, y]/(y 2 x 2k+2, we have (v 0, u 1 = x and (0, u k+1 1 = (x k+1 y/2. We conclude that x k 1 and y 1 span the cotangent space, and thus separate tangent vectors, at the sngularty. Fnally, the followng elementary observaton s the key to analyzng the stablty of Hlbert ponts of C. Lemma 3.9. H 0( C, ω C s a multplcty-free Aut(C-representaton and the bass {x, y } k =1 s compatble wth ts rreducble decomposton. Proof. Note that G m Aut(C acts on x wth weght and on y wth weght. Thus H 0( C, ω C decomposes nto g = 2k dstnct characters of Gm Bcanoncal case, odd genus: The rosary wth G m -acton. In ths secton we wll construct, n every odd genus, a sngular curve C whose bcanoncal embeddng satsfes the hypotheses of Proposton 2.4. For any odd nteger g 3, we defne C to be the curve, called a rosary n [HH08, Secton 8.1], obtaned from a set of (g 1 P 1 s ndexed by Z g 1 and havng unformzers u at 0 and v at (so that u = 1/v by cyclcally dentfyng v wth u +1 to specfy g 1 tacnodes. Note that G m D g 1 Aut(C, where the dhedral group D g 1 permutes the components and G m = Spec C[t, t 1 ] acts by u t ( 1 u. We should remark that n the case of even genus, one may stll defne the curve C, but C does not admt G m -acton and does not satsfy the hypotheses of Proposton 2.4. Thus, n what follows, we always assume g odd. u k+1 1 Lemma (a A bass for H 0( C, ω C s gven by the followng dfferentals: ( ω =..., 0, du, du +1, 0,..., Z g 1, η = u 2 +1 ( du0, du 1,..., du g 2. u 0 u 1 u g 2 (b A bass for H 0( C, ω 2 C s gven by the followng dfferentals: x = ω 2, Z g 1, y = ω η, Z g 1, z = ω 1 ω, Z g 1. Proof. Usng dualty on sngular curves as n [Ser88, Ch.IV] or [BHPVdV04, Ch.II.6], t s straghtforward to verfy that each dfferental from (a s a Rosenlcht dfferental and hence s an element of H 0( C, ω C. Snce these g dfferentals are lnearly ndependent, Part (a s establshed. Part (b follows mmedately: The (3g 3 dfferentals from (b are products of elements n H 0( C, ω C and are easly seen to be lnearly ndependent. Lemma ω C s very ample for odd g 5 and ωc 2 s very ample for odd g 3..

13 FINITE HILBERT STABILITY OF (BICANONICAL CURVES 13 Proof. We prove that C s canoncally embedded for g 5. Frst, observe that ω C embeds each P 1 as a conc n P g 1, and that the plane spanned by the th conc meets only the planes spanned by the cyclcally adjacent concs, and meets each of these only at the correspondng tacnode. Ths shows that ω C separates ponts and tangent vectors at smooth ponts. To see that ω C separates tangent vectors at the tacnode obtaned by the dentfcaton v = u +1, note that the local generator of ω C at ths tacnode s ω. Locally around the tacnode, we have η = (v, u +1 ω and ω +1 = (0, u 2 +1 ω. Under the dentfcaton C[(v, u +1, (0, u 2 +1 ] C[x, y]/ ( y(x 2 y, we have (v, u +1 = x and (0, u 2 +1 = y. We conclude that η and ω +1 span the cotangent space, and thus separate tangent vectors, at the tacnode. A straghtforward computaton shows that ωc 2 s also very ample for g = 3. We fnsh by notng that C s hyperellptc n genus 3 and thus s not canoncally embedded. The G m -acton on H 0( C, ω C s gven by t ω = t ( 1 ω, t η = η. The G m -acton on H 0( C, ω 2 C s gven by x (t 2 ( 1 x, y t ( 1 y, z z. We defne the weght of a monomal to be ts G m -weght. Proposton Both H 0( C, ω C and H 0 ( C, ω 2 C are multplcty-free representatons of G m Z g 1 Aut(C. Moreover, the bass {ω 0,..., ω g 2, η} s compatble wth the rreducble decomposton of H 0( C, ω C, and the bass {x, y, z : Z g 1 } s compatble wth the rreducble decomposton of H 0( C, ω 2 C. Proof. The acton of Z g 1 D g 1 on the span of {ω } g 2 =0 (resp., {x } g 2 =0, {y } g 2 =0, {z } g 2 =0 corresponds to the regular representaton of Z g 1 and s thus multplcty-free. Snce the weght of ω s ±1 and of η s 0 (resp., the weght of x s ±2, of y s ±1, and of z s 0, t follows that H 0( C, ω C (resp., H 0 ( C, ωc 2 s a multplcty-free representaton of Gm Z g 1. The followng lemmas are elementary and so we omt the proofs. Lemma The multplcaton map Sym m H 0( C, ω C H 0 ( C, ω m C s surjectve. A set B of degree m monomals n ω 0,..., ω g 2, η forms a monomal bass of H 0( C, ω m C f and only f the followng condtons are satsfed: (1 B contans the (g 1 monomals {ω m}g 2 =0 of weght ±m, (2 B contans the (g 1 monomals {ω m 1 η} g 2 =0 of weght ±(m 1, (3 B contans (g 1 lnearly ndependent monomals of each weght 2 m j m 2. The reader may wsh to check, as an example, that {ω j ηm j } g 2 =0 and {ωj+1 ω 1 η m j 2 } g 2 =0 gve 2g 2 lnearly ndependent monomals, wth (g 1 monomals of weghts j and j each. Thus, takng the unon of all these monomals, together wth {ω m}g 2 =0 and {ωm 1 η} g 2 =0 gves a monomal bass of H 0( C, ωc m. Lemma The multplcaton map Sym m H 0( C, ωc 2 H 0 ( C, ωc 2m s surjectve. A set B of degree m monomals n {x } g 2 =0, {y } g 2 =0, {z } g 2 =0 forms a monomal bass of H0( C, ωc 2m f and only f the followng condtons are satsfed: (1 B contans the (g 1 monomals {x m }g 2 =0 of weght ±2m, (2 B contans the (g 1 monomals {x m 1 y} g 2 =0 of weght ±(2m 1, (3 B contans (g 1 lnearly ndependent monomals of each weght 2 2m j 2m 2.

14 14 ALPER, FEDORCHUK, AND SMYTH 4. Monomal bases and semstablty 4.1. Canoncally embedded rbbon. Let C denote the balanced rbbon as defned n Secton 3.1. In ths secton, we prove the odd genus case of the frst part of our Man Result. Theorem 4.1. If C PH 0( C, ω C s a canoncally embedded balanced rbbon, then the Hlbert ponts [C] m are semstable for all m 2. Corollary 4.2. Suppose C PH 0( C, K C s a canoncally embedded generc smooth curve of odd genus. Then the m th Hlbert pont of C s semstable for every m 2. Proof of Corollary 4.2. Qute generally, the locus of semstable ponts (Hg,1 m ss H m g,1 s open [MFK94]. Snce H m g,1 s an rreducble varety whose generc pont s the m th Hlbert pont of a canoncally embedded smooth genus g curve, t remans to fnd a sngle semstable pont n Hg,1. m The balanced rbbon C deforms to a smooth canoncal curve by [Fon93] and Proposton 3.5 shows that [C] m Hg,1. m Applyng Theorem 4.1 fnshes the proof. We have already seen that there s a dstngushed bass {x 0,..., x k, y k+1,..., y 2k } of H 0( C, ω C on whch Gm Aut(C acts wth dstnct weghts (Proposton 3.3. Accordng to Lemma 2.5, to prove Theorem 4.1 t suffces to fnd a set of monomal bases such that an effectve lnear combnaton of ther ρ-weghts s 0 wth respect to every one-parameter subgroup ρ: G m SL(g. For ease of exposton, we wll treat the cases m = 2 and m 3 separately Monomal bases of H 0( C, ωc 2. Frst, we defne two monomal bases, B + and B, of H 0( C, ωc 2 as follows. We defne B + to be the set of quadratc monomals dvsble by one of x 0, x k, or y 2k. More precsely, (4.1 B + := We defne B as follows: { {x 0 x } k =0, {x 0 y } 2k =k+1, {x kx } k =1, {x k y } 2k =k+1, {y 2kx } k 1 =1, {y 2ky } 2k =k+1 (4.2 B := {x 2 } k =0, {y 2 } 2k =k+1, {x x +1 } k 1 =0, x ky k+1, {y y +1 } 2k 1 {x y +k } =1 k 1, {x y +k+1 } k 1 =0 =k+1, Lemma 4.3. B + and B are monomal bases of H 0( C, ω 2 C. For any one-parameter subgroup ρ actng on (x 0,..., y 2k dagonally wth weghts (ρ 0,..., ρ 2k the ρ-weghts of B + and B are: w ρ (B + = (g 2(ρ 0 + ρ k + ρ 2k, w ρ (B = 2(ρ 0 + ρ k + ρ 2k. Proof. Usng Lemma 3.6, one easly checks that B + and B are monomal bases. To compute the weght of B + observe that varables {x, y k+ } k 1 =1 each occur 3 tmes and varables {x 0, x k, y 2k } each occur g + 1 tmes n Dsplay (4.1. It follows that. k 1 w ρ (B + = 3 (ρ + ρ k+ + (g + 1(ρ 0 + ρ k + ρ 2k = (g 2(ρ 0 + ρ k + ρ 2k, =1 where the last equalty follows from the relaton 2k =0 ρ = 0. }.

15 FINITE HILBERT STABILITY OF (BICANONICAL CURVES 15 Smlarly, varables {x, y k+ } k 1 =1 each occur 6 tmes and varables {x 0, x k, y 2k } each occur 4 tmes n Dsplay (4.2. It follows that k 1 w ρ (B = 6 (ρ + ρ k+ + 4(ρ 0 + ρ k + ρ 2k = 2(ρ 0 + ρ k + ρ 2k, =1 where the last equalty agan follows from the relaton 2k =0 ρ = 0. Corollary 4.4. The 2 nd Hlbert pont of C s semstable. Proof. We have 2w ρ (B + + (g 2w ρ (B = 0 for any ρ: G m SL(g actng dagonally on the dstngushed bass. The clam follows by Lemma Monomal bases of H 0( C, ω m C for m 3. Fndng monomal bases n hgher degrees s slghtly more cumbersome than n the case m = 2. Frst, we wll need three monomal bases n every degree m 3. Second, the precse form of one of these bases depends on the resdue of g = 2k + 1 modulo 4. Nevertheless, the proof s conceptually no dfferent than n the case m = 2. Fnally, we work throughout wth m fxed and each bass used n degree m s defned ndependently as a set of degree m monomals, though we have, for smplcty, suppressed the dependence on m n our notaton. We begn by defnng two hgher-degree analogues of the bass B + from Secton Defnton 4.5. We defne B + 1 to be the set of degree m monomals n the deal (x 0, x k m 1 (x 0,..., x k 1, y k+1,..., y 2k + (x k, y 2k m 1 (x 0,..., x k 1, y k+1,..., y 2k + x m k. We defne B + 2 to be the set of degree m monomals n the deal (x 0, y 2k m 1 (x 1,..., x k 1, y k+1,..., y 2k 1 +x k (x 0, y 2k m 2 (x 1,..., x k 1, y k+1,..., y 2k 1 + (x 0, y 2k m + x k (x 0, y 2k m 1 + x 2 k (x 0, y 2k m 2 + x 3 k (x 0, y 2k m 3 Lemma 4.6. B 1 + and B+ 2 are monomal bases of H0( C, ωc m. For any one-parameter subgroup ρ actng on (x 0,..., y 2k dagonally wth weghts (ρ 0,..., ρ 2k the ρ-weghts of B 1 + and B+ 2 are: w ρ (B 1 + = ( (m 1 2 (g 1 (2m 3 ( m(m 1 ρ k + (g 1 1 (ρ 0 + ρ 2k, 2 w ρ (B + 2 = ( (m 1(g 1 + (2m 5 ρ k + ( (m 1 2 (g 1 (2m 3 (ρ 0 + ρ 2k. Proof. Usng Lemma 3.6, t s easy to see that B 1 + and B+ 2 are monomal bases. Next, note that ( n B 1 + the varable x k appears (m 1 2 (g 1+2 tmes, varables x 0 and y 2k each appear m 2 (g 1 + 2m 2 tmes, and varables x1,..., x k 1, y k+1,..., y 2k 1 each appear 2m 1 tmes. Recallng that 2k =0 ρ = 0, we deduce the formula for w ρ (B 1 +. The ρ-weght of B 2 + s computed analogously by observng that n B+ 2 the varable x k appears (m 1(g 1 + (4m 6 tmes, varables x 0 and y 2k each appear (m 1 2 (g tmes, and varables x 1,..., x k 1 and y k+1,..., y 2k 1 each appear 2m 1 tmes. Next, we construct hgher-degree analogues of the bass B from Secton Throughout the constructon, we let ι be the nvoluton exchangng x and y 2k and leavng x k fxed.

16 16 ALPER, FEDORCHUK, AND SMYTH Let l = k/2. We ntroduce the followng sets of monomals: {{ x m S 0 := k, x 0 y 2k x m 2 } k f m s odd, { x m k, x l y 2k l x m 2 } k f m s even. { } S 1 := x m d x d +1 : 0 k 1, 0 d m 1 { x m 1 d x d } +1 y +k+1 : 0 l 2, 0 d m 1 S 2 := x m 1 d l 1 x d l y l+k : 0 d m 2 The defnton of the next set of monomals depends on party of k. If k = 2l, we defne { } S 3 := x m 1 d x d +1 y k+2l 1 : l k 2, 1 d m 2 ; and f k = 2l + 1, we defne { x m 1 d x d } +1 y k+2l 1 : l k 3, 1 d m 2 S 3 := x m 2 d k 2 x d k 1 x l y 3l+1 : 0 d m 2 We proceed to defne { } S 4 := x m 2 d k 1 x d k (x 0y 2k : 0 d m 4 { } x k 1 x (m 1/2 l y (m 1/2 2k l f m s odd, S 5 := { } x k 1 x k x (m 2/2 l y (m 2/2 2k l f m s even. Defnton 4.7. We defne a set B of degree m monomals by 5 B ( := S 0 S ι(s. =1 Lemma 4.8. For m 3, B s a monomal bass of H 0( C, O C (m = H 0( C, ωc m. For any ρ: G m SL(g actng on (x 0,..., y 2k dagonally wth weghts (ρ 0,..., ρ 2k we have { w ρ (B (m 2 3m + 5(ρ 0 + ρ 2k (5m 10ρ k f m s odd, = (m 2 3m + 6(ρ 0 + ρ 2k (5m 12ρ k f m s even. Proof. Although the precse defnton of B depends on the party of k, our proof of the lemma does not. Thus we suppress the party of k n what follows. To prove that B s a monomal bass, we make use of the dentfcaton of H 0( C, ω m C wth functons n C[u, ɛ]/(ɛ 2 made n Secton 3.1. To begn, observe that B s nvarant under ι. Snce ι maps a monomal of u-degree d to a monomal of u-degree 2mk d, t suffces, n vew of Lemma 3.6, to show that B contans one monomal of each u-degree d = 0,..., k and two lnearly ndependent monomals of each u-degree d = k+1,..., mk. To do ths, note that S 0 conssts of two lnearly ndependent monomals of u-degree km; that S 1 conssts by the Rbbon Product Lemma 3.4 of exactly pure powers of u of each u-degree d = 0,..., mk 1; and that S 2 S 3 S 4 S 5 contans exactly one monomal of each u-degree d = k +1,..., mk 1 wth a non-zero ɛ term. Ths fnshes the proof that B s a monomal bass. To compute the ρ-weght of B, we observe that n S 1 {x m k } ι(s 1 all varables wth the excepton of x 0 and y 2k occur the same number of tmes, namely 2 m 1 d=1 d + m = m2

17 tmes, whle x 0 and y 2k each occur m of S 1 {x m k } ι(s 1 s FINITE HILBERT STABILITY OF (BICANONICAL CURVES 17 2k 1 m 2 =1 ρ + d=1 d = m(m + 1/2 tmes. It follows that the ρ-weght m(m + 1 (ρ 0 + ρ 2k = m(m 1(ρ 0 + ρ 2k /2, 2 where the last equalty follows from 2k =0 ρ = 0. Smlarly, one can easly see that n the remanng monomals of B each of the varables x 1,..., x k 1, y k+1,..., y 2k occurs exactly m(m 1 tmes; each of the varables x 0 and y 2k occurs { (m 2 + 3m 10/2 (f m s odd (m 2 tmes; + 3m 12/2 (f m s even and x k occurs { m 2 6m + 10 m 2 6m + 12 (f m s odd (f m s even tmes. Usng 2k =0 ρ = 0, t follows that the total ρ-weght of these remanng monomals s (m2 5m + 10 (ρ 0 + ρ 2k (5m 10ρ k 2 f m s odd, (m2 5m + 12 (ρ 0 + ρ 2k (5m 12ρ k 2 f m s even. The clam follows. Lemma 4.9. There exst c 0, c 1, c 2 Q (0, such that c 0 w ρ (B + c 1 w ρ (B c 2w ρ (B + 2 = 0 for all one-parameter subgroups of SL(g actng on the bass (x 0,..., y 2k dagonally. Proof. We need to show that w ρ (B gven by Lemma 4.8 and consdered as the lnear functon n (ρ 0,..., ρ 2k s the negatve of an effectve lnear combnaton of w ρ (B 1 + and w ρ (B 2 + gven by Lemma 4.6. In the case of odd m, the clam holds because the nequaltes m(m 1(g 1 2 2(m 1 2 (g 1 2(2m 3 (m2 3m + 5 (m 12 (g 1 (2m 3 5m 10 (m 1(g 1 + (2m 5, are satsfed for all g, m 3. In the case of even m, we requre the same nequaltes save that the mddle term s replaced by (m2 3m+6 5m 12. Proof of Theorem 4.1. The case of m = 2 was handled n Corollary 4.4. If m 3, the clam follows from Lemma 4.9 and Lemma Canoncally embedded A 2k+1 -curve. Let C denote the balanced double A 2k+1 -curve as defned n Secton 3.2. In ths secton, we prove the even genus case of the frst part of our Man Result. Snce H 0( C, ω C s a multplcty-free representaton of Gm Aut(C by Lemma 3.9, we can apply the Kempf-Morrson Crteron (Proposton 2.4 to prove semstablty of C. Namely, to prove that [C] m s semstable, t suffces to check that for every one-parameter subgroup ρ: G m SL(g actng dagonally on the dstngushed bass {x 1,..., x k, y 1,..., y k } wth nteger weghts λ 1,..., λ k, ν 1,..., ν k, there exsts a monomal bass of H 0( C, ωc m of non-postve ρ-weght. Explctly, ths means that we must exhbt a

18 18 ALPER, FEDORCHUK, AND SMYTH set B of (2m 1(2k 1 degree m monomals n the varables {x, y } k =1 wth the propertes that: (1 B maps to a bass of H 0( C, ωc m va Sym m H 0( C, ω C H 0 ( C, ωc m. (2 B has non-postve ρ-weght, that s, f B = {e l } (2m 1(2k 1 l=1 and e l = k then (2m 1(2k 1 k (a l λ + b l ν 0. l=1 =1 =1 xa l Theorem If C PH 0( C, ω C s a canoncally embedded balanced double A2k+1 -curve, then the Hlbert ponts [C] m are semstable for all m 2. As an mmedate corollary of ths result, we obtan a proof of Theorem 1.3 (1 and hence of Theorem 1.2: Corollary 4.11 (Theorem 1.3 (1. A generc canoncally embedded smooth trgonal curve of even genus has semstable m th Hlbert pont for every m 2. Proof of Corollary. Recall from Proposton 3.8 that the canoncal embeddng of the balanced double A 2k+1 -curve C les on a balanced surface scroll n P 2k 1 n the dvsor class (3, k + 1. It follows that C deforms flatly to a smooth curve n the class (3, k + 1 on the scroll. Such a curve s a canoncally embedded smooth trgonal curve of genus 2k. The semstablty of a generc deformaton of C follows from the openness of semstable locus. Proof of Theorem Recall from Lemma 3.7 that H 0( C, ωc m = span{ωj } mk j=m span{η j } mk j=m span{χ l } k(m 1 1 l= k(m 1+1. Now, gven a one-parameter subgroup ρ as above, we wll construct the requste monomal bass B as a unon B = B ω B η B χ, where B ω, B η, and B χ are collectons of degree m monomals whch map onto the bases of the subspaces spanned by {ω j } mk j=m, {η j} mk j=m, and {χ l} k(m 1 1 l= k(m 1+1, respectvely. To construct B ω and B η, we use Kempf s proof of the stablty of Hlbert ponts of a ratonal normal curve. More precsely, consder the component C 0 of C wth the unformzer u 0 at 0. Clearly, ω C C0 O P 1(k 1. The restrcton map H 0( C, ω C H 0 ( P 1, O P 1(k 1 dentfes {x } k =1 wth a bass of H0( P 1, O P 1(k 1 gven by {1, u 0,..., u0 k 1 }. Under ths dentfcaton, the subspace span{ω j } mk j=m s dentfed wth H0( P 1, O P 1(m(k 1. Set λ := k =1 λ /k. Gven a one-parameter subgroup ρ: G m SL(k actng on {x 1,..., x k } dagonally wth weghts (λ 1 λ,..., λ k λ, Kempf s result on the semstablty of a ratonal normal curve n P k 1 [Kem78, Corollary 5.3], mples the exstence of a monomal bass B ω of H 0( P 1, O P 1(m(k 1 wth non-postve ρ-weght. Under the above dentfcaton, B ω s a monomal bass of span{ω j } mk j=m of ρ-weght at most m(mk m + 1λ. Smlarly, f ν := k =1 ν /k, we deduce the exstence of a monomal bass B η of span{η j } mk j=m whose ρ- weght s at most m(mk m + 1ν. Snce λ + ν = 0, t follows that the total ρ-weght of B ω B η s non-postve. y b l,

19 FINITE HILBERT STABILITY OF (BICANONICAL CURVES 19 Thus, to construct a monomal bass B of non-postve ρ-weght, t remans to construct a monomal bass B χ of non-postve ρ-weght for the subspace span{χ l } (m 1k 1 l= (m 1k 1 H0( C, ω m C. In Lemma 4.12, proved below, we show the exstence of such a bass. Thus, we obtan the desred monomal bass B and fnsh the proof. Note that f we defne the weghted degree by deg(x = and deg(y =, then a set B χ of 2k(m 1 1 degree m monomals n {x 1,..., x k, y 1,..., y k } maps to a bass of span{χ l } (m 1k 1 l= (m 1k 1 f and only f t satsfes the followng two condtons: (1 Each monomal has both x and y terms, (2 Each weghted degree from (m 1k 1 to (m 1k + 1 occurs exactly once. We call such a set of monomals a χ-bass. The followng combnatoral lemma completes the proof of Theorem Lemma Suppose ρ: G m SL(2k s a one-parameter subgroup whch acts on {x 1,..., x k, y 1,..., y k } dagonally wth nteger weghts λ 1,..., λ k, ν 1,..., ν k satsfyng k =1 (λ + ν = 0. Then there exsts a χ-bass wth non-postve ρ-weght. Proof of Lemma 4.12 for m = 2. Take the frst χ-bass to be B 1 := {{x y k } 1 k 1, {x y k +1 } 1 k }. In ths bass, all varables except x k and y k occur twce and x k, y k occur once each. Thus w ρ (B 1 = 2(λ λ k 1 + 2(ν ν k 1 + λ k + ν k = (λ k + ν k. Take the second χ-bass to be We have B 2 := {{x k y } 1 k, {x y k } 1 k 1 }. k 1 w ρ (B 2 = k(λ k + ν k + (λ + ν = (k 1(λ k + ν k. =1 We conclude that for any one-parameter subgroup ρ, we have (k 1w ρ (B 1 + w ρ (B 2 = 0. It follows that ether B 1 or B 2 gves a χ-bass of non-postve weght. Proof of Lemma 4.12 for m 3. We wll prove the Lemma by exhbtng one collecton of χ-bases whose ρ-weghts sum to a postve multple of λ k + ν k and a collecton of χ-bases whose ρ-weghts sum to a negatve multple of λ k + ν k. Snce, for any gven one-parameter subgroup ρ, we have ether λ k + ν k 0 or λ k + ν k 0, t follows at once that one of our χ-bases must have non-postve weght. Throughout ths secton, we let ι be the nvoluton exchangng x and y. We begn by wrtng down χ-bases maxmzng the occurrences of x k and y k whle balancng the occurrences of the other varables. Defne T 1 as the set of all degree m monomals havng both x and y terms that belong to the deal (x k, y k m 1 (x 1,..., x k, y 1,..., y k.

20 20 ALPER, FEDORCHUK, AND SMYTH The ρ-weght of T 1 s ( ( m 1 k(m 1 + (2k 1 2 Note that T 1 msses only the m 2 weghted degrees k 1 (λ k + ν k + (m 1 (λ + ν. =1 k(m 3, k(m 5,..., k(m 5, k(m 3. For each s = 1,..., k 1, defne a set of m 2 monomals havng exactly these mssng degrees by T 2 (s := {x m 2 d k y d k (x k sx s : 1 d m 2} For each s, the sets T 1 T 2 (s and T 1 ι(t 2 (s are χ-bases. Usng k =1 (λ + ν = 0, one sees at once that the sum of the ρ-weghts of such bases, as s ranges from 1 to k 1, s a postve multple of (λ k + ν k. We now wrte down bases mnmzng the occurrences of x k and y k. We handle the case when k s even and odd separately. Case of even k: If k = 2l, we defne the followng set of monomals where the weghted degrees range from k(m 1 1 to m: { x m 1 d x d } 1 y k+1 : l + 2 k, 0 d m 1 S 1 := x m 1 d x d 1 y 1 : 2 l + 1, 0 d m 3 In the set S 1 ι(s 1, the varables x k and y k occur (m 2 m ( m 2 tmes, xl+1 and y l+1 occur (m 2 m 1 tmes, x l and y l occur (m 2 m m tmes, and x 1 and y 1 occur m 2 m (( m 2 1 tmes whle all of the other varables occur m 2 m tmes. To complete S 1 ι(s 1 to a χ-bass, we defne, for each s = 1,..., k 1, the followng set of monomals where the weghted degrees range from m 1 to 1 m: S 2 (s := x l+1 y l x m 2 1 x l y l (x s y s x m : for 0 2 m 2, x l y l (x s y s y m : for 0 2 < m 2, (x k y s y k s (x s y s x m : for 0 2 m 3, (x k y s y k s (x s y s y1 m 2 3 : for 0 2 < m 3, y l+1 x l y1 m 2 For each s = 1,..., k 1, the sets S 1 ι(s 1 S 2 (s and S 1 ι(s 1 ι(s 2 (s are χ-bases. We compute that n the unon k 1 ( S1 ι(s 1 S 2 (s ( S 1 ι(s 1 ι (S 2 (s s=1 of 2(k 1 χ-bases the varables x k and y k each occurs tmes whle all of the other varables occur 2(k 1(m 2 m (k 1(m 2 2m + 2 2(k 1(m 2 m + (m 2(m 1