On GIT quotients of Hilbert and Chow schemes of curves
aa r X i v : . [ m a t h . AG ] J a n ON GIT QUOTIENTS OF HILBERT AND CHOW SCHEMES OF CURVES
GILBERTO BINI, MARGARIDA MELO, FILIPPO VIVIANI
Abstract.
The aim of this note is to announce some results on the GIT problem for theHilbert and Chow scheme of curves of degree d and genus g in the projective space ofdimension d − g , whose full details will appear in [BMV]. In particular, we extend theprevious results of L. Caporaso up to d > g −
2) and we observe that this is sharp. In therange 2(2 g − < d < (2 g − Motivation
One of the first successful applications of Geometric Invariant Theory (GIT for short) wasthe construction of the moduli space M g of smooth curves of genus g ≥ M g via stable curves, carried out by Mumford ([Mum77]) and Gieseker([Gie82]). Indeed, the moduli space of stable curves was constructed as a GIT quotient ofthe locally closed subset of a suitable Hilbert scheme (as in [Gie82]) or Chow scheme (as in[Mum77]) parametrizing n -canonically embedded curves, for n ≥ n , specially in connection with the so called Hassett-Keel program whose ultimategoal is to find the minimal model of M g via the successive constructions of modular birationalmodels of M g (see [FS11] and [AH] for nice overviews). The first work in this direction is dueto Schubert, who described in [Sch91] the GIT quotient of the locus of 3-canonically embeddedcurves (of genus g ≥
3) in the Chow scheme as the coarse moduli space M ps g of pseudo-stablecurves (or p-stable curves for short), i.e., reduced, connected, projective curves with finite Date : October 11, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
GIT, Hilbert scheme, Chow scheme, stable curves, pseudo-stable curves, compact-ified universal Jacobian.The first named author has been partially supported by “FIRST” Universit`a di Milano and by MIURPRIN 2010 - Variet`a algebriche: geometria, aritmetica e strutture di Hodge. The second named author waspartially supported by CMUC and FCT (Portugal) through European program COMPETE/FEDER, by theFCT project
Espa¸cos de Moduli em Geometria Alg´ebrica (PTDC/MAT/111332/2009), by the FCT project
Geometria Alg´ebrica em Portugal (PTDC/MAT/099275/2008) and by the Funda¸c˜ao Calouste Gulbenkianprogram “Est´ımulo `a investiga¸c˜ao 2010”. The third named author was partially supported by CMUC and FCT(Portugal) through European program COMPETE/FEDER and by the FCT project
Espa¸cos de Moduli emGeometria Alg´ebrica (PTDC/MAT/111332/2009). automorphism group, whose only singularities are nodes and ordinary cusps, and which haveno elliptic tails. Indeed, it was shown by Hyeon-Morrison in [HM10] that one gets the sameGIT quotient for the Hilbert scheme of 3-canonically embedded curves and for the Hilbert orChow scheme of 4-canonically embedded curves. Later, Hasset-Hyeon constructed in [HH09] amodular map T : M g → M ps g which on geometric points sends a stable curve onto the p-stablecurve obtained by contracting all its elliptic tails to cusps. Moreover the authors of loc. cit.identified the map T with the first contraction in the Hassett-Keel program for M g . Finally theGIT quotient of the Hilbert and Chow scheme of 2-canonically embedded curves was studiedin great detail by Hassett-Hyeon in [HH]. For some partial results on the GIT quotient forthe Hilbert scheme of 1-canonically embedded curves, see the works of Alper-Fedorchuk-Smyth([AFSa], [AFSb]) and of Fedorchuk-Jensen ([FJ]).From the point of view of constructing new projective birational models of M g , it is of coursenatural to restrict the GIT analysis to the locally closed subset inside the Hilbert or the Chowscheme parametrizing n -canonical embedded curves. However, the problem of describing thewhole GIT quotient seems very natural and interesting too. The first result in this directionis the breakthrough work of Lucia Caporaso [Cap94], where she describes the GIT quotientof the Hilbert scheme of connected curves of genus g ≥ d ≥ g −
2) in P d − g .The GIT quotient that she obtains is indeed a modular compactification J d,g of the universalJacobian J d,g , which is the moduli scheme parametrizing pairs ( C, L ) where C is a smoothcurve of genus g and L is a line bundle on C of degree d . Note that recently Li-Wang in [LW]have given a different proof of the Caporaso’s result for d ≫ Problem(I):
Describe the GIT quotient of the Hilbert and Chow scheme of curves of genus g and degree d in P d − g , as d decreases with respect to g . Ideally, one would like then to interpret the different GIT quotients obtained (as d decreaseswith respect to g ) as first steps in a suitable “Hassett-Keel” program for Caporaso’s compact-ified universal Jacobian J d,g (see also Question B and the discussion following it). We hope tocome back to this project in a future work.In order to describe our results, we need to introduce some notation.2. Setup
We work over an algebraically closed field k (of arbitrary characteristic). For an integer g ≥ d , denote by Hilb d the Hilbert scheme of curves of degree d andarithmetic genus g in P d − g = P ( V ); denote by Chow d the Chow scheme of 1-cycles of degree d in P d − g and by Ch : Hilb d → Chow d the surjective map sending a one dimensional subscheme [ X ⊂ P d − g ] ∈ Hilb d to its 1-cycle.The linear algebraic group SL(V) ∼ = SL d − g+1 acts naturally on Hilb d and Chow d in such a way N GIT QUOTIENTS OF HILBERT AND CHOW SCHEMES OF CURVES 3 that Ch is an equivariant map. The action of SL(V) on Hilb d and Chow d can be naturallylinearized as follows.For any m ≫
0, setting P ( m ) := md + 1 − g , the Hilbert scheme Hilb d admits a SL(V)-equivariant embedding(1) j m : Hilb d ֒ → Gr( P ( m ) , Sym m V ∨ ) ֒ → P P ( m ) ^ Sym m V ∨ := P , where Gr( P ( m ) , Sym m V ∨ ) is the Grassmannian variety parametrizing P ( m )-dimensional quo-tients of Sym m V ∨ , which embeds in P (cid:16)V P ( m ) Sym m V ∨ (cid:17) via the Pl¨ucker embedding. Explic-itly, the map j m sends [ X ⊂ P d − g ] ∈ Hilb d into j m ([ X ⊂ P d − g ]) := (cid:2) Sym m V ∨ ։ H ( X, O X ( m )) (cid:3) ∈ Gr( P ( m ) , Sym m V ∨ ) ֒ → P ( P ( m ) ^ Sym m V ∨ ) . We refer to [Mum66, Lect. 15] for details. From the embedding (1), we get an ample SL(V)-linearized line bundle Λ m := j ∗ m O P (1) (for m ≫
0) and we denote byHilb s,md ⊆ Hilb ps,md ⊆ Hilb ss,md ⊆ Hilb d the locus of points that are, respectively, stable, polystable and semistable with respect toΛ m . It is well-known (see [HH, Sec. 3.6]) that Hilb s,m , Hilb ps,md and Hilb ss,m are constant for m ≫ Hilb sd := Hilb s,md for m ≫ , Hilb psd := Hilb ps,md for m ≫ , Hilb ssd := Hilb ss,md for m ≫ . If [ X ⊂ P d − g ] ∈ Hilb sd (resp. Hilb psd , resp. Hilb ssd ), we say that [ X ⊂ P d − g ] is Hilbert stable (resp.
Hilbert polystable , resp.
Hilbert semistable ).The Chow scheme Chow d admits a SL(V)-equivariant embedding(2) Chow d i ֒ → P ( ⊗ Sym d V ∨ ) := P ′ obtained by sending a 1-cycle Z of degree d in P d − g into the hyperplane of ⊗ Sym d V ∨ generatedby all the elements F ⊗ G ∈ Sym d V ∨ such that Z ∩ { F = 0 } ∩ { G = 0 } 6 = ∅ . We refer to[Mum66, Lec. 16] for details. From the embedding (2), we get an ample SL(V)-linearized linebundle Λ := i ∗ O P ′ (1) and we denote byChow sd ⊆ Chow psd ⊆ Chow ssd ⊆ Chow d the locus of points of Chow d that are, respectively, stable, polystable or semistable with respectto Λ. We say that [ X ⊂ P d − g ] ∈ Hilb d is Chow semistable (resp.
Chow polystable , resp.
Chowstable ) if its image Ch([ X ⊂ P d − g ]) belongs to Chow ssd (resp. Chow psd , resp. Chow sd ). Therelation between Hilbert (semi)stability and Chow (semi)stability is given by the followingchain of open inclusions (see [HH, Prop. 3.13])(3) Ch − (Chow sd ) ⊆ Hilb sd ⊆ Hilb ssd ⊆ Ch − (Chow ssd ) ⊆ Hilb d . GILBERTO BINI, MARGARIDA MELO, FILIPPO VIVIANI
In particular, there is a natural morphism of GIT-quotientsHilb ssd //SL ( V ) → Chow ssd //SL ( V ) . Note that, in general, there is no obvious relation between Hilb psd and Ch − (Chow psd ): forexample, according to [HH, Prop. 11.6 and Prop. 11.8], there are 2-canonical curves that areHilbert polystable but not Chow polystable and conversely.We can now reformulate Problem(I) in the following form. Problem(II):
Describe the points [ X ⊂ P d − g ] ∈ Hilb d that are Hilbert or Chow (semi, poly)stable, as d decreases with respect to g . The aim of this note is to announce some partial results on the above Problem(II). Fulldetails will appear in [BMV]. 3.
Results
Our partial answer to the above Problem(II) will require some conditions on the singularitiesof X and some conditions on the multidegree of the line bundle O X (1). Let us introduce therelevant definitions. Definition 1. (i) A curve X is said to be quasi-stable if it is obtained from a stable curve Y by “blowingup” some of its nodes, i.e. by taking the partial normalization of Y at some of its nodesand inserting a P connecting the two branches of each node.(ii) A curve X is said to be quasi-p-stable if it is obtained from a p-stable curve Y by “blowingup” some of its nodes (as before) and “blowing up” some of its cusps, i.e. by taking thepartial normalization of Y at some of its cusps and inserting a P tangent to the branchpoint of each cusp.Given a quasi-stable or a quasi-p-stable curve X , we call the P ’s inserted by blowing up nodesor cusps of Y the exceptional components , and we denote by X exc ⊂ X the union of all of them. Definition 2.
Let X be a quasi-stable or a quasi-p-stable curve of genus g ≥ L be aline bundle on X of degree d . We say that:(i) L is balanced if for each subcurve Z ⊂ X the following inequality (called the basic in-equality) is satisfied(*) (cid:12)(cid:12)(cid:12)(cid:12) deg Z L − d g − Z ( ω X ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | Z ∩ Z c | , where | Z ∩ Z c | denotes the length of the 0-dimensional subscheme of X obtained as thescheme-theoretic intersection of Z with the complementary subcurve Z c := X \ Z .(ii) L is properly balanced if L is balanced and the degree of L on each exceptional componentof X is 1.(iii) L is strictly balanced if L is properly balanced and the basic inequality (*) is strict exceptpossibly for the subcurves Z such that Z ∩ Z c ⊂ X exc . N GIT QUOTIENTS OF HILBERT AND CHOW SCHEMES OF CURVES 5 (iv) L is stably balanced if L is properly balanced and the basic inequality (*) is strict exceptpossibly for the subcurves Z such that Z or Z c is entirely contained in X exc .The inequality (*) first appeared in the work of Mumford [Mum77, Prop. 5.5] and Gieseker[Gie82, Prop. 1.0.11]. See also [Cap94, Sec. 3.1], where (*) is called the “Basic Inequality”.We are now ready to state the main results of [BMV]. Our first result deals with high valuesof the degree d . Theorem A.
Consider a point [ X ⊂ P d − g ] ∈ Hilb d with d > g − and g ≥ ; assumemoreover that X is connected. Then the following conditions are equivalent:(i) [ X ⊂ P d − g ] is Hilbert semistable (resp. polystable, resp. stable);(ii) [ X ⊂ P d − g ] is Chow semistable (resp. polystable, resp. stable);(iii) X is quasi-stable and O X (1) is balanced (resp. strictly balanced, resp. stably balanced).In each of the above cases, X ⊂ P d − g is non-degenerate and linearly normal, and O X (1) isnon-special.Moreover, the Hilbert or Chow GIT quotient is geometric (i.e. all the Hilbert or Chowsemistable points are stable) if and only if gcd(2 g − , d − g + 1) = 1 . The above Theorem was proved by Caporaso in [Cap94] for d ≥ g −
2) and for Hilbert(semi-, poly-)stability. We remark that the hypothesis d > g −
2) in the above TheoremA is sharp: in [HM10] it is proved that a 4-canonical p-stable curve (which in particular canhave cusps) is Hilbert stable while a 4-canonical stable curve with an elliptic tail is not Hilbertsemistable.We then investigate what happens if d ≤ g −
2) and we get a complete answer in the case2(2 g − < d < (2 g −
2) and g ≥ Theorem B.
Consider a point [ X ⊂ P d − g ] ∈ Hilb d with g − < d < (2 g − and g ≥ ;assume moreover that X is connected. Then the following conditions are equivalent:(i) [ X ⊂ P d − g ] is Hilbert semistable (resp. polystable, resp. stable);(ii) [ X ⊂ P d − g ] is Chow semistable (resp. polystable, resp. stable);(iii) X is quasi-p-stable and O X (1) is balanced (resp. strictly balanced, resp. stably balanced).In each of the above cases, X ⊂ P d − g is non-degenerate and linearly normal, and O X (1) isnon-special.Moreover, the Hilbert or Chow GIT quotient is geometric (i.e. all the Hilbert or Chowsemistable points are stable) if and only if gcd(2 g − , d − g + 1) = 1 . We note that the conditions on the degree d and the genus g in the above Theorem B aresharp. Indeed, if d = 2(2 g −
2) then it follows from [HH, Thm. 2.14] that there are 2-canonicalHilbert stable curves having arbitrary tacnodes and not only tacnodes obtained by blowing upa cusp as in Definition 1(ii). On the other hand, if d = (2 g −
2) (resp. d > (2 g − X ⊂ P d − g ] ∈ Hilb d such that X is aquasi-p-stable but not p-stable curve is not Chow stable (resp. Chow semistable) . Finally, if We thank Fabio Felici for pointing out to us the relevance of [Gie82, Prop. 1.0.6, Case 2].
GILBERTO BINI, MARGARIDA MELO, FILIPPO VIVIANI g = 3 then Hyeon-Lee proved in [HL07] that a 3-canonical irreducible p-stable curve with onecusp is not Hilbert polystable.As an application of Theorem B, we get a new compactification of the universal Jacobian J d,g over the moduli space of p-stable curves of genus g . To this aim, consider the category fiberedin groupoids J ps d,g over the category of schemes, whose fiber over a scheme S is the groupoidof families of quasi-p-stable curves over S endowed with a line bundle whose restriction to thegeometric fibers is properly balanced. Theorem C.
Let g ≥ and d ∈ Z .(1) J ps d,g is a smooth, irreducible Artin stack of finite type over k and of dimension g − .Moreover J ps d,g is universally closed and weakly separated (in the sense of [ASvdW] ).(2) J ps d,g admits an adequate moduli space J ps d,g (in the sense of [Alp] ), which is a normalirreducible projective variety of dimension g − containing J d,g as an open subvari-ety. Moreover, if char( k ) = 0 , then J ps d,g has rational singularities, hence it is Cohen-Macauly.(3) There exists a commutative digram J ps d,g / / Ψ ps (cid:15) (cid:15) J ps d,g Φ ps (cid:15) (cid:15) M ps g / / M ps g where Ψ ps is surjective, universally closed and weakly separated (in the sense of [ASvdW] )and Φ ps is surjective and projective with equidimensional fibers of dimension g .(4) If char( k ) = 0 or char( k ) = p > is bigger than the order of the automorphismgroup of any p-stable curve of genus g , then for any X ∈ M ps g , the fiber (Φ ps ) − ( X ) isisomorphic to Jac d ( X ) / Aut( X ) , where Jac d ( X ) is the Simpson’s compactified Jacobianof X parametrizing S -equivalence classes of rank , torsion-free sheaves on X that areslope-semistable with respect to ω X .(5) If g − < d < (2 g − then J ps d,g ∼ = [ H d /GL ( r + 1)] and J ps d,g ∼ = H d /GL ( r + 1) ,where H d ⊂ Hilb d is the open subset consisting of points [ X ⊂ P d − g ] ∈ Hilb d such that X is connected and [ X ⊂ P d − g ] is Hilbert semistable (or equivalently, Chow semistable). The proof of the above Theorems will appear in [BMV].4.
Open questions
The above results leave unsolved some natural questions for further investigation, that wediscuss briefly here.The first question is of course the following
Question A.
Describe the Hilbert and Chow (semi-,poly-)stable points of
Hilb d in the casewhere (2 g − ≤ d ≤ g − . N GIT QUOTIENTS OF HILBERT AND CHOW SCHEMES OF CURVES 7
Indeed, as observed before, Theorems A and B do not hold for these values of d . In [BMV,Thm. 5.1], we proved the following partial result: if [ X ⊂ P d − g ] is Chow semistable with X connected then O X (1) is balanced and X is a reduced curve whose singularities are at mostnodes, cusps and tacnodes at which two components of X meet, one of which is a line of P d − g .Further progresses have been made by Fabio Felici in [Fel].By analogy with the contraction map T : M ps g → M g constructed by Hassett-Hyeon in[HH09], the following problem arises very naturally. Question B.
Construct a map e T : J d,g → J ps d,g fitting into the following commutative diagram J d,g e T / / Φ s (cid:15) (cid:15) J ps d,g Φ ps (cid:15) (cid:15) M g T / / M ps g , where J d,g Φ s −−→ M g is Caporaso’s compactification of the universal Jacobian over M g . More generally, one would like to set up a “Hassett-Keel” program for J d,g and give aninterpretation of the above map e T as the first step in this program.Finally, the reader may have noticed that in Theorems A and B we have characterized points[ X ⊂ P d − g ] ∈ Hilb d that are Hilbert or Chow (semi,poly)stable under the assumption that X is connected. Indeed, it is easy to prove that, at least for d > g − Question C.
Are there connected components inside the Hilbert or Chow GIT semistable locusmade entirely of non-connected curves?
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Compositio Math. (1991),297–313. E-mail address : [email protected] Current address : Dipartimento di Matematica, Universit`a degli Studi di Milano, Via C. Saldini 50,20133 Milano, Italy.
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