On universal Severi varieties of low genus K3 surfaces
aa r X i v : . [ m a t h . AG ] M a y On universal Severi varieties of low genus K surfaces Ciro Ciliberto and Thomas Dedieu
Abstract
We prove the irreducibility of universal Severi varieties parametrizing irreducible, reduced, nodalhyperplane sections of primitive K g , with 3 ≤ g ≤ g = 10. Introduction
F. Severi was one of the first algebraic geometers who stressed the importance, for moduli and enumerativeproblems, of studying the families V d,g of irreducible, nodal, plane curves of degree d and geometric genus g (see e.g. [29, Anhang F]). In particular he proved that the varieties V d,g all have the expected dimension d + g − d minus the number ofnodes) and asserted, but did not succeed to prove, that they are irreducible, a result due to J. Harris in[16]. For this reason, the V d,g ’s have been called Severi varieties .The notion of Severi variety can be extended to families of nodal curves on any surface, and analogousirreducibility problems naturally arise. These are in general hard questions, even for rational surfaces.For instance, irreducibility is known to hold for Hirzebruch surfaces [32] and for rational curves on DelPezzo’s surfaces [31], with one notable (and understood) exception for Del Pezzo’s surfaces of degree 1.On the other hand, for surfaces with positive canonical bundle, Severi varieties have in general a quiteunpredictable behaviour: examples are given in [6] of surfaces with reducible Severi varieties, and evenwith components of Severi varieties of dimension different from the expected one.In this note we concentrate on Severi varieties on K universal Severivarieties parametrizing irreducible, reduced, nodal curves on primitive K g . Conjecturally, all these varieties should be irreducible (see § ≤ g ≤ g = 10. This partially affirmatively answers a question posedby the second author in [13] and it is, as far as we know, the first irreducibility result for Severi varietiesof K δ of nodes of the K P g as surfaces of degree2 g −
2, fill up the whole component of nodal degenerate canonical curves with δ nodes in the Hilbertscheme (see Proposition 2.6 or [14]). Secondly, using a degeneration technique due to Pinkham [24], weprove that all components of a certain flag Hilbert scheme pass through some cone points , where, on theother hand, we are able to prove smoothness of the flag Hilbert scheme, which is then irreducible at thosepoints. Both ideas are inspired by [9, 11].In § K § § § § §
2. We take aHilbert schematic viewpoint, which we set up in § § § § Acknowledgements
The second author wishes to thank the Groupement de Recherche europ´eenItalo-Fran¸cais en G´eom´etrie Alg´ebrique (CNRS and INdAM) for funding his stay at the university ofRoma Tor Vergata during part of the preparation of this work. K surfaces and their Severi varieties A K surface X is a smooth complex projective surface with Ω X ∼ = O X and h ( X, O X ) = 0. A primitive K surface of genus g is a pair ( X, L ), where X is a K L is an indivisible, nef line bundle1n X , such that | L | is without fixed component and L = 2 g − g ≥ | L | is base point free, and the morphism ϕ | L | determined by this linear system is birational if and only if L > | L | does not contain any hyperelliptic curve (hence g ≥ ϕ | L | is a surface of degree 2 g − P g , with canonical singularities, and whose general hyperplanesection is a canonical curve of genus g (see [27]).For all g ≥
2, we can consider the moduli stack B g of primitive K g , which issmooth, of dimension 19 (see [1, 23]). For ( X, L ) very general in B g , the Picard group of X is generatedby the class of L , and L is very ample if g ≥ Given a K X, L ) of genus g and two integers k and h , consider V k,h ( X, L ) := { C ∈ | kL | irreducible and nodal with g ( C ) = h } , where g ( C ) is the geometric genus of C , i.e. the genus of its normalization, so that C has g − h nodes. V k,h ( X, L ), called the ( k, h ) –Severi variety of ( X, L ) (or simply
Severi variety if there is no danger ofconfusion), is a functorially defined, locally closed subscheme of the projective space | kL | of dimension1 + k ( g −
1) =: p a ( k ), which is the arithmetic genus of the curves in | kL | . We will drop the index k if k = 1 and we may drop the indication of the pair ( X, L ) if there is no danger of confusion.
Theorem 1.1
Let k ≥ and ≤ h ≤ p a ( k ) . The variety V k,h , if not empty, is smooth of dimension h .If ( X, L ) is general in B g , then V k,h is not empty. The first assertion is classical and standard in deformation theory (see [29] and, more recently, e.g.[6, 13, 30]). The second part is a consequence of the main theorem in [7] (see also Mumford’s theorem in[1, pp. 365–367]).If (
X, L ) is general, V k, is reducible, consisting of a finite number of points (for the degree of V k, ,see [4, 33]). One might instead expect that if ( X, L ) is general and h ≥
1, then V k,h is irreducible. Thisis trivially true for h = p a ( k ) and not difficult for h = p a ( k ) − h gets lower. This conjecture, if true, certainly will not be easy to prove.As a first approximation, one may propose a weaker irreducibility conjecture concerning universal Severivarieties (see [13]), which we now recall. For any g ≥ k ≥ ≤ h ≤ p a ( k ), one can consider a stack V gk,h (see [14, Proposition 4.8]), calledthe universal Severi variety , which is pure and smooth of dimension 19 + h , endowed with a morphism φ gk,h : V gk,h → B ◦ g , where B ◦ g is a suitable dense open substack of B g . The morphism φ gk,h is smooth onall components of V gk,h , and its fibres are described in the following diagram: V gk,h ⊃ φ gk,h (cid:15) (cid:15) V k,h ( X, L ) (cid:15) (cid:15) B ◦ g ∋ ( X, L )Thus a point of V gk,h can be regarded as a pair ( X, C ) with (
X, L ) ∈ B g and C ∈ V k,h ( X, L ).One can conjecture that all universal Severi varieties V gk,h are irreducible . This does not imply theirreducibility of the pointwise Severi varieties V k,h ( X, L ), even if (
X, L ) is general in B g . The conjecturerather means that the monodromy of the morphism φ gk,h transitively permutes the components of thefibre V k,h ( X, L ), for (
X, L ) ∈ B g general. This makes sense even if h = 0, when the pointwise Severivariety V k, ( X, L ) is certainly reducible.In addition to its intrinsic interest, this conjecture is motivated by the results in [13], where it is shownthat (a weak version of) it implies the non–existence of rational map f : X X with deg( f ) > K X, L ) of a given genus g . Very recently a proof of this result, based on quite delicatedegeneration argument, has been proposed by Xi Chen [8].2 .4 The moduli map There is a natural moduli map µ gk,h : V gk,h → M h , where M h is the moduli stack of curves of genus h . The case k = 1, h = g has been much studied. It is related to the behaviour of the Wahl map w C : V H ( C, ω C ) → H (cid:0) C, ω C (cid:1) of a smooth curve C of genus g , to extension properties of canonicalcurves and to the classification of Fano varieties of the principal series and of
Mukai varieties . Wewill not dwell recalling all results on this subject, deferring the reader to the current literature (see, inchronological order, [20, 34, 3, 21, 11, 12, 22, 9, 10, 5]). Only recently the nodal case h < g , k = 1,received the deserved attention. We recall the following theorem. Theorem 1.2
Assume ≤ g ≤ and ≤ h ≤ g . For any irreducible component V of V gh , the modulimap µ gh | V : V → M h is dominant, unless g = h = 10 . The case h = g is in the series of papers [20, 21, 22] (see also [5]). The rest is in [14]. Remark 1.3
As stated in [14], the theorem applies only for h ≥
2. The case h = 0 is trivially true. Theproof in [14] applies to the case h = 1 if 3 ≤ g ≤
11 and g = 10 as well. The case h = 1 , g = 10 is notcovered by the original argument (see also [14, last lines of the proof of Theorem 5.5]), but can also befixed. We do not dwell on this here.In the recent paper [15], the moduli map µ gk,h has been studied also for g ≥
13, any k and h sufficientlylarge with respect to g , proving that, as one may expect, µ gk,h is generically finite to its image in thesecases. The remaining cases for g, h, k are very interesting and still widely open. The aim of this paper is to prove the following result, which affirmatively answers the conjecture in § Theorem 2.1
For ≤ g ≤ , g = 10 and ≤ h ≤ g , the universal Severi variety V gh is irreducible. By adopting a Hilbert schematic viewpoint and inspired by [11], we find a flag Hilbert scheme F g,h ,with a rational map F g,h V gh dominating all components of V gh , and we prove that F g,h is irreducible(see Theorem 2.2). To show this, we exhibit smooth points of F g,h which are contained in all irreduciblecomponents of F g,h (see § For any g ≥
3, we let B g be the component of the Hilbert scheme of surfaces in P g whose general pointparametrizes a primitive K g . An open subset of B g is a PGL( g + 1 , C )-bundle overthe open subset of B g corresponding to pairs ( X, L ) with very ample L . The variety B g is thereforeirreducible of dimension g + 2 g + 19.Let C g be the component of the Hilbert scheme of curves in P g whose general point parametrizes a degenerate canonical curve of genus g , i.e. a smooth canonical curve of genus g lying in a hyperplane of P g . An open subset of C g is a |O P g (1) | × PGL( g, C )–bundle over the open subset of M g parametrizingnon–hyperelliptic curves, so C g is irreducible of dimension g + 4 g − F g be the component of the flag Hilbert scheme of P g (see [17, 28]) whose general point is a pair( X, C ) with X ∈ B g general and C ∈ C g a general hyperplane section of X . An open subset of F g is a P g –bundle over an open subset of B g . As such it is irreducible of dimension g + 3 g + 19.Let 0 ≤ h ≤ g . We denote by C g,h the Zariski closure of the locally closed, functorially defined, subsetof C g formed by irreducible, nodal, genus h curves. It comes with a moduli map c g,h : C g,h M h , whichis dominant. Up to projective transformations, the fibre over a curve C ∈ M h is a dense open subsetof Sym δ (cid:0) Sym ( C ) (cid:1) , with δ = g − h , and is therefore irreducible. So C g,h is irreducible, of dimension g + 4 g − − δ .We let F g,h be the inverse image of C g,h under the projection F g → C g . We have a natural dominantmap m g,h : F g,h V gh . Any irreducible component F of F g,h dominates B g via the restriction of theprojection F g → B g (see §§ F ) = dim ( F g,h ) = g + 3 g + 19 − δ, δ = g − h . We let p g,h : F g,h → C g,h be the natural projection and we use the shorternotation p g for p g,g .Because of the existence of the dominant map m g,h , the following implies Theorem 2.1. Theorem 2.2
Let ≤ g ≤ , g = 10 , and ≤ h ≤ g . Then F g,h is irreducible. For the proof, we need to recall a few facts, collected in the next two subsections.
The following lemma relies on a well known construction of Pinkham [24, (7.7)], and is based on the factthat smooth K projectively Cohen–Macaulay , see [19, 27]. Lemma 2.3
Let ( X, C ) ∈ F g,h with X a smooth K surface. Let X C be the cone over C from a point v in P g off the hyperplane in which C sits. Then one can flatly degenerate ( X, C ) to ( X C , C ) inside thefibre F C of p g,h over C . Proof.
Let H be the hyperplane containing C . Choose homogeneous coordinates ( x : . . . : x g ) such that v = (1: 0 : . . . : 0) and H is given by x = 0. Consider the projective transformation ω t , t = 0, such that ω t ( x : . . . : x g ) = ( tx : x : . . . : x g ). Set X t = ω t ( X ). Then ( X t , C ) ∈ F C for all t = 0. Since X isprojectively Cohen–Macaulay, X C , with its reduced structure, is the flat limit of X t when t tends to 0. (cid:3) The fibre F C of p g,h equals the fibre of p g , whose tangent space at the point ( X, C ) is isomorphicto H ( X, N X/ P g ( − § cone point ( X C , C ) (the proof is the same as in [24, Theorem 5.1], and relies on the fact that C is projectivelyCohen–Macaulay, see [25, 18, 26]). Lemma 2.4
Let C be a reduced and irreducible, not necessarily smooth, degenerate canonical curve in P g , of arithmetic genus g . Let X C be the cone over C from a point in P g off the hyperplane in which C sits. For all i ≥ , one has H (cid:0) X C , N X C / P g (cid:0) − i )) ∼ = M k ≥ i H (cid:0) C, N C/ P g − ( − k ) (cid:1) . (2.1)Next we need to bound from above the dimensions of the cohomology spaces appearing in the right–hand–side of (2.1). We use semi–continuity, and a special type of canonical curves for which they can becomputed. A graph curve of genus g is a stable curve of genus g consisting of 2 g − g − trivalent graph, consisting of 2 g − g − C is a graph curve and itsdualizing sheaf ω C is very ample, then C can be canonically embedded in P g − as a union of 2 g − canonical graph curve . Proposition 2.5 [11]
For ≤ g ≤ , g = 10 , there exists a genus g canonical graph curve Γ g in P g − ,sitting in the image of p g , such that the dimensions of the spaces of sections of negative twists of thenormal bundle are given in the following table: h (cid:0) N Γ g / P g − ( − k ) (cid:1) \ g k = 1 10 13 15 16 16 15 14 12 k = 2 6 5 3 1 0 0 0 0 k = 3 3 1 0 0 0 0 0 0 k = 4 1 0 0 0 0 0 0 0 k ≥ for every g hence X k ≥ h (cid:0) Γ g , N Γ g / P g − ( − k ) (cid:1) = 23 − g. (2.2)4 .4 Proof of the main theorem Here we prove Theorem 2.2 and therefore also Theorem 2.1. The first step is the following:
Proposition 2.6
Let g and h be two integers such that ≤ g ≤ , g = 10 , and ≤ h ≤ g . Let F bea component of F g,h and let ( X, C ) ∈ F be a general point. Then all components of the fibre F C of p g,h over C have dimension − g , and the restriction of p g,h to F is dominant onto C g,h . Proof.
Note that X is general in B g (see § F C equals the fibre of p g , whose tangentspace at ( X, C ) is isomorphic to H ( X, N X/ P g ( − C of X . So this is like computing the tangent space to the fibre of p g at a general point ( X, ¯ C )of F g , with ¯ C general in C g by the case h = g of Theorem 1.2.By degenerating to the cone point ( X ¯ C , ¯ C ), by Lemma 2.4, and by upper–semi–continuity, we have h ( X, N X/ P g ( − ≤ P k ≥ h ( ¯ C, N ¯ C/ P g − ( − k )). By further degenerating to one of the graph curves inProposition 2.5, and taking into account (2.2), we have h ( X, N X/ P g ( − ≤ − g (this argument hasbeen extracted from [9, § F C at ( X, C ). Since23 − g = dim ( F ) − dim ( C g,h )(see § F C at ( X, C ), and the restriction of p g,h at F is dominant. (cid:3) With a similar argument we can finish the:
Proof of Theorem 2.2.
Let F i , 1 ≤ i ≤
2, be distinct components of F g,h . Let C ∈ C g,h be a generalpoint. By Proposition 2.6, there are points ( X i , C ) ∈ F i , and they can be assumed to be general pointson two distinct components F i of F C , 1 ≤ i ≤
2. By Lemma 2.3, both F and F contain the cone point( X C , C ). We will reach a contradiction by showing that ( X C , C ) is a smooth point of F C .Since C is general in C g,h and C g,h clearly contains the graph curves Γ g of Proposition 2.5, by upper–semicontinuity h ( X C , N X C / P g ( − F C at ( X C , C ), concluding the proof. (cid:3) Remark 2.7 (i) Proposition 2.6 gives a quick alternative proof of the part h < g of Theorem 1.2 when g = 10, which is based on the part h = g .(ii) The argument does not work for g = 10. In fact, if C is any curve in the image of p lying on asmooth K h (cid:0) X, N X/ P g ( − (cid:1) = 14, see [12, Lemma 1.2]. The analogue of Proposition2.6 in this case is that all components of a general fibre of p ,h have dimension 14. So the image of p g,h has codimension 1 in C g,h . Luckily, and as one could expect, Theorem 1.2 ensures that the modulimap c g,h dominates M h for 0 ≤ h ≤
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Universit´e Paul Sabatier, Institut de Math´ematiques de Toulouse, 118 route de Narbonne, 31062Toulouse Cedex 9, France [email protected]@math.univ-toulouse.fr