On -gonal loci in Severi varieties on general surfaces and rational curves on hyperkähler manifolds
In this paper we study the gonality of the normalizations of curves in the linear system of a general primitively polarized complex surface of genus . We prove two main results. First we give a necessary condition on for the existence of a curve in with geometric genus whose normalization has a . Secondly we prove that for all numerical cases compatible with the above necessary condition, there is a family of nodal curves in of genus carrying a and of dimension equal to the expected dimension . Relations with the Mori cone of the hyperkähler manifold are discussed.
Let be a primitively polarized complex surface of genus , i.e. , and is a globally generated, indivisible, divisor (or line bundle) with . The main objective of this paper is to study the gonality of the normalization of curves, specifically of nodal curves, in the linear system , when is general in its moduli space, that is, belongs to a Zariski open dense subset.
Let be the Severi variety of curves with nodes. It is a classical result that is a nonempty, locally closed, smooth variety of dimension , which is the geometric genus of the curves in . The moduli morphism is finite to its image (see Proposition LABEL:prop:fff below).
We consider the subvariety of curves whose normalizations carry a . By Brill-Noether theory, if then so the interesting range is . A count of parameters, carried out in §LABEL:ssec:loci, suggests that the expected dimension of is . In any event, if nonempty, has dimension at least (see Proposition LABEL:degarg).
Our first main result is Theorem LABEL:thm:noexist, which yields a necessary condition for the normalization of a curve of geometric genus (with any type of singularities) on a primitively polarized surface of genus with no reducible curves in (e.g., with ) to possess a , namely that
and is the usual Brill-Noether number. In the case of ’s, setting , and , the above necessary condition reads
Theorem LABEL:thm:noexist is a strong improvement of [fkp, Thm. 1.4] and its proof is based on the vector bundle approach à la Lazarsfeld [L].
Our second main result deals with nonemptiness and dimension of , and with properties of its general member, and proves that the bound (1) is optimal:
Let be a general primitively polarized surface of genus and let and be integers satisfying and . Set . Then:
if and only if (1) holds;
when nonempty, has an irreducible component of the expected dimension whose general element is an irreducible curve with normalization of genus such that ;
in addition, when (resp. ), any (resp. the general) on has simple ramification and all nodes of are non-neutral with respect to it.
As for statement (i), we note that nonemptiness when is even, and has been proved by Voisin [vo, pf. of Cor. 1, p. 366] by a totally different approach.
Theorem 0.1 yields that, for fixed , the general curve in (some component of) the Severi variety has the gonality of a general genus curve, i.e. , but, for all satisfying (1), there are substrata of dimension of curves of lower gonality . This is in contrast to the case , where the gonality is constant and equal to . Parts (ii) and (iii) of Theorem 0.1 also yield that has gonality and enjoys properties of a general curve of such a gonality.
The proof of Theorem 0.1 relies on a rather delicate degeneration argument. Indeed, the general can be specialized to the case where contains a smooth rational curve of degree and such that can be embedded into by the linear system . This can be in turn degenerated to a union of two smooth rational normal surfaces and intersecting transversally along a smooth elliptic curve of degree and such that the rational curve specializes to the negative section of and specializes to the line bundle . This is proved in §LABEL:ssec:unionscr. In §§LABEL:sec:chains and LABEL:sec:limnod we describe nodal curves on the limit surface that fill up limit components of . In §LABEL:S:kn we describe possible limits of ; this latter analysis is one of the crucial points in the paper, and, to the best of our knowledge, is a nontrivial novelty. If nonempty, these limit varieties have the expected dimension, and this yields nonemptiness and expected dimension for the general containing a smooth rational curve of degree as above, and therefore also for the general as in the statement of the theorem (see Proposition LABEL:degargrel). In §LABEL:sec:finalexistence we show nonemptiness of the limit varieties and the required properties for the in the range (1).
We note that our two–step degeneration seems to be new and has the property of independent interest that the stable model of the general hyperplane section of the limit surface is an irreducible rational nodal curve. We believe that this technique can be useful in other contexts. Specifically, for surfaces this can be used to study Severi varieties of nodal curves, also in for .
Besides its intrinsic interest for Brill-Noether theory and moduli problems, the subject of this paper is related to Mori theory and rational curves on the -dimensional hyperkähler manifold parametrizing -dimensional length -subschemes of the surface . A curve on with a on its normalization determines a rational curve on . For the importance of rational curves on hyperkähler manifolds see, e.g., [H1, H2, Bou, HT, HT2, HTint, W1, W2, WW] and § LABEL:S:ratcur2. In particular, rational curves determine the nef and ample cones.
The curves on in Theorem 0.1 determine a family of rational curves in of dimension , which is the expected dimension of any family of rational curves on a -dimensional hyperkähler manifold. In §LABEL:S:ratcur we determine their classes in (see Lemma LABEL:classR); in this computation the properties of the stated in part (iii) of Theorem 0.1 play an essential role. The lower is, the closer the class is to the boundary of the Mori cone. As a consequence, we obtain necessary conditions for a divisor in to be nef or ample (see Proposition LABEL:prop:conoampio). For infinitely many we prove that the classes of the rational curves in we obtain from Theorem 0.1 with minimal satisfying (1) (which we call optimal classes) generate extremal rays of the Mori cone of (see Corollary LABEL:cor:intminima and Proposition LABEL:prop:intzero2). After the appearance of the first version of this paper on the web, this has been verified also in the cases , where , in [BM] (see Proposition LABEL:prop:BM). To determine the Mori cone of for all one would have to extend our results to the nonprimitive cases , . This is a difficult task, but should in principle be possible to treat with similar methods. We plan to do this in future research.
In §LABEL:S:ratcur2 we also relate our work to some interesting conjectures of Hassett and Tschinkel on the Mori cone of (see in particular Remark LABEL:rem:HT) and of Huybrechts and Sawon on Lagrangian fibrations (see in particular Corollary LABEL:cor:neclagfib).
Throughout this paper we work over . As usual, and as we did already in this Introduction, we may sometimes abuse notation and identify divisors with the corresponding line bundles, indifferently using the additive and the multiplicative notation.
This paper is the result of a long term collaboration between the authors. A substantial part of it was accomplished while they were both visiting the Mittag-Leffler Institute in Stockholm during the programme Algebraic Geometry with a view towards applications in the spring of 2011. Both authors are very grateful to this institution for the warm and fruitful hospitality.
The authors thank K. Ranestad and F. Flamini for useful conversations, L. Benzo for comments on the first version of this paper, B. Hassett for useful correspondence, A. Bayer and E. Macrì for informing us about and sending us their recent preprints [BM, BM-MMP] and for correspondence about them, and the referee for the careful reading of the paper and useful suggestions.
The first author is a member of the G.N.S.A.G.A. of the Istituto Nazionale di Alta Matematica “F. Severi”.
1. Severi varieties, surfaces and -gonal loci
1.1. Severi varieties and -gonal loci
Let be a connected, projective surface with normal crossing singularities and let be a base point free, complete linear system of Cartier divisors on whose general element is a connected curve with at most nodes as singularities, located at the singular points of . We will set .
For any integer , we denote by the locally closed subscheme of parametrizing the universal family of curves having only nodes as singularities, exactly of them (called the marked nodes) off the singular locus of , and such that the partial normalization at these nodes is connected (i.e., the marked nodes are not disconnecting nodes). We set . If is smooth the ’s are called Severi varieties of -nodal curves in on . We use the same terminology in our more general setting.
Let be an integer. We denote by the moduli space (or stack) of smooth curves of genus , whose dimension is . We recall that is quasi-projective and admits a projective compactification , parametrizing all connected stable curves of arithmetic genus .
One has the moduli morphism