smoothing of limit linear series on curves and metrized complexes of pseudocompact type
We investigate the connection between Osserman limit series [Oss] (on curves of pseudocompact type) and Amini-Baker limit linear series [AB15] (on metrized complexes with corresponding underlying curve) via a notion of pre-limit linear series on curves of the same type. Then, applying the smoothing theorems of Osserman limit linear series, we deduce that, fixing certain metrized complexes, or for certain types of Amini-Baker limit linear series, the smoothability is equivalent to a certain “weak glueing condition”. Also for arbitrary metrized complexes of pseudocompact type the weak glueing condition (when it applies) is necessary for smoothability. As an application we confirm the lifting property of specific divisors on the metric graph associated to a certain regular smoothing family, and give a new proof of [CJP15, Theorem 1.1] for vertex avoiding divisors, and generalize loc.cit. for divisors of rank one in the sense that, for the metric graph, there could be at most three edges (instead of two) between any pair of adjacent vertices.
The theory of limit linear series has been developed by Eisenbud and Harris in [EH86] for handling the degeneration of linear series on smooth curves as the curves degenerate to reducible curves (of compact type). It has been applied to prove results involving moduli space of curves, such as the Brill-Noether theorem ([GH80]), the Gieseker-Petri theorem ([Gie82]), and that moduli spaces of curves of sufficiently high genus are of general type ([HM82] and [EH87]), etc.
A complete generalization of Eisenbud-Harris theory has remained open. Earlier approaches can be found in papers of Eduardo Esteves such as [Est98] and [EM02]. Recently, Amini and Baker [AB15] introduced a notion of metrized complexes, which is roughly speaking a finite metric graph together with a collection of marked curves , one for each vertex , such that the marked points of is in bijection with the edges of incident on . This can be considered as an enrichment of both metric graphs and nodal curves. The concept of limit linear series on a metrized complex is proposed in loc.cit. as well as the specialization map. While the Amini-Baker limit linear series satisfies the specialization theorem, it is unclear how to prove a general theorem of their smoothing behaviors. Related results can be found in [LM14], where a sufficient and necessary condition for the smoothability (with respect to certain base field) of a (saturated) limit linear series of rank one is given.
On the other hand, Brian Osserman developed in [Oss] a notion of limit linear series on curves possibly not of compact type, which is a generalization of Eisenbud and Harris limit linear series. He also proved the specialization theorem, as well as a smoothing theorem for curves of pseudocompact type (see below for definition), which states that a limit linear series is smoothable if the moduli space is of expected dimension at the corresponding point. This improves the smoothing theorem in the compact-type case in the sense that it applies for possibly non-refined limit linear series.
In the present paper we investigate the smoothing of Amini-Baker limit linear series by studying their connection with Osserman limit linear series.
Let be a curve of pseudocompact type, which is a curve whose dual graph is obtained from a tree, denoted by , by adding edges between adjacent vertices. A chain structure on is roughly a integer-valued length function on . This induces a metric graph as well as a metrized complex with underlying graph (see §2 for details). Given a chain structure , let be the curve obtained from by inserting a chain of projective lines at the node of for all . Let be the dual graph of . An Osserman limit linear series (which we also refer to as a limit linear series on ) then consists of a certain line bundle on and a collection of linear systems on each component of that satisfies certain (multi)vanishing conditions, and an Amini-Baker limit linear series (which we also refer to as a limit linear series on ) consists of a divisor on with a collection of linear spaces of rational functions on each component of satisfying certain rank conditions. The multidegree of a limit linear series on is the multidegree of on , which induces a divisor on as we identify with the metric graph obtained from by assigning length to every edge. We also call a limit linear series on of multidegree if its underlying divisor on is (up to linear equivalence) induced by .
In the following we fix the multidegree of a limit linear series on and . Note that, by the definition of Osserman limit linear series, is assumed to be “admissible”, namely, when restricting to each component of , the corresponding divisor of is effective and of at most degree one, see also Definition 2.1.
In [Oss17] the author constructed a map from the set of limit linear series on to the set of limit linear series on . It is also proved in loc.cit. that can be defined over the set of pre-limit linear series (a weakened version of Osserman limit linear series), in this case we show in Section 3 that is a bijection. This essentially says that a limit linear series on carries the same data as a pre-limit linear series on . We also give a necessary condition for a pre-limit linear series (hence for a limit linear series on ) to be lifted to a limit linear series on , which is called the weak glueing condition.
In Section we consider the smoothing of limit linear series on . We show in Theorem 4.7 that a necessary condition is the weak glueing condition. On the other hand, it is proved in [Oss16] that for a special case of the moduli space of limit linear series (again of multidegree ) is of expected dimension, hence every limit linear series on is smoothable. We show in this case that the weak glueing condition is sufficient for a pre-limit linear series to be lifted to a limit linear series, and use the “equivalence” between pre-limit linear series on and limit linear series on to give a smoothing theorem for the latter (see Theorem 4.8 for details):
Suppose we have such that the induced metric graph has few edges and general edge lengths, and that the components of are strongly Brill-Noether general, then a limit linear series on is smoothable if and only if it satisfies the weak glueing condition.
Additionally, for arbitrary and with strongly Brill-Noether general components we consider a family of such that the induced divisor on is “randomly distributed” on , as in Theorem 4.4. More precisely, for any edge , let be the components of corresponding to edges that lies over . Given a certain direction on , the divisor gives an integer in , and we consider such that are distinct modulo . We show that a pre-limit linear series automatically lifts to a limit linear series on , and the dimension of the moduli space of limit linear series on is as expected. In this case any limit linear series on is smoothable.
In section we consider the problem of lifting divisors on the metric graph to the generic fiber of any regular smoothing family (Definition 2.9) with special fiber with rational components and dual graph . When is a generic chain of loops, it is proved by reducing to the so-called vertex avoiding divisors that every rational divisor on is liftable (see [CJP15] for details). For as in Theorem 1.1, again since the weak glueing condition is sufficient for a pre-limit linear series to be lifted to a limit linear series on , and the dimension counting shows that every limit linear series on is smoothable, we are able to prove the following theorem, which gives an alternate approach of lifting (rational) vertex avoiding divisors on a generic chain of loops, by lifting the divisor on to a pre-limit linear series on that satisfies the weak glueing condition.
Let be as in Theorem 1.1 and be as above. Suppose further that only has rational components, and that is a chain. Then every rational divisor on of rank less than or equal to lifts to a divisor on of the same rank. In addition, if is a generic chain of loops then every rational vertex avoiding divisor lifts to a divisor on with the same rank.
See Theorem 5.1 and Theorem 5.4 for more precise statements. Note that, ignoring the field condition (see below), the first part of the theorem confirms [CDPR12, Conjecture 1.5] and [CJP15, Theorem 1.1] for divisors of rank one, and generalizes the conditions in loc.cit. for the graph in the sense that there could be at most three edges (instead of two) between any pair of adjacent vertices.
1.1. Conventions and Notations.
All curves we consider are proper, (geometrically) reduced and connected, and at worst nodal. All nodal curves are split. All irreducible components of a nodal curve are smooth.
In the sequel we let be a complete discrete valuation ring with valuation and residue field . Let be the fraction field of , which is a non-archimedean field with the induced norm . Let be the completion of the algebraic closure of and the valuation ring of . Note that is still algebraically closed, the valuation/norm extends uniquely to , and has residue field if is algebraically closed (cf. [Con08, §1.1]).
Let be a curve with dual graph . For let be the irreducible component of corresponding to . For that is incident on let be the node of corresponding to and the preimage (of the normalization map of ) of in . Let be the graph obtained from as follows: for each pair of adjacent vertices and replace all edges connecting and by a single edge. For we denote
where in the first expression runs through all edges of that lies over .
For a curve (graph, metric graph, metrized complex) denote the space of divisors on . If is a curve let be the space of rational functions on ; if is a graph (metric graph, metrized complex) denote the space of rational functions on by . Given a rational function on denote the associated divisor.
Let be a curve. Let be an invertible sheaf on where and take a nonzero section . We denote the effective divisor associated to that is rationally equivalent to . For a divisor and denote the coefficient of in , denote . For a rational function let be the vanishing order of at . Hence if is considered as a rational function then we have .
Let be a metric graph with underlying graph . For and incident on and , let be the outgoing slope of at along the tangent direction corresponding to . For let be the sum of outgoing slopes of over all tangent directions of .
Acknowledgements. The author would like to thank Brian Osserman for introducing this problem and for helpful conversations, and thank Sam Payne for the idea of the proof of Theorem 5.4.
We recall some relative notions about Osserman and Amini-Baker limit linear series. Let be a graph without loops. Recall that a chain structure on is a function . Let be the corresponding metric graph with edge length defined by , and the graph obtained from by inserting vertices between the vertices adjacent to for all . Then can also be obtained from by associating unit edge lengths to all of . We use or instead of for different choices of vertex sets of .
2.1. Some notions for graph theory.
An admissible multidegree of total degree on consists of a function together with a tuple such that .
Correspondingly, we define the notion of (integral) edge-reduced divisors, which is closely related to admissible multidegrees, as follows:
A divisor on is edge-reduced if the restriction of on each connected component of is either empty or an effective divisor of degree one. We say that is rational (resp. integral) if is supported on rational (resp. integral) points.
Suppose is directed, there is a natural bijection between the set of integral edge-reduced divisors on (of degree ) and the set of admissible multidegrees on (of degree ). Precisely, for integral and edge-reduced, we set and for with tail let be the distance between the point in and if and otherwise, where is the edge of corresponding to . Denote for any admissible multidegree .
For each pair of an edge and an adjacent vertex , let if is the tail of and otherwise. We have the following definition of twisting:
Let be an admissible multidegree on and . For each incident on we do the following operation:
(1) If increase by where is the other vertex of ;
(2) If decrease by ;
(3) Increase by .
The resulting admissible multidegree is called the twist of at . The negative twist of at is the admissible multidegree such that the twist of at is equal to .
Let be an admissible multidegree. We denote by the directed graph with vertex set consisting of all admissible multidegrees obtained from by sequences of twists, and with an edge from to if is obtained from by twisting at some vertex . Given and (not necessarily distinct), let denote the path in obtained by starting at and twisting successively at each .
In the sense of an integral edge-reduced divisor on twisting at (where we choose as the vertex set of ) is just firing all chips at edges incident on away from by distance 1 (assume that we put in advance one chip of at the position of in for all incident on such that ). Hence the definition of twisting is independent of the direction of , and we get a linearly equivalent divisor after twisting.
Consider a graph consists of two vertices and connected by three edges and . Take a chain structure with , and a direction from to . Let where , , , and . Then is as in the left of the following graphs, with each number represents the coefficient of the corresponding node in .
After twisting at we get where , , , and . The induced is given in the right graph.
Let denote the graph obtained from by contracting edges between every pair of adjacent vertices into one single edge. We say that is a multitree if is a tree. Any graph with two vertices is a multitree. For multitrees we also consider partial twists:
Let be a multitree, and a pair of an edge and an adjacent vertex of , and an admissible multidegree on . The twist of at is the admissible multidegree obtained from by doing the operations as in Definition 2.3 for all edges in that lies over .
Again for an integral edge-reduced divisor on twisting at is just firing the chips at edges over away from by distance . Note that the twists are commutative, and remains the same after twisting at all vertices of . Hence the twists are invertible, as the negative twist at is the same as the composition of the twists at all . In addition, for multitrees twisting at is the same as twisting at all vertices in the connected component of that contains . If is the other vertex of then twisting at is the inverse of twisting at .
Let be a multitree. An admissible multidegree is concentrated on if there is an ordering on starting at , and such that for each subsequent vertex , we have that becomes negative in vertex after taking the composition of the negative twists at all previous vertices. A tuple of admissible multidegrees is tight if is concentrated at for all , and for all incident on vertices and , we have that is obtained from by twisting times at for some .
If has only two vertices and then being concentrated on is the same as being negative at after twisting at . An example of a tight tuple of admissible multidegrees is given by the reduced divisors:
Given a vertex , a divisor on is -reduced if: (1) is effective on ; (2) for every nonempty subset , there is some such that (the coefficient of in ) is strictly smaller than the number of edges from to .
Given a point , a divisor is -reduced if: (1) is effective on ; (2) for any closed connected subset there is a point such that is strictly less than the number of tangent directions of at .
For every divisor (resp. ) and (resp. ) there is a unique divisor (resp. ) such that (resp. ) is linearly equivalent to and -reduced (resp. -reduced). One easily checks that a divisor is reduced if and only if it is reduced as a divisor on .
Let be a multitree. Let be the admissible multidegree such that is -reduced. Then is the unique admissible multidegree in which is concentrated on and nonnegative on all , and is a tight tuple.
The first conclusion follows directly from [Oss17, Corollary 3.9]. According to Dhar’s burning algorithm ([Luo11, Algorithm 2.5]), for and in adjacent to , we have that is obtained from by twisting times at , where is the largest number (could be negative) such that the resulting multidegree is effective on . ∎
2.2. Limit linear series on curves of pseudocompact type.
In this subsection we recall the definition of limit linear series by Osserman in [Oss16] for curves of pseudocompact type. Note that the notion of limit linear series for more general curves is given in [Oss, Definition 2.21].
We say that is a smoothing family if is the spectrum of a DVR, and:
(1) is flat and proper;
(2) the special fiber of is a (split) nodal curve;
(3) the generic fiber is smooth;
(4) admits sections through every component of .
If further is regular we say that is a regular smoothing family.
Note that if then by completeness condition (4) in the above definition is satisfied automatically. Let be a curve over with dual graph . For , let be the corresponding irreducible component of and the closure of the complement of .
An enriched structure on consists of the data, for each a line bundle on , satisfying:
(1) for any we have and ;
Note that an enriched structure is always induced by any regular smoothing of (cf. [Oss, Proposition 3.10]).
Take a chain structure on and an admissible multidegree . Let be the nodal curve obtained from by, for each , inserting a chain of projective lines at the corresponding node, and the dual graph of . We say that a divisor on is of multidegree if (considered as a divisor on ) it is equal to , and a line bundle on is of multidegree if its associated divisor on is.
Given an enriched structure on and a tuple of admissible multidegrees in such that is concentrated on and a line bundle on of multidegree , we get a tuple of line bundles . Roughly speaking, suppose , as a divisor on , is obtained from by consecutively fire chips at a set of vertices, then is obtained from by tensoring with for all . Note that is of multidegree . See [Oss16, §2] for details of this construction.
A curve over is of pseudocompact type if its dual graph is a multitree.
Now suppose is a curve of pseudocompact type, and the tuple is tight. For each pair of an edge and an adjacent vertex of , let be the effective divisors on defined by setting and
where is induced by . Intuitively records the chips we lose (in every direction) at vertex when twisting times at from .
In Example 2.4, let and be obtained from by twisting three times at . It is easy to check that is a tight tuple. Straightforward calculation shows that
For convenience we call the twisting divisors associated to . In order to define limit linear series, we have the following glueing isomorphism:
[Oss16, Proposition 2.14] Let be a curve of pseudocompact type and be as above. Denote . Take vertices and of connected by an edge . Then for we have isomorphisms
Note that in the above proposition, if then . In particular we have for all .
(1) Let be a smooth curve and . Let be a sequence of effective divisors on . We say is critical for if .
(2) Suppose further that and . Given a on , we define the multivanishing sequence of along to be the sequence
where a value appears in the sequence times if for some we have and , and .
(3) Given nonzero, the order of vanishing along is where is maximal so that .
One checks easily that is critical for if and only if is critical for . We are now able to state the definition of limit linear series on a curve of pseudocompact type (cf. [Oss16, Definition 2.16]):
Suppose we have a tuple with a line bundle of multidegree on , and each is a -dimensional space of global sections of as in Proposition 2.13. For each pair in where is a vertex of , let be the multivanishing sequence of along . Then is a limit linear series of multidegree with respect to on if for any with vertices and we have:
(I) for if with critical for , then ;
(II) there exists bases of and of such that
and similarly for , and for all such that (where as in (I)) we have where we consider and and is as in Proposition 2.13.
Let . Using the identification of the two linear spaces induced by , the condition (2) in the above definition is equivalent to that the space
has at least dimension .
2.3. Limit linear series on metrized complexes.
In this subsection we introduce Amini and Baker’s construction in [AB15] of metrized complexes and limit linear series on them. Suppose for now that is algebraically closed. Recall that a metrized complex of curves over consists of the following data: (1) a metric graph with underlying graph ; (2) for each vertex of a smooth curve over ; (3) for each vertex of , a bijection between the edges of incident on (recall that is assumed loopless in the beginning of this section) and a subset of . For consistency of symbols we assume that is of integral edge lengths.
The geometric realization of is the union of the edges of and the collection of curves , with each endpoint of identified with . The following is a geometric realization of a metrized complex which has rational for all and whose underlying graph is .
Given a curve with dual graph and a chain structure on , we can associate a metrized complex : let be the metric graph with underlying graph and edge lengths given by , let be the component and (hence ). If the edge lengths are all we denote by instead.
A divisor on is a finite formal sum of points in . Denoting , we can naturally associate a divisor on , called the -part of , as well as, for each , a divisor on called the -part of as follows:
Note that we have .
A nonzero rational function on is the data of a rational function (the -part) on and nonzero rational functions (the -part) on for each . The divisor associated to is defined to be
where is as follows: if then ; if then ; if then . In particular the -part of is equal to .
Similarly to the divisors on graphs we have the following definitions:
Suppose has integral edge lengths. A divisor on is rational, integral, and edge-reduced if is. Similarly, let be the chain structure on induced by and an admissible multidegree on , then is of multidegree if .
Divisors of the form are called principal. Two divisors in are called linearly equivalent if they differ by a principal divisor. The rank of a divisor is the largest integer such that is linearly equivalent to an effective divisor for all effective divisor of degree on .
The definition of the rank of can be refined by restricting the set of rational functions on that induce linear equivalence, as follows:
Suppose we are given, for each , a non-empty -linear subspace of . Denote by the collection of all . We define the -rank of to be the maximum integer such that for every effective divisor of degree , there is a nonzero rational function on with for all such that .
Note that by definition we always have . We are now able to define limit linear series on metrized complexes:
A limit linear series of degree and rank on is a (equivalence class of) pair consisting of a divisor of degree and a collection of -dimensional subspaces for all , such that . Two pairs and are considered equivalent if there is a rational function on such that and for all .
A limit linear series on is of multidegree if there is a representative such that is so.
Let be a smooth curve over . Recall that a strongly semistable model for is a flat and integral proper relative curve over whose generic fiber is isomorphic to and whose special fiber is a curve in our setting (or a strongly semistable curve over as in [AB15, §4.1]). Given a strongly semistable model , again there is a associated metrized complex where the underlying graph is the dual graph of the special fiber and is the component of and . Moreover, the length of is val for some such that the local equation of at the node of is . Equivalently, consider the natural reduction map induced by the bijection between and ; this extends to a map and is the modulus of the open (analytic) annulus in , where is the Berkovich analytification of . Note that any metrized complex with edge lengths contained in can be constructed from strongly semistable models (cf. [AB15, Theorem 4.1]).
There is a canonical embedding of into as well as a canonical retraction map , which induces by linearity a specialization map which maps to the set of rational points of . For , if then is a nonsingular closed point of in . We thus have the specialization map of divisors given by the linearly extension of:
On the other hand, let be a point of type . The completed residue field of has transcendence degree one over and corresponds to a curve over . Given a nonzero rational function on , choose such that . We denote as the image of in , which is well defined up to scaling by . We call the normalized reduction of . Note that the normalized reduction of a -vector space is a -vector space of the same dimension, whereas the normalized reduction of a basis of is not necessarily a basis of the normalized reduction of . See [AB15, Lemma 4.3] for details.
Given and as above. The specialization of is a nonzero rational function on whose -part is the restriction to of the piecewise linear function