Linear series on metrized complexes of algebraic curves

Abstract

A metrized complex of algebraic curves over an algebraically closed field $\kappa$ is, roughly speaking, a finite metric graph $\Gamma$ together with a collection of marked complete nonsingular algebraic curves $C_v$ over $\kappa$, one for each vertex $v$ of $\Gamma$; the marked points on $C_v$ are in bijection with the edges of $\Gamma$ incident to $v$. We define linear equivalence of divisors and establish a Riemann–Roch theorem for metrized complexes of curves which combines the classical Riemann–Roch theorem over $\kappa$ with its graph-theoretic and tropical analogues from Amini and Caporaso (Adv Math 240:1–23, 2013); Baker and Norine (Adv Math 215(2):766–788, 2007); Gathmann and Kerber (Math Z 259(1):217–230, 2008) and Mikhalkin and Zharkov (Tropical curves, their Jacobians and Theta functions. Contemporary Mathematics 203–231, 2007), providing a common generalization of all of these results. For a complete nonsingular curve $X$ defined over a non-Archimedean field $\mathbb {K}$, together with a strongly semistable model $\mathfrak {X}$ for $X$ over the valuation ring $R$ of $\mathbb {K}$, we define a corresponding metrized complex $\mathfrak {C}\mathfrak {X}$ of curves over the residue field $\kappa$ of $\mathbb {K}$ and a canonical specialization map $\tau ^{\mathfrak {C}\mathfrak {X}}_*$ from divisors on $X$ to divisors on $\mathfrak {C}\mathfrak {X}$ which preserves degrees and linear equivalence. We then establish generalizations of the specialization lemma from Baker (Algebra Number Theory 2(6):613–653, 2008) and its weighted graph analogue from Amini and Caporaso (Adv Math 240:1–23, 2013), showing that the rank of a divisor cannot go down under specialization from $X$ to $\mathfrak {C}\mathfrak {X}$. As an application, we establish a concrete link between specialization of divisors from curves to metrized complexes and the theory of limit linear series due to Eisenbud and Harris (Invent Math 85:337–371, 1986). Using this link, we formulate a generalization of the notion of limit linear series to curves which are not necessarily of compact type and prove, among other things, that any degeneration of a $\mathfrak {g}^r_d$ in a regular family of semistable curves is a limit $\mathfrak {g}^r_d$ on the special fiber.

Appendix: Rank-determining sets for metrized complexes

We retain the terminology from Sect. 2. Let $\mathfrak {C}$ be a metrized complex of algebraic curves, $\Gamma$ the underlying metric graph, $G=(V,E)$ a model of $\Gamma$ and $\{C_v\}_{v\in V}$ the collection of smooth projective curves over $\kappa$ corresponding to $\mathfrak {C}$. In this section, we generalize some basic results concerning rank-determining sets [30, 34] from metric graphs to metrized complexes by following and providing complements to the arguments of [3] (to which we refer for a more detailed exposition).

Let $\mathcal {R}$ be a set of geometric points of $\mathfrak {C}$ (i.e., a subset of $\bigcup _{v\in V} C_v(\kappa )$). The set $\mathcal {R}$ is called rank-determining if for any divisor $\mathcal {D}$ on $\mathfrak {C}$, $r_\mathfrak {C}(\mathcal {D})$ coincides with $r_{\mathfrak {C}}^\mathcal {R}(\mathcal {D})$, defined as the largest integer $k$ such that $\mathcal {D} - \mathcal {E}$ is linearly equivalent to an effective divisor for all degree $k$ effective divisors $\mathcal {E}$ on $\mathfrak {C}$ with support in $\mathcal {R}$. In other words, $\mathcal {R}$ is rank-determining if in the definition of rank given in Sect. 2, one can restrict to effective divisors $\mathcal {E}$ with support in $\mathcal {R}$.

The following theorem is a common generalization of (a) Luo’s theorem [34] (see also [30]) that $V$ is a rank-determining set for any loopless model $G=(V,E)$ of a metric graph $\Gamma$ and (b) the classical fact (see [34] for a proof) that for any smooth projective curve $C$ of genus $g$ over $\kappa$, every subset of $C(\kappa )$ of size $g+1$ is rank-determining.

Theorem 6.1

Let $\mathfrak {C}$ be a metrized complex of algebraic curves, and suppose that the given model $G$ of $\Gamma$ is loopless. Let $\mathcal {R}_v \subset C_v(\kappa )$ be a subset of size $g_v+1$ and let $\mathcal {R}= \cup _{v\in V} \mathcal {R}_v$. Then $\mathcal {R}$ is a rank-determining subset of $\mathfrak {C}$.

Let $\mathcal {D}$ be a divisor on $\mathfrak {C}$. For any point $P \in \Gamma$, let $\mathcal {D}^P$ be the quasi-unique $P$-reduced divisor on $\mathfrak {C}$ linearly equivalent to $\mathcal {D}$, and denote by $D^P_\Gamma$ (resp. $D^P_v$) the $\Gamma$-part (resp. $C_v$-part) of $\mathcal {D}^P$.

Lemma 6.2

A divisor $\mathcal {D}$ on $\mathfrak {C}$ has rank at least one if and only if

(1):

For any point $P$ of $\Gamma$, $D^P_\Gamma ( P )\ge 1$, and

(2):

For any vertex $v\in V(G)$, the divisor $D^v_v$ has rank at least one on $C_v$.

Proof

The condition $r_\mathfrak {C}(\mathcal {D}) \ge 1$ is equivalent to requiring that $r_\mathfrak {C}(\mathcal {D} - \mathcal {E} ) \ge 0$ for every effective divisor $\mathcal {E}$ of degree $1$ on $\mathfrak {C}$. For $P \in \Gamma {\setminus }V$, the divisor $\mathcal {D} -( P )$ has non-negative rank if and only if $D^P_\Gamma ( P )\ge 1$ (by Lemma 3.11). Similarly, for $v \in V$ and $x\in C_v(\kappa )$, the divisor $\mathcal {D} - (x)$ has non-negative rank in $\mathfrak {C}$ if and only if $D^v_\Gamma (v) \ge 1$ and $D^v_v - (x)$ has non-negative rank on $C_v$ (by Lemma 3.11). These are clearly equivalent to (1) and (2). $\square$

Lemma 6.3

A subset $\mathcal {R}\subseteq \bigcup _{v\in V} C_v(\kappa )$ which has non-empty intersection with each $C_v(\kappa )$ is rank-determining if and only if for every divisor $\mathcal {D}$ of non-negative rank on $\mathfrak {C}$, the following two assertions are equivalent:

(i):

$r_\mathfrak {C}(\mathcal {D}) \ge 1$.

(ii):

For any vertex $u \in V$, and for any point $z \in \mathcal {R}\, \cap \, C_u(\kappa )$, $D_u^u -(z)$ has non-negative rank on $C_u$.

Proof

In view of Lemma 6.2, for a rank-determining set the two conditions (i) and (ii) are equivalent. Suppose now that (i) and (ii) are equivalent for any divisor $\mathcal {D}$ on $\mathfrak {C}$. By induction on $r$, we prove that $r_\mathfrak {C}(\mathcal {D})\ge r$ if and only if for every effective divisor $\mathcal {E}$ of degree $r$ with support in $\mathcal {R}$, $r_\mathfrak {C}(\mathcal {D} - \mathcal {E}) \ge 0$. This will prove that $\mathcal {R}$ is rank-determining.

The case $r=1$ follows by the hypothesis and Lemma 6.2. Supposing now that the statement holds for some integer $r \ge 1$, we prove that it also holds for $r+1$.

Let $\mathcal {D}$ be a divisor with the property that $r_\mathfrak {C}(\mathcal {D} -\mathcal {E}) \ge 0$ for every effective divisor $\mathcal {E}$ of degree $r+1$ with support in $\mathcal {R}$. Fix an effective divisor $\mathcal {E}$ of degree $r$ with support in $\mathcal {R}$. By the base case $r=1$, the divisor $\mathcal {D} - \mathcal {E}$ has rank at least $1$ on $\mathfrak {C}$ because $r_\mathfrak {C}(\mathcal {D} -\mathcal {E} -(x))\ge 0$ for any $x\in \mathcal {R}$. Thus $r_\mathfrak {C}(\mathcal {D} - (x) - \mathcal {E}) \ge 0$ for any point of $|\mathfrak {C}|$. This holds for any effective divisor $\mathcal {E}$ of degree $r$ with support in $\mathcal {R}$, and so from the inductive hypothesis we infer that $\mathcal {D} - (x)$ has rank at least $r$ on $\mathfrak {C}$. Since this holds for any $x \in |\mathfrak {C}|$, we conclude that $\mathcal {D}$ has rank at least $r+1$. $\square$

Let $\mathcal {D}$ be a divisor of degree $d$ and non-negative rank on $\mathfrak {C}$, and let $D_\Gamma$ and $D_v$ be the $\Gamma$ and $C_v$-parts of $\mathcal {D}$, respectively. Define

$$\begin{aligned} |D_\Gamma | :=\{E\ge 0 \,|\,\, E\in \mathrm{Div }(\Gamma ) \text { and } E\sim D_\Gamma \}. \end{aligned}$$

Note that $|D_\Gamma |$ is a non-empty subset of the symmetric product $\Gamma ^{(d)}$ of $d$ copies of $\Gamma$.

Consider the reduced divisor map ${\mathrm {Red}}^{\mathfrak {C}}_\mathcal {D} : \Gamma \rightarrow \Gamma ^{(d)}$ which sends a point $P\in \Gamma$ to $D^P_\Gamma$, the $\Gamma$-part of the $P$-reduced divisor $\mathcal {D}^P$. The following theorem extends [3, Theorem 3] to divisors on metrized complexes.

Theorem 6.4

For any divisor $\mathcal {D}$ of degree $d$ and non-negative rank on $\mathfrak {C}$, the reduced divisor map ${\mathrm {Red}}^{\mathfrak {C}}_\mathcal {D} : \Gamma \rightarrow \Gamma ^{(d)}$ is continuous.

Proof

This is based on an explicit description of the reduced divisor map ${\mathrm {Red}}^{\mathfrak {C}}_\mathcal {D}$ in a small neighborhood around any point of $\Gamma$, similar to the description provided in [3, Theorem 3] in the context of metric graphs. We merely give the description by providing appropriate modifications to [3, Theorem 3], referring to loc. cit. for more details.

Let $P$ be a point of $\Gamma$ and let $\vec \mu$ be a (unit) tangent direction in $\Gamma$ emanating from $P$. For $\epsilon >0$ sufficiently small, we denote by $P+\epsilon \vec \mu$ the point of $\Gamma$ at distance $\epsilon$ from $P$ in the direction of $\vec \mu$. We will describe the restriction of ${\mathrm {Red}}^{\mathfrak {C}}_\mathcal {D}$ to the segment $[P, P+\epsilon \vec \mu ]$ for sufficiently small $\epsilon >0$. One of the two following cases can happen:

• (1) For all sufficiently small $\epsilon >0$, the $P$-reduced divisor $\mathcal {D}^P$ is also $(P+\epsilon \vec \mu )$-reduced. In this case, the map ${\mathrm {Red}}^{\mathfrak {C}}_\mathcal {D}$ is constant (and so obviously continuous) on a small segment $[P, P+\epsilon _0 \vec \mu ]$ with $\epsilon _0>0$.

• (2) There exists a cut $S$ in $\Gamma$ which is saturated with respect to $\mathcal {D}^P$ such that $P \in \partial S$ and $P+\epsilon \vec \mu \not \in S$ for all sufficiently small $\epsilon > 0$.

We note that there is a maximum saturated cut $S$ (i.e., containing any other saturated cut) with the property described in (2) (see the proof of [3, Theorem 3] for details). In the following $S$ denotes the maximum saturated cut with property (2). In this case, there exists an $\epsilon _0>0$ such that for any $0<\epsilon <\epsilon _0$, the reduced divisor $\mathcal {D}^{P+\epsilon \vec \mu }$ has the following description (the proof mimics that of [3, Theorem 3] and is omitted).

Let $\mu ,\vec \mu _1,\dots ,\vec \mu _s$ be all the different tangent vectors in $\Gamma$ (based at the boundary points $P, x_1,\dots ,x_s \in \partial S$, respectively) which are outgoing from $S$. (It might be the case that $x_i=x_j$ for two different indices $i$ and $j$). Let $\gamma _0>0$ be small enough so that for any point $x \in \partial S$ and any tangent vector $\vec \nu$ to $\Gamma$ at $x$ which is outgoing from $S$, the entire segment $(x,x+\gamma _0\vec \nu \,]$ lies outside $S$ and does not contain any point of the support of $D_\Gamma$.

For any $0<\gamma < \gamma _0$ and any positive integer $\alpha$, we will define below a rational function $f_{\Gamma }^{(\gamma ,\alpha )}$ on $\Gamma$. Appropriate choices of $\gamma = \gamma (\epsilon )$ and $\alpha$ will then give the ($P+\epsilon \vec \mu$)-reduced divisor $\mathcal {D}^{P+\epsilon \vec \mu } = \mathcal {D}^P + \mathrm {div}(\mathfrak {f}^{\gamma ,\alpha })$, for any $\epsilon < \epsilon _0 := \frac{\gamma _0}{\alpha }$, where $\mathfrak {f}^{\gamma ,\alpha }$ is the rational function on $\mathfrak {C}$ given by $f_\Gamma ^{\gamma ,\alpha }$ on $\Gamma$ and $f_v =1$ on each $C_v$.

For $0<\gamma <\gamma _0$ and integer $\alpha \ge 1$, define $f_\Gamma ^{(\gamma ,\alpha )}$ as follows:

• $f_\Gamma ^{(\gamma ,\alpha )}$ takes value zero at any point of $S$;

• On any outgoing interval $[x_i,x_i+ \gamma \vec \mu _i]$ from $S$, $f_\Gamma ^{(\gamma ,\alpha )}$ is linear of slope $-1$;

• The restriction of $f_\Gamma ^{(\gamma ,\alpha )}$ to the interval $[P,P+ (\frac{\gamma }{\alpha }) \vec \mu \,]$ is linear of slope $-\alpha$;

• $f_\Gamma ^{(\gamma ,\alpha )}$ takes value $-\gamma$ at any other point of $\Gamma$.

Note that the values of $f_\Gamma ^{(\gamma ,\alpha )}$ at the points $(x_i+ \gamma \vec \mu _i)$ and $P+ (\frac{\gamma }{\alpha }) \vec \mu$ are all equal to $-\gamma$, so $f_\Gamma ^{(\gamma ,\alpha )}$ is well-defined.

It remains to determine the values of $\alpha$ and $\gamma$. Once the value of $\alpha$ is determined, $\gamma$ will be defined as $\alpha \epsilon$ so that the point $P+(\frac{\gamma }{\alpha })\vec \mu$ coincides with the point $P+\epsilon \vec \mu$. We consider the following two cases, depending on whether or not $P$ is a vertex of $G$:

• If $P \in \Gamma {\setminus }V$, then $\alpha = D^P_\Gamma ( P ) - \mathrm {outdeg}_S ( P ) +1$. (Note that since $S$ is saturated with respect to $D_\Gamma ^P$, we have $D^P_\Gamma ( P ) \ge \mathrm {outdeg}_S ( P )$ and thus $\alpha \ge 1$.)

• If $P=v$ for a vertex $v\in V(G)$, let $e_0, e_1, \dots , e_l$ be the outgoing edges at $v$ with respect to $S$, and consider the points $x^{e_0}_v, x^{e_1}_v, \dots , x^{e_l}_v$ in $C_v(\kappa )$ indexed by these edges. Suppose in addition that $e_0$ is the edge which corresponds to the tangent direction $\vec \mu$. Since $S$ is a saturated cut with respect to $D^v_\Gamma$, the divisor $D_v - \mathrm {div}_v(\partial S) = D_v - \sum _{i=0}^l (x^{e_i}_v)$ has non-negative rank in $C_v$. Define $\alpha$ to be the largest integer $n\ge 1$ such that $D_v - n (x_v^{e_0}) - \sum _{i=1}^l (x^{e_i}_v)$ has non-negative rank.

Now for any $0\le \epsilon <\epsilon _0= \frac{\gamma _0}{\alpha }$, the divisor $\mathcal {D}^{P+\epsilon \vec \mu }$ is $(P+\epsilon \vec \mu )$-reduced. (The argument is similar to [3, Proof of Theorem 3].) It follows immediately that the reduced divisor map is continuous on the interval $[P, P+\epsilon _0 \vec \mu )$, and the result follows. $\square$

We are now ready to give the proof of Theorem 6.1.

Proof of Theorem 6.1

By Lemma 6.3, it is enough to check the equivalence of the following two properties for any divisor $\mathcal {D}$ on $\mathfrak {C}$:

• (i) $r_\mathfrak {C}(\mathcal {D}) \ge 1$.

• (ii) For any $u \in V$ and any point $z \in \mathcal {R}_u = \mathcal {R}\, \cap \, C_u(\kappa )$, the divisor $D_u^u -(z)$ has non-negative rank on $C_u$.

It is clear that (i) implies (ii). So we only need to prove that (ii) implies (i). In addition, by Lemma 6.2, Property (i) is equivalent to:

• (1) for any point $P$ of $\Gamma$, $D^P_\Gamma ( P )\ge 1$; and

• (2) for any vertex $v\in V(G)$, the divisor $D^v_v$ has rank at least one on $C_v$.

So it suffices to prove that $\mathrm{(ii)}\Rightarrow (1)$ and $(2)$. Since cardinality of $\mathcal {R}_v$ is $g_v+1$, $\mathcal {R}_v$ is rank-determining in $C_v$. Therefore, (ii) implies $(2)$. We now show that $(2)$ implies $(1)$. Let $\Gamma _0$ be the set of all $P\in \Gamma$ such that $D^P_\Gamma ( P )\ge 1$. By the continuity of the map ${\mathrm {Red}}^{\mathfrak {C}}_\mathcal {D}$, $\Gamma _0$ is a closed subset of $\Gamma$. In addition, since $D^v_v$ has rank at least one on $C_v$ and $D_\Gamma ^v( v ) = \deg (D_v^v)$ for every vertex $v\in C$, we have $V \subset \Gamma _0$. This shows that $\Gamma {\setminus }\Gamma _0$ is a disjoint union of open segments contained in edges of $G$. Suppose for the sake of contradiction that $\Gamma _0 \subsetneq \Gamma$, and let $I = (P,Q)$ be a non-empty segment contained in the edge $\{u,v\}$ of $G$ such that $I \cap \Gamma _0 = \emptyset$.

Claim. ${\mathrm {Red}}^{\mathfrak {C}}_\mathcal {D}$ is constant on the closed interval $[P,Q]$.

To see this, note that for any point $Z \in [P,Q]$ and any tangent direction $\vec \mu$ for which $Z+\epsilon \vec \nu \in [P,Q]$ for all sufficiently small $\epsilon >0$, we are always in case (1) in the description of ${\mathrm {Red}}^{\mathfrak {C}}_\mathcal {D}$. Otherwise, there would be an integer $\alpha >0$ such that $\mathcal {D}^{Z+\epsilon \vec \mu } = \mathcal {D}^{Z} + \mathrm {div}(\mathfrak {f}^{(\epsilon \alpha , \alpha )})$ for all sufficiently small $\epsilon > 0$. In particular, this would imply (by the definition of $f^{(\eta ,a)}$) that $D_\Gamma ^{Z+\epsilon \vec \mu }( Z+\epsilon \vec \mu ) = \alpha \ge 1$, which implies that $Z+\epsilon \vec \nu \in \Gamma _0$, a contradiction. This proves the claim.

A case analysis (depending on whether $P$ and $Q$ are vertices or not) shows that for a point $Z \in (P,Q)$, the cut $S = \Gamma \setminus (P,Q)$ is saturated for $\mathcal {D}^P =\mathcal {D}^Q$. Since $\mathcal {D}^Z = \mathcal {D}^P=\mathcal {D}^Q$, and $S$ does not contain $Z$, this contradicts the assumption that $\mathcal {D}^Z$ is $Z$-reduced. $\square$

Theorem 6.1 has the following direct corollaries.

Corollary 6.5

Let $\mathcal {G}$ be a subgroup of $\mathbb {R}$ which contains all the edge lengths in $G$. For any divisor $\mathcal {D} \in \mathrm{Div }(\mathfrak {C})_\mathcal {G}$, we have

$$\begin{aligned} r_{\mathfrak {C}, \mathcal {G}}(\mathcal {D}) = r_\mathfrak {C}(\mathcal {D}). \end{aligned}$$

Proof

Fix a rank-determining set $\mathcal {R}\subset \cup _{v\in V} C_v(\kappa )$ as in Theorem 6.1. Since $\mathcal {R}$ is rank-determining and any effective divisor $\mathcal {E}$ with support in $\mathcal {R}$ obviously belongs to $\mathrm{Div }(\mathfrak {C})_\mathcal {G}$, to prove the equality of $r_{\mathfrak {C}, \mathcal {G}}(\mathcal {D})$ and $r_\mathfrak {C}(\mathcal {D})$ it will be enough to show that the two statements $r_{\mathfrak {C}, \mathcal {G}}(\mathcal {D})\ge 0$ and $r_\mathfrak {C}(\mathcal {D})\ge 0$ are equivalent. Obviously, the former implies the latter, so we only need to show that if $r_\mathfrak {C}(\mathcal {D})\ge 0$ then $r_{\mathfrak {C}, \mathcal {G}}(\mathcal {D})\ge 0$. Let $v$ be a vertex of $G$ and $\mathcal {D}^{v}$ the $v$-reduced divisor linearly equivalent to $\mathcal {D}$. By Lemma 3.11, $r_\mathfrak {C}(\mathcal {D})\ge 0$ is equivalent to $r_{C_v}(D^{v}_{v})\ge 0$. Now let $\mathcal {D}$ be an element of $\mathrm{Div }(\mathfrak {C})_\mathcal {G}$ with $r_{C_v}(D^v_v)\ge 0$. Since $v\in V$ and $\mathcal {G}$ contains all the edge-lengths in $G$, it is easy to see that $\mathcal {D}$ and $\mathcal {D}^v$ differ by the divisor of a rational function $\mathfrak {f}$ with support in $\mathrm{Div }(\mathfrak {C})_{\mathcal {G}}$. In other words, $\mathcal {D} \sim \mathcal {D}^v$ in $\mathrm{Div }(\mathfrak {C})_\mathcal {G}$. Since $\mathcal {D}^v$ is linearly equivalent to an effective divisor in $\mathrm{Div }(\mathfrak {C})_\mathcal {G}$ (with constant rational function on $\Gamma$), we conclude that $r_{\mathfrak {C},\mathcal {G}}(D)\ge 0$. $\square$

Corollary 6.6

Let $\mathfrak {C}X_0$ be the regularization of a strongly semistable curve $X_0$ over $\kappa$. Let $\mathcal {L}$ be a line bundle on $X_0$ corresponding to a divisor $\mathcal {D} \in \mathrm{Div }(\mathfrak {C})$. Then $r_{c} (\mathcal {L}) = r_{\mathfrak {C}X_0}(\mathcal {D})$.

Proof

This follows from the previous corollary with $\mathcal {G} = \mathbb {Z}$. $\square$