Limit canonical series

TL;DR

The paper develops a comprehensive framework to describe limits of spaces of Abelian differentials along degenerations of smooth curves to nodal curves general in topology. It combines tropicalization, submodular polymatroid methods, and a canonical fan on the edge-length space to organize all possible limit canonical series into a finite, projective parameter space called the variety of limit canonical series. Central innovations include tilings of the standard simplex by brick polytopes (via UpMin transforms) and a GIT/Mumford-type construction that yields a projective parameter space capturing all limits across degenerations, with explicit residue/pole data and gluing conditions. The work extends Eisenbud–Harris and Esteves–Medeiros results beyond compact type to arbitrary nodal topologies and provides a roadmap for tropical compactifications of moduli spaces in this context. This framework yields practical, effective descriptions of limits, residues, and their interrelations, with broad implications for Weierstrass data and canonical maps in degenerating families.

Abstract

We describe the limits of canonical series along families of curves degenerating to a nodal curve which is general for its topology, in the weak sense that the branches over nodes on each of its components are in general position. We define a fan structure on the space of edge lengths on the dual graph of the limit curve, and construct a projective variety parametrizing the limits, organized in strata associated to the cones of this fan. This extends to all topologies the works by Eisenbud-Harris (Invent. Math. 87: 496-515, 1987) on curves of compact type and Esteves-Medeiros (Invent. Math. 149: 267-338, 2002) on two-component curves.

Figures (10)

• Figure 1: A level graph $(G,\pi)$ with two levels: $V { \sp{} } !_\pi=\{1,2\}$ with order $1<2$. The arrows $u { \sp{} } !_1 u { \sp{} } !_4, u { \sp{} } !_1 u { \sp{} } !_5, u { \sp{} } !_2 u { \sp{} } !_4, u { \sp{} } !_2 u { \sp{} } !_5, u { \sp{} } !_3 u { \sp{} } !_5$ are upward. The edge $\{ u { \sp{} } !_2, u { \sp{} } !_3\}$ is horizontal.
• Figure 2: The polytope $\mathbf P { \sp{} } !_ \varphi { \sp{} }$ on the left is the one associated to the submodular function $\varphi { \sp{} }$ defined in Example \ref{['ex:4lines-polytope']}. The drawing shows its projection to $\mathbb R { \sp{} } !^3$, using the last three coordinates. It has $7$ visible and $4$ hidden facets. The origin $O$ lies in the interior of $\mathbf P { \sp{} } !_ \varphi { \sp{} }$. The red tetrahedron with vertices $O, A, B$ and $C$ is the intersection of $\mathbf P { \sp{} } !_ \varphi { \sp{} }$ with $\Delta { \sp{} } !_3 \subset \mathbb R { \sp{} } !_{\geq 0}^4$ (projected down again to $\mathbb R { \sp{} } !^3$): it is the polytope $\mathbf P { \sp{} } !_ \chi { \sp{} }$ associated to the UpMin transform $\chi { \sp{} } = \mathrm{UpMin}( \varphi { \sp{} } )$. The rescaled figure on the right shows the position of $\mathbf P { \sp{} } !_ \chi { \sp{} }$ within $\Delta { \sp{} } !_3$ (the big tetrahedron in blue with vertices $O, X, Y, Z$).
• Figure 3: A level graph with two levels on the left, and the corresponding upward and horizontal arrows on the right. The set $\mathbb E { \sp{} } !_I$ for $I=\{ u { \sp{} } !_2, u { \sp{} } !_3\}$ is depicted in red.
• Figure 4: The second case appearing in the proof of Proposition \ref{['prop:compatibility-residue']}, with $h'(w)= h(w)+ \lambda { \sp{} } =n$. The level graph given by $h'$ is depicted on the left and the one by $h$ on the right. We have $s=k=1$, with horizontal $b_1$ in red and upward $a_1$ in blue depicted on the left. Both have tail $w$. The connected component $\Xi$ in $G[ V { \sp{} } !_{h<n}]$ on the right consists of the two blue vertices and the top azure vertex. $\bar{b} { \sp{} } !_1$ becomes upward for $h$ on the right, and $a_1$ becomes horizontal.
• Figure 5: A level graph $(G,\pi)$ on the left, $G { \sp{} } !_\pi=( V { \sp{} } !_\pi, E { \sp{} } !_\pi)$ is given in the middle, and $G { \sp{} } !^\dagger_\pi=( V { \sp{} } !_\pi, E { \sp{} } !^\dagger_\pi)$ on the right. Ghost edges are dashed.

Theorems & Definitions (148)

• Theorem 1.2
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Example 2.4
• Proposition 2.5
• proof