Characterizing $(d,h)$-elliptic stable irreducible curves

TL;DR

This paper studies degenerations of curves that are $(d,h)$-elliptic, i.e., limits of smooth curves admitting a degree $d$ map to a genus $h$ curve. The authors develop and leverage the theory of admissible covers, extending it with a pseudo-admissible variant and gluing constructions to connect different covering data, thereby obtaining concrete criteria for ellipticity in families. A central contribution is a normalization-based irreducible-curve criterion that describes how the normalization $C_n$ and the configuration of nodes control $(d,h)$-ellipticity, including explicit blocks and ramification data that determine the genus shift. This framework clarifies how $(d,h)$-elliptic curves degenerate in the moduli of stable curves and yields practical corollaries for one-node irreducible curves and implications for hyperelliptic bases.

Abstract

We use admissible covers to characterize irreducible stable curves that are $(d,h)$-elliptic, that is, that are limits of smooth curves admiting finite maps of degree-$d$ to smooth curves of genus $h\geq 1$.

Figures (1)

• Figure 1: $(d,h+1)$-ellipticity for $d=3$

Theorems & Definitions (24)

• Theorem 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof