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

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 $h\geq 1$.

💡 Research Summary

The paper investigates the limits of smooth curves that admit a finite morphism of degree d to a smooth curve of genus h (≥ 1), a property the authors call “(d,h)-elliptic”. Using the theory of admissible covers, the authors give a complete characterization of stable irreducible curves that are (d,h)-elliptic.

First, the authors recall the classical notion of a d‑gonal curve (a degree‑d map to ℙ¹) and generalize it to (d,h)-elliptic curves, i.e., curves admitting a degree‑d map to a smooth curve of genus h. They note that for a general curve of genus g ≥ 2 such a map does not exist when d ≥ 2, so understanding the possible degenerations is a natural problem.

The core tool is the notion of an admissible cover, originally introduced by Harris–Mumford and later refined by many authors. An admissible cover π : C → B of degree d requires that C and B are nodal, that the branch points are simple away from the nodes, that the pre‑image of the singular locus of B coincides with the singular locus of C, and that the target curve B, together with the smooth branch points, is a stable pointed curve. The authors relax condition (3) (stability of the target) to a weaker condition (3′) that the target has no internal nodes, thereby defining a “pseudo‑admissible cover”.

Theorem 1 (citing