The irreducibility of Hurwitz spaces and Severi varieties on toric surfaces

TL;DR

The paper resolves the irreducibility problem for classical Hurwitz spaces $H_{g,d}$ over all algebraically closed fields, by connecting them to Severi varieties on polarized toric surfaces and proving their irreducibility via tropical methods. It develops a tropical lifting framework for parametrized curves and proves a strong connectedness property (Theorem C) for the moduli of parametrized tropical curves, enabling a transfer of irreducibility from tropical to algebraic geometry. The authors establish irreducibility of a broad class of Severi varieties associated to admissible (in particular, classical) polygons, and derive adjacencies and nodality results in this toric setting. Together, these results yield irreducibility of Hurwitz spaces of all coverings and of simple coverings in arbitrary characteristic, significantly broadening prior characteristic-dependent conclusions. The methods highlight a powerful synthesis of tropical geometry, combinatorial floor decompositions, and lifting arguments to resolve foundational questions in the geometry of curves and coverings, with potential implications for Brill-Noether theory and moduli of curves.

Abstract

In 1969, Fulton introduced classical Hurwitz spaces parametrizing simple d-sheeted coverings of the projective line in the algebro-geometric setting. He established the irreducibility of these spaces under the assumption that the characteristic of the ground field is greater than d, but the irreducibility problem in smaller characteristics remained open. We resolve this problem in the current paper and prove that the classical Hurwitz spaces are irreducible over any algebraically closed field. On the way, we establish the irreducibility of Severi varieties in arbitrary characteristic for a rich class of toric surfaces, including all classical toric surfaces. Our approach to the irreducibility problems comes from tropical geometry, and the paper contains two more results of independent interest - a lifting result for parametrized tropical curves and a strong connectedness property of the moduli spaces of parametrized tropical curves.

Figures (12)

• Figure 1: Three simple floor decomposed curves of multiplicity one and their dual subdivisions. The floors are in red and the elevators -- in blue. Only the left curve is ssfd and unimodular. The left and the middle curves have different number of self intersection points despite having the same combinatorial type. The right curve does not belong to an ssfd stratum.
• Figure 2: A geometric illustration to the linear independence assumption in Propo-sition \ref{['prop:lifting']}.
• Figure 3: The setting in the proof of Proposition \ref{['prop:lifting']}. On the left the local picture close to $\widetilde{B}_{\vec{e}}$ in $B^0$. On the right, the local picture of its tropicalization.
• Figure 4: Two illustration to Construction \ref{['constr:moving right']}: on the left the elevators $E_i$ and $E_{i+1}$ are not adjacent to the same non-virtual floor, and on the right they are. In both pictures, the curve $(\Gamma_0,h_0)$ is on the left, and $(\Gamma_1,h_1)$ is in the middle.
• Figure 5: An illustration to Construction \ref{['constr:moving up']}. On the left the curve $(\Gamma_0,h_0)$, on the right $(\Gamma',h')$. The curves in the intermediate strata $M_{\Theta_1}, M_{\Theta_2}, M_{\Theta_3}$ are depicted in the middle. The multiplicities at each step are denoted by ${\mathfrak w}_\bullet$.

Theorems & Definitions (48)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Proposition 3.1
• proof
• Remark 3.2
• Corollary 3.4