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This is a Quantitative Study study.

October 7, 2025Open Access

On some boundary divisors in the moduli spaces of stable Horikawa surfaces with K²=2pg-3

Key Points

D describes a boundary divisor for stable surfaces with K^2=2p_g-3, showing interesting target attributes among surface types.
A general point of D correlates with surfaces having singularity type 1/4(1,1), confirming component characteristics.
Closure of D intersects all components in the Gieseker moduli space, highlighting its significance in the broader framework.
D has one or two irreducible components based on p_g's residue mod 4, indicating a unique modular structure.

Abstract

We describe the normal stable surfaces with K²=2pg-3 and pg>14 whose only non canonical singularity is a cyclic quotient singularity of type 1/4k (1, 2k-1) and the corresponding locus D inside the KSBA moduli space of stable surfaces. More precisely, we show that: (1) a general point of any irreducible component of D corresponds to a surface with a singularity of type 1/4 (1, 1), (2) the closure of D is a divisor contained in the closure of the Gieseker moduli space of canonical models of surfaces with K²=2pg-3 and intersects all the components of such closure, and (3) the KSBA moduli space is smooth at a general point of D. In addition, we show that D has 1 or 2 irreducible components, depending on the residue class of pg modulo 4.

On some boundary divisors in the moduli spaces of stable Horikawa surfaces with K²=2pg-3

Key Points

Abstract

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