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On some boundary divisors in the moduli spaces of stable Horikawa surfaces with $K^2=2p_g-3$
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We describe the normal stable surfaces with K^2=2p_g-3 and p_g>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 K^2=2p_g-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 p_g modulo 4.
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Normal stable degenerations of Noether-Horikawa surfaces
Every Q-Gorenstein smoothable normal stable Horikawa surface falls into one of seven explicit families, and its global smoothability is controlled by a single local condition at its elliptic double cone singularities.
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