Unimodal singularities and boundary divisors in the KSBA moduli of a class of Horikawa surfaces

Abstract

Smooth minimal surfaces of general type with , , and constitute a fundamental example in the geography of algebraic surfaces, and the 28‐dimensional moduli space of their canonical models admits a modular compactification via the minimal model program. We describe eight new irreducible boundary divisors in such compactification parameterizing reducible stable surfaces. Additionally, we study the relation with the GIT compactification of and the Hodge theory of the degenerate surfaces that the eight divisors parameterize.

I‐surfaces from surfaces with one exceptional unimodal point

“…We now turn to the exceptional unimodal double points of type 𝑍 𝑛 or 𝑊 𝑛 , which are listed in Table 4. The analysis proceeds in a similar vain as in Section 3, but, due to the nature of the singularities, the analysis is a bit simpler, because the smooth birational models are special K3 surfaces already described in [17].…”

“…This paper is inspired by a recent work of Gallardo, et al [17], so let us briefly set out the context. Classically, the coarse moduli space 𝔐 1,3 of canonical models of surfaces of general type with 𝐾 2 𝑋 = 1 and 𝜒(𝒪 𝑋 ) = 𝜒(𝑋) = 3 is an irreducible and unirational variety of dimension 28.…”

“…Gallardo et al used a different approach in [17]: instead of working inside the realm of stable surfaces they consider degenerations to non-log-canonical surfaces and consider the stable replacement of the limit surface, whose existence is guaranteed by the properness of the stable compactification.…”

“…These divisors parameterize reducible surfaces, where one component is a so-called K3-tail, the two-dimensional equivalent of an elliptic tail on a stable curve. Because their approach starts with explicit equations, they cannot exclude that other degenerations with a unique singular point of the given type exist [17,Remark 4.5]. In addition, they did not provide an explicit geometric description of the second component in case of the singularities 𝐸 12 , 𝐸 13 , 𝐸 14 .…”

“…1. If 𝑊 has a unique singular point 𝑤 which is an exceptional unimodal double point, then 𝑊 is in the closure of one of the families constructed in [17]. 2.…”