Existence of standard graded almost Gorenstein non-Gorenstein cone singularities for fixed genus

Determine, for each integer g, whether there exists a standard graded ring R(C, D) associated with a smooth curve C of genus g and a divisor D such that R(C, D) is not Gorenstein but is almost Gorenstein, with g(C) = g.

Background

Section 6 studies cone singularities R(C, D) built from a smooth curve C and divisor D, relates almost Gorensteinness to graded structures, and uses Stanley’s theorem to analyze canonical modules. After presenting examples and Higashitani’s criterion for standard graded rings, the authors pose a genus-parametrized existence question for non-Gorenstein but almost Gorenstein standard graded rings of this form.

They note a negative answer for g = 2 later in the section, suggesting scarcity of such examples but leaving the general existence pattern across genera unresolved.

References

Question 6.11. For what g, does there exist R(C, D), which is standard graded, not Gorenstein, and almost Gorenstein with g(C) = g? We will show later that there is no such R if g = 2. We think that a standard graded ring R with g ≥ 2, which is not Gorenstein and almost Gorenstein are very few.

— A Geometric description of almost Gorensteinness for two-dimensional normal singularities  (2410.23911 - Okuma et al., 2024) in Question 6.11, Section 6.1