Converse characterization of closed points in M_g^{2g+1} for n = 0

Determine whether every special curve (i.e., an A_{2g+1}-stable curve whose worse-than-nodal singularities and low-genus honestly hyperelliptic tails/bridges occur only as attached atoms) is a closed point in the moduli stack M_g^{2g+1} when there are no marked points (n = 0). Equivalently, ascertain whether the converse direction in the closed-point characterization of Theorem thm:closed_points_Mgnr extends to the case n = 0 and r = 2g + 1 despite the presence of non-reductive stabilizers.

Background

Theorem thm:closed_points_Mgnr identifies necessary conditions for a point C in M_{g,n}^r to be closed and proves these conditions are also sufficient when either n ≥ 1 or r ≤ 2g. The notion of “special” curves captures those A_r-stable curves where all worse-than-nodal singularities appear as attached atoms (even or odd) and any low-genus honestly hyperelliptic tail or bridge is itself an atom.

In the unpointed case with r = 2g + 1, the stack may have non-reductive stabilizers at closed points. This obstructs the standard use of Θ-degenerations to detect all degenerations of k-points, making it unclear whether the sufficiency part of the theorem remains valid. Establishing the converse in this specific setting would complete the characterization of closed points in M_g^{2g+1}.