Stability of tri-canonical curves of genus two
Let \(\overline{M}_2\) be the moduli space of stable curves of genus \(2\) and \(\overline{M}_2^{\text{ps}}\) be the moduli space of pseudo-stable curves of genus \(2\). Recall that a complete curve is pseudo-stable if it is connected, reduced, has only ordinary nodes and cusps as singularities, genus \(1\) subcurves meet the rest of the curve in \(2\) or more points and genus \(0\) subcurves meet the rest of the curve in \(3\) or more points. In the paper under review it is shown that: 1) A tri-canonically embedded complete curve of genus \(2\) is Chow semistable if and only if it is pseudo-stable, 2) There is a natural isomorphism between \(\overline{M}_2^{\text{ps}}\) and the compact moduli space of binary sextics, and 3) \(\overline{M}_2^{\text{ps}}\) is the log canonical model of the pair \((\overline{M}_2, \alpha \Delta _0+\frac{1+\alpha}2\Delta _1 +\frac 1 2 \Xi)\) for \(7/10<\alpha \leq 9/11\).
scientific article; zbMATH DE number 5118873