Abstract
We completely classify tri-canonically embedded curves of genus two that are Chow semistable, and identify the moduli space of them with the compact moduli space of binary sextics. This moduli space is the log canonical model for the pair \(\big(\overline{M}_2,\alpha\Delta_0+\frac{1+\alpha}{2}\Delta_1+\frac{1}{2}\Xi\big)\) for 7/10 \(< \alpha \le\) 9/11 whose log canonical divisor pulls back to \(K_{\overline{M}_2}+\alpha\delta\) on the moduli stack
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References
Deligne P., Mumford D. (1969) The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math. 36: 75–109
Eisenbud D.: Commutative Algebra, vol. 150 of Graduate Texts in Mathematics. Springer, Berlin Heidelberg New York, 1995. With a view toward algebraic geometry
Hassett B.: Classical and minimal models of the moduli space of curves of genus two. In: Geometric methods in algebra and number theory, vol. 235 of Progress in mathematics pp. 160–192. Birkhäuser, Boston (2005)
Hassett B., Hyeon D.: Log canonical models for the moduli space of curves: First divisorial contraction. arXiv:math.AG/0607477
Hassett B., Hyeon D., Lee Y.: Stability computation via Gröbner basi (2006) Preprint
Kollár J. (1987) The structure of algebraic threefolds: an introduction to Mori’s program. Bull. Amer. Math. Soc. (N.S.) 17(2): 211–273
Mumford D. Stability of projective varieties. Enseignement Math. (2) 23(1–2), 39–110 (1977)
Mumford D. Fogarty J., Kirwan F.: Geometric invariant theory, vol. 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete (2), [Results in Mathematics and Related Areas (2)], Springer, Berlin Heidelberg New York, 3rd edn. (1994)
Schubert D. (1991) A new compactification of the moduli space of curves. Compositio Math. 78(3): 297–313
Viehweg E. (1989) Weak positivity and the stability of certain Hilbert points. Invent. Math. 96(3): 639–667