By Wolfgang Franz

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14 (Mathematical Society of Japan, Tokyo, 2004) 45. K. Ohno, Some inequalities for minimal fibrations of surfaces of general type over curves. J. Math. Soc. Jpn. 44(4), 643–666 (1992) 46. K. Paranjape, S. Ramanan, On the Canonical Ring of a Curve. Algebraic Geometry and Commutative Algebra, vol. II (Kinokuniya, Tokyo, 1988), pp. 503–516 4 for surfaces of maximal Albanese dimension. 47. R. Pardini, The Severi inequality K 2 Invent. Math. 159(3), 669–672 (2005) 48. D. Schubert, A new compactification of the moduli space of curves.

Higher codimensional varieties: Lee [32] proved that a subvariety F Pr of degree d is Chow semistable as far as the log canonical threshold of its Chow form is greater or equal to rC1 d (resp. > for stability). In [13] both the Chow and the Hilbert stability of curves of degree d and arithmetic genus g in Pd g are studied. Symh V ! X; D h //: In this case Hilbert and Chow stability have been proved to be equivalent by Fogarty [21] and Mabuchi [33]. There are beautiful results due to Donaldson, Ross, Thomas and many others relating asymptotic Chow stability to differential geometry properties, such that the existence of a constant scalar curvature metric.

Advanced Studies in Pure Mathematics, vol. 10 (Kinokuniya/NorthHolland/Elsevier, Tokyo/Amsterdam/New York, 1987), pp. 449–476 38. A. Moriwaki, Semi-stably polarized fiber spaces and Bogomolov-Gieseker type inequality. J. Math. Kyoto Univ. 32(4), 843–872 (1992) 39. A. Moriwaki, A sharp slope inequality for general stable fibrations of curves. J. Reine Angew. Math. 480, 177–195 (1996) 40. I. Morrison, Projective stability of ruled surfaces. Invent. Math. 56(3), 269–304 (1980) 41. I. Morrison, Stability of Hilbert Points of Generic K3 Surfaces, vol.