×

Bound of automorphisms of surfaces of general type. II. (English) Zbl 0841.14011

[For part I see Ann. Math., II. Ser. 139, No. 1, 51-77 (1994; Zbl 0811.14011).]
The author finds a precise bound for the number of automorphisms of a nonsingular projective algebraic surface of general type over a field of characteristic zero. It is equal to \((42 K_S)^2\) where \(K_S\) is the canonical class of the surface. This is a truly remarkable result which generalizes Hurwitz’s bound \(42 \deg K_S = 84 (g - 1)\) for the number of automorphisms of a Riemann surface \(S\) of genus \(g > 1\). The author also finds the cases when equality takes place. This occurs if and only if \(S \cong (C \times C)/N\), where \(C\) is a curve with \(|\operatorname{Aut} (S) |= 84 (g - 1)\) and \(N\) is a normal subgroup of \(\operatorname{Aut} (C \times C)\) acting freely on the product and preserving the two projections of \(C \times C\). The proof of the result is a very tedious classification of singular rational surfaces obtained by dividing the surface \(S\) by its automorphism group (the same idea as in the case of curves).

MSC:

14H37 Automorphisms of curves
14J50 Automorphisms of surfaces and higher-dimensional varieties

Citations:

Zbl 0811.14011
PDFBibTeX XMLCite