On the horizontal base locus of multiples of tautological sheaves of projectivised cotangent bundles. II
The present paper is a continuation of part I [cf. Comment. Math. Helv. 66, No. 4, 580-589 (1991; Zbl 0769.14013)]. Let \(S\) be a minimal algebraic surface of general type, \(V\) the projectivised cotangent bundle. The horizontal base components of high multiples of \(L\), the tautological line bundle of \(V\), is studied. In the earlier paper a positive number \(t_ 0\), which depends on the coefficient of a base component along \(\pi^* K_ S\) in a standard decomposition of the component in \(\text{Pic} (V)\), is defined and bounds using Chern numbers of \(S\) are given. The problem of computing \(t_ 0\) is proposed. A partial solution to the problem is given in the present article. When \(L\) (or \(\Omega_ S)\) is nef, it is proved that \(t_ 0 = (c^ 2_ 1 - c_ 2)/c^ 2_ 1\). The techniques employed are based upon the Riemann- Roch theory for nef and big divisors. As an application the conjecture of Green and Griffiths in the case \(c^ 2_ 1 > {3 \over 2} c_ 2\) is verified; i.e. the images of nontrivial holomorphic curves in \(S\) are algebraically degenerated.
minimal algebraic surface of general type
scientific article; zbMATH DE number 492749
scientific article