Contributions to Riemann-Roch on projective 3-folds with only canonical singularities and applications. (English) Zbl 0662.14026

Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, part 1, Proc. Symp. Pure Math. 46, 221-231 (1987).
[For the entire collection see Zbl 0626.00011.]
A result of M. Reid [same proceedings, Proc. Symp. Pure Math. 46, 345-414 (1987; Zbl 0634.14003)] gives a formula for the plurigenera of a 3-fold X with canonical singularities in terms of the characteristic numbers of X and an explicit formula on an “equivalent basket” of cyclic quotient singularities.
Here the explicit formula is manipulated to a more convenient form, and it is shown that vanishing of plurigenera restricts the number and type of singularities in the basket. It is shown that if \(\chi\) (\({\mathcal O}_ X)<0\) then \(P_ 2\geq 4\); if \(\chi\) (\({\mathcal O}_ X)=0\), \(P_ 2\geq 1\) and \(P_ 4>2\); and if \(\chi\) (\({\mathcal O}_ X)=1\) then \(P_{12}\geq 1\) and \(P_{24}\geq 2\). These results rest on extensive numerical computations.
Reviewer: C.T.C.Wall


14J30 \(3\)-folds
14J17 Singularities of surfaces or higher-dimensional varieties
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14B05 Singularities in algebraic geometry