Plurigenera of 3-folds and weighted hypersurfaces

• Main Author: Fletcher, Anthony Robert
• Published: University of Warwick 1988
• Subjects: 510; QA Mathematics
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.254415

Chapter I gives basic results and definitions for nonsingular varieties, normal varieties and canonical singularities. In Chapter II we give alternative forms of the Riemann-Roch formula for projec­tive 3-folds with at worst canonical singularities. We show for a canonical 3-fold X with x(Ox) « 1 that Pl2(X) > 1. Pu(X) > 2 and K3x > (1⁄160)3. The last section of Chapter II shows that the record of pluridata representing a canonical 3-fold is unique. In Chapter III we find necessary and sufficient conditions for weighted complete intersections of codimensions 1 and 2 to be quasismooth. We also give conditions for quasismooth surface and 3-fold intersections of codimension 1 and 2 to have at worst only isolated canonical singularities. We produce lists of such complete inter­sections in two different ways: one using these conditions for quasismoothness and having only isolated canonical singularities and the second deducing the degrees of the generators and relations from the plurigenera via the Poincaré series of the canonical ring.