Base conditions and covariant systems in an algebraic threefold

Abstract

If F is a free linear system of surfaces in an algebraic threefold V which is either non-singular or possesses only normal singularities, then F has Jacobian and adjoint surfaces, J2(F) and A2(F), and Jacobian and adjoint curve systems, J1(F) and A1(F), such thatwhere X2, X2 are the canonical systems of surfaces and curves on V, and X1(F) is the canonical system of curves of F. The imposition of base elements (points or curves) Ei, of assigned multiplicities λi, on F defines a system F1 which we may represent formally by the equationand it is natural to enquire how the Jacobian systems of F1 differ from those of F, and how we may define adjoint systems A2(F1) and A1(F1) which cut on F1 its canonical curves and sets respectively.