ON THE DISCRIMINANT VARIETY OF A PROJECTIVE MANIFOLD

Let L be a very ample line bundle on a smooth, n-dimensional, connected, complex, projective manifold, X. Let D denote the discriminant variety of (X, L), i. e. the set D subset-of Absolute value of L of singular divisors. The defect of (X, L), def(X, L), is defined to be 1 less than the codimension of D in Absolute value of L. In this article pairs, (X, L), as above with def(X, L) > 0 are studied. The main tool used is a result of Beltrametti, Sommese, and Wisniewski showing that if (X, L) is as above with def(X, L) > 0, then there is a contraction of an extremal ray, PHI: X --> Y with Y normal and projective, and K(X) + ((n + def(X, L))/2 + 1) L congruent-to O(X). Using this result the classification of pairs (X, L) with positive defect is reduced to the study of Fano manifolds of positive defect. As one application the complete classification of pairs (X, L) with positive defect and dimX less-than-or-equal-to 10 is given. Previously classification had been done in dimX less-than-or-equal-to 6 by Ein, with partial results in dimX = 7 by Ein and Lanteri-Struppa.