On the discriminant variety of a projective manifold

Abstract

Let L be a very ample line b ndle on a smooth, «-dimensional, connected, complex, projective manifold, X. Let Q) denote the discriminant variety of (X, L), i.e. the set @ a \L\ of Singular divisors. The defect of (X, L), def (X, L), is defined to be l less than the codimension of Q) in \L\. In this article pairs, (X, L), s above with def (T, L) > 0 are studied. The main tool used is a result of Beltrametti, Sommese, and Wisniewski showing that if (X, L) is s above with def(A", L) > 0, then there is a contraction of an extremal ray, Φ: Χ-+ Υ with Υ normal and projective, and K x + ((n + def (T, L))/2 + 1)L ^ 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 < 10 is given. Previously classification had been done in aimX < 6 by Ein, with partial results in dim ' = 7 by Ein and Lanteri-Struppa.

A simple combinatorial criterion for projective toric manifolds with dual defect

“…We refer to [5], where Conjecture 1.4 first appeared, for more background. As was shown in [3], any dual defective n-dimensional polarized manifold satisfies µ > n+2 2…”

“…The rational function D A equals the alternating product of the principal determinants associated to all the facial subsets of A. In Example 2.5 of Chapter 11 of [15], an example of a non-simple lattice polytope (the hypersimplex ∆ (3,6)) is given such that D A is not a polynomial. Still, Di Rocco calculated in [24] that c(∆(3, 6)) > 0.…”

The NEF Value and Defect of Homogeneous Line Bundles

“…In this paper a general formula is derived for the nef value of L when X is a homogeneous space equivariantly imbedded in P^ by the sections of 7, see Theorem 2.2. It is then an easy matter to tabulate the exact values for t(X , L) when Pic(X) =■ Z, see Corollary 2.4.As is shown in [2,3], there is a connection between the nef value, t(X , L), and the codimension of the variety X' c P^ of hyperplanes tangent to X, known as the dual or discriminant variety of X. The defect of (X, L) is defined to be def(X, L) = codim X' -1.…”

“…As is shown in [2,3], there is a connection between the nef value, t(X , L), and the codimension of the variety X' c P^ of hyperplanes tangent to X, known as the dual or discriminant variety of X. The defect of (X, L) is defined to be def(X, L) = codim X' -1.…”

“…Most smooth varieties have defect 0. If def(X, L) > 0, then the defect is determined by the nef value [2], def(X, L) = 2(x(X, L) -1) -dimX.…”