Let X be an irreducible smooth complex projective variety of dimension n, L a nontrivial spanned line bundle on X, V \subseteq H^0(X,L) a vector subspace od global sections spanning L with dim(V)=N+1. The discriminant variety D(X,V) parameterizes the singular elements of |V|. By Bertini's theorem one can write dim(D(X,V))=N-1-k with k a non-negative integer. An upper bound on k in terms of N, n, and the dimension of the fibers of the map defined by V is established. Moreover, triplets (X,L,V) for which k is near the maximum are classified.
On the discriminant of spanned line bundles / A. Lanteri, R. Munoz - In: Projective Varieties with Unexpected Properties / [a cura di] C. Ciliberto, T. Geramita, B. Harbourne, R.M. Miro-Roig, K. Ranestad. - Berlin : Walter de Gruyter, 2005. - ISBN 3110181606.
On the discriminant of spanned line bundles
A. Lanteri;
2005
Abstract
Let X be an irreducible smooth complex projective variety of dimension n, L a nontrivial spanned line bundle on X, V \subseteq H^0(X,L) a vector subspace od global sections spanning L with dim(V)=N+1. The discriminant variety D(X,V) parameterizes the singular elements of |V|. By Bertini's theorem one can write dim(D(X,V))=N-1-k with k a non-negative integer. An upper bound on k in terms of N, n, and the dimension of the fibers of the map defined by V is established. Moreover, triplets (X,L,V) for which k is near the maximum are classified.
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